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Luc_Faucheux_2021
THE RATES WORLD โ€“ Part VI
Cleaning up a bunch of loose ends
1
Luc_Faucheux_2021
That deck
2
ยจ After a bunch of decks, we take here a breather to revisit some of the assumptions/results,
and finish up a number of sections that we had left unfinished
ยจ Something to say about the notation / progression of those decks.
ยจ I tried very hard to do it in a progressive manner, and so the formalism and notations
became more complicated but also more complete as we went on.
ยจ So in many ways the โ€simpleโ€ notation that I used at the beginning were potentially
confusing. Many apologies for that, but that was intended in order to demonstrate as we go
along the need for more complicated notation, as opposed to just dump it at the beginning
in a very formal manner
ยจ Hopefully you will have found the journey interesting and enlightning, and maybe more alive
than a formal class, which again this is not. This is merely a bunch of notes that I put down
in a Powerpoint in a selfish purpose so that I can more easily find them and retrieve them,
and hopefully this helps you reading and understanding real serious and formal textbooks on
the subject.
Luc_Faucheux_2021
Letโ€™s play a game.
Letโ€™s see if you can spot the mistakes in the
next section
(*) Gilles Franchini found them under 2 minutes
3
Luc_Faucheux_2021
Rules of the game
ยจ The result is correct
ยจ The derivation is wrong
ยจ There are a bunch of mistakes
ยจ I will highlight and explain the mistakes in the next deck
ยจ There is a prize for the first one to point out all the mistakes
ยจ Gilles Franchini is disqualified from this contest as he found them under 2 minutes
4
Luc_Faucheux_2021
Redoing the useful relationship with the
Isometry property of the ITO integral
5
Luc_Faucheux_2021
Useful relationship through isometry
ยจ ๐”ผ exp โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = exp[โˆซ
!"#
!"$ %
&
๐‘“ ๐‘  &. ๐‘‘๐‘ ]
ยจ We know that this is true, we are trying to re-derive it in another manner, by using the
regular Taylor expansion: exp(๐‘‹) = โˆ‘'"#
'"( %
'!
. ๐‘‹'
ยจ ๐”ผ exp โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = ๐”ผ{โˆ‘'"#
'"( %
'!
. โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
'
}
ยจ ๐”ผ exp โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = โˆ‘'"#
'"( %
'!
. ๐”ผ{ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
'
}
ยจ We surmise that because of the Isometry property of the ITO integral, all the odd terms in ๐‘˜
equal 0 and all the even terms in ๐‘˜ = 2๐‘ are equal to:
ยจ ๐”ผ{ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
&*+%
} = 0
ยจ ๐”ผ{ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
&*
} = โˆซ
!"#
!"$
๐‘“& ๐‘  . ๐‘‘๐‘ 
*
6
Luc_Faucheux_2021
Useful relationship through isometry - II
ยจ To convince ourselves of this, it pays to expand the integral as the usual limit of a sum.
ยจ Remember, since we are using ITO calculus, we are using the LHS (Left Hand Side) for the
value of the function in the interval/mesh/bucketing for the function to be evaluated
ยจ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = โˆซ
!"#
!"$
๐‘“ ๐‘  . ([). ๐‘‘๐‘Š ๐‘ 
ยจ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = lim
,โ†’(
โˆ‘."#
.",
๐‘“ ๐‘ . . {๐‘Š ๐‘ .+% โˆ’ ๐‘Š(๐‘ .)}
ยจ ๐”ผ{โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  } = ๐”ผ{lim
,โ†’(
โˆ‘."#
.",
๐‘“ ๐‘ . . {๐‘Š ๐‘ .+% โˆ’ ๐‘Š(๐‘ .)} }
ยจ ๐”ผ{โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  } = lim
,โ†’(
โˆ‘."#
.",
๐‘“ ๐‘ . . ๐”ผ{๐‘Š ๐‘ .+% โˆ’ ๐‘Š(๐‘ .)}
ยจ ๐”ผ{โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  } = lim
,โ†’(
โˆ‘."#
.",
๐‘“ ๐‘ . . 0 = 0
7
Luc_Faucheux_2021
Useful relationship through isometry - III
ยจ [โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  ]&= [โˆซ
!"#
!"$
๐‘“ ๐‘  . ([). ๐‘‘๐‘Š ๐‘  ]&
ยจ [โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  ]&= [lim
,โ†’(
โˆ‘."#
.",
๐‘“ ๐‘ . . {๐‘Š ๐‘ .+% โˆ’ ๐‘Š(๐‘ .)} ]&
ยจ [โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  ]%
= lim
&โ†’(
โˆ‘)"#
)"&
๐‘“ ๐‘ ) . ๐‘Š ๐‘ )*+ โˆ’ ๐‘Š ๐‘ ) . [lim
,โ†’(
โˆ‘-"#
-",
๐‘“ ๐‘ - . {๐‘Š ๐‘ -*+ โˆ’ ๐‘Š(๐‘ -)} ]
ยจ [โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  ]%
= lim
,โ†’(
โˆ‘-"#
-",
lim
&โ†’(
โˆ‘)"#
)"&
๐‘“ ๐‘ ) . ๐‘“ ๐‘ - . ๐‘Š ๐‘ )*+ โˆ’ ๐‘Š ๐‘ ) . {๐‘Š ๐‘ -*+ โˆ’ ๐‘Š(๐‘ -)}
ยจ ๐”ผ ๐‘Š ๐‘ .+% โˆ’ ๐‘Š ๐‘ . = 0
ยจ ๐”ผ ๐‘Š ๐‘ .+% โˆ’ ๐‘Š ๐‘ . . [๐‘Š ๐‘ /+% โˆ’ ๐‘Š(๐‘ /)] = ๐›ฟ.,/. [๐‘ /+% โˆ’ ๐‘ /]
ยจ Where we are using the usual Kronecker notation:
ยจ ๐›ฟ.,/ = 1 if ๐‘– = ๐‘—
ยจ ๐›ฟ.,/ = 0 if ๐‘– โ‰  ๐‘—
8
Luc_Faucheux_2021
Useful relationship through isometry - IV
ยจ ๐”ผ ๐‘Š ๐‘ .+% โˆ’ ๐‘Š ๐‘ . . [๐‘Š ๐‘ /+% โˆ’ ๐‘Š(๐‘ /)] = ๐›ฟ.,/. [๐‘ /+% โˆ’ ๐‘ /]
ยจ [โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  ]%
= lim
,โ†’(
โˆ‘-"#
-",
lim
&โ†’(
โˆ‘)"#
)"&
๐‘“ ๐‘ ) . ๐‘“ ๐‘ - . ๐‘Š ๐‘ )*+ โˆ’ ๐‘Š ๐‘ ) . {๐‘Š ๐‘ -*+ โˆ’ ๐‘Š(๐‘ -)}
ยจ ๐”ผ{ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
%
} = ๐”ผ{ lim
,โ†’(
โˆ‘-"#
-",
lim
&โ†’(
โˆ‘)"#
)"&
๐‘“ ๐‘ ) . ๐‘“ ๐‘ - . ๐‘Š ๐‘ )*+ โˆ’ ๐‘Š ๐‘ ) . {๐‘Š ๐‘ -*+ โˆ’ ๐‘Š(๐‘ -)} }
ยจ ๐”ผ{ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
%
} = lim
,โ†’(
โˆ‘-"#
-",
lim
&โ†’(
โˆ‘)"#
)"&
๐‘“ ๐‘ ) . ๐‘“ ๐‘ - . ๐”ผ ๐‘Š ๐‘ )*+ โˆ’ ๐‘Š ๐‘ ) . [๐‘Š ๐‘ -*+ โˆ’ ๐‘Š(๐‘ -)]
ยจ ๐”ผ{ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
%
} = lim
,โ†’(
โˆ‘-"#
-",
lim
&โ†’(
โˆ‘)"#
)"&
๐‘“ ๐‘ ) . ๐‘“ ๐‘ - . ๐›ฟ),-. [๐‘ -*+ โˆ’ ๐‘ -]
ยจ ๐”ผ{ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
%
} = lim
,โ†’(
โˆ‘-"#
-",
๐‘“ ๐‘ - . ๐‘“ ๐‘ - . [๐‘ -*+ โˆ’ ๐‘ -]
ยจ ๐”ผ{ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
%
} = โˆซ
!"#
!"$
๐‘“ ๐‘  %
. ๐‘‘๐‘ 
9
Luc_Faucheux_2021
Useful relationship through isometry - V
ยจ ๐”ผ{ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
&
} = โˆซ
!"#
!"$
๐‘“ ๐‘  &. ๐‘‘๐‘ 
ยจ This is the usual Isometry property of the ITO integral
ยจ Note that this would NOT apply in Stratonovitch calculus
ยจ So really we should write it as:
ยจ ๐”ผ{ โˆซ
!"#
!"$
๐‘“ ๐‘  . ([). ๐‘‘๐‘Š ๐‘ 
&
} = โˆซ
!"#
!"$
๐‘“ ๐‘  &. ๐‘‘๐‘ 
ยจ We can try to generalize to the higher orders: ๐”ผ{ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
'
}
ยจ In particular can we write the following ?
ยจ ๐”ผ{ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
&*
} = โˆซ
!"#
!"$
๐‘“& ๐‘  . ๐‘‘๐‘ 
*
10
Luc_Faucheux_2021
Useful relationship through isometry - VI
ยจ ๐”ผ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
'
= ๐”ผ{ lim
,โ†’(
โˆ‘."#
.",
๐‘“ ๐‘ . . {๐‘Š ๐‘ .+% โˆ’ ๐‘Š(๐‘ .)}
'
}
ยจ In the case where ๐‘˜ is odd, we have ๐‘˜ = 2๐‘ + 1
ยจ ๐”ผ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
&*+%
= ๐”ผ{ lim
,โ†’(
โˆ‘."#
.",
๐‘“ ๐‘ . . {๐‘Š ๐‘ .+% โˆ’ ๐‘Š(๐‘ .)}
&*+%
}
ยจ That leaves us with a product of (2๐‘ + 1) limits of sums that looks something like that:
ยจ ๐”ผ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
&*+%
= ๐”ผ{โˆ'"%
'"&*+%
lim
,/โ†’(
โˆ‘./"#
./",/
๐‘“ ๐‘ ./
. {๐‘Š ๐‘ ./+% โˆ’ ๐‘Š(๐‘ ./
)} }
ยจ That is a product of sums
ยจ We can switch to the sum of products, even though the notation gets a little ugly:
11
Luc_Faucheux_2021
Useful relationship through isometry - VII
ยจ ๐”ผ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
&*+%
= ๐”ผ{โˆ'"%
'"&*+%
lim
,/โ†’(
โˆ‘./"#
./",/
๐‘“ ๐‘ ./
. {๐‘Š ๐‘ ./+% โˆ’ ๐‘Š(๐‘ ./
)} }
ยจ So instead of indexing the elements of the sum by ๐‘–' we will now index the elements of the
products by ๐‘˜.
ยจ ๐”ผ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
&*+%
= ๐”ผ{lim
,โ†’(
โˆ‘."#
.", โˆ'0"%
'0"&*+%
๐‘“ ๐‘ '0
. {๐‘Š ๐‘ '012
โˆ’ ๐‘Š(๐‘ '0
)} }
ยจ Note that the index for the stochastic jump went from ๐‘ ./+% to ๐‘ '012
ยจ We can now have the expectation enter into the sum:
ยจ ๐”ผ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
&*+%
= lim
,โ†’(
โˆ‘."#
.",
๐”ผ{โˆ'0"%
'0"&*+%
๐‘“ ๐‘ '0
. {๐‘Š ๐‘ '012
โˆ’ ๐‘Š(๐‘ '0
)}}
ยจ This is where the properties of Brownian motion will greatly simplify the product, just like in
the previous slide for the simple square case (isometry property)
ยจ Note: we need to also spend some time to explain why it is called โ€œisometryโ€
12
Luc_Faucheux_2021
Useful relationship through isometry - VIII
ยจ ๐”ผ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
&*+%
= lim
,โ†’(
โˆ‘."#
.",
๐”ผ{โˆ'0"%
'0"&*+%
๐‘“ ๐‘ '0
. {๐‘Š ๐‘ '012
โˆ’ ๐‘Š(๐‘ '0
)}}
ยจ And we know that:
ยจ ๐”ผ ๐‘Š ๐‘ .+% โˆ’ ๐‘Š ๐‘ . . [๐‘Š ๐‘ /+% โˆ’ ๐‘Š(๐‘ /)] = ๐›ฟ.,/. [๐‘ /+% โˆ’ ๐‘ /]
ยจ So we are just left with:
ยจ ๐”ผ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
%&'(
= lim
)โ†’+
โˆ‘,"#
,")
๐”ผ{โˆ-!"(
-!"%&'(
๐›ฟ!"#
,!"!
๐‘“ ๐‘ -!
. ๐‘“ ๐‘ -#
. ๐‘ -!$%
โˆ’ ๐‘ -!
&
. {๐‘Š ๐‘ -!$%
โˆ’ ๐‘Š(๐‘ -!
)}}
ยจ Since:
ยจ ๐”ผ ๐‘Š ๐‘ .+% โˆ’ ๐‘Š ๐‘ . = 0
ยจ For the odd term we are left with:
ยจ ๐”ผ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
&*+%
= 0
13
Luc_Faucheux_2021
Useful relationship through isometry - IX
ยจ In the case where ๐‘˜ is even, we have ๐‘˜ = 2๐‘
ยจ ๐”ผ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
&*
= ๐”ผ{ lim
,โ†’(
โˆ‘."#
.",
๐‘“ ๐‘ . . {๐‘Š ๐‘ .+% โˆ’ ๐‘Š(๐‘ .)}
&*
}
ยจ That leaves us with a product of (2๐‘ + 1) limits of sums that looks something like that:
ยจ ๐”ผ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
&*
= ๐”ผ{โˆ'"%
'"&*
lim
,/โ†’(
โˆ‘./"#
./",/
๐‘“ ๐‘ ./
. {๐‘Š ๐‘ ./+% โˆ’ ๐‘Š(๐‘ ./
)} }
ยจ That is a product of sums
ยจ We can switch to the sum of products, even though the notation gets a little ugly:
14
Luc_Faucheux_2021
Useful relationship through isometry - X
ยจ ๐”ผ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
&*
= ๐”ผ{โˆ'"%
'"&*
lim
,/โ†’(
โˆ‘./"#
./",/
๐‘“ ๐‘ ./
. {๐‘Š ๐‘ ./+% โˆ’ ๐‘Š(๐‘ ./
)} }
ยจ So instead of indexing the elements of the sum by ๐‘–' we will now index the elements of the
products by ๐‘˜.
ยจ ๐”ผ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
&*
= ๐”ผ{lim
,โ†’(
โˆ‘."#
.", โˆ'0"%
'0"&*
๐‘“ ๐‘ '0
. {๐‘Š ๐‘ '012
โˆ’ ๐‘Š(๐‘ '0
)} }
ยจ Note that the index for the stochastic jump went from ๐‘ ./+% to ๐‘ '012
ยจ We can now have the expectation enter into the sum:
ยจ ๐”ผ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
&*
= lim
,โ†’(
โˆ‘."#
.",
๐”ผ{โˆ'0"%
'0"&*
๐‘“ ๐‘ '0
. {๐‘Š ๐‘ '012
โˆ’ ๐‘Š(๐‘ '0
)}}
15
Luc_Faucheux_2021
Useful relationship through isometry - XI
ยจ ๐”ผ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
&*
= lim
,โ†’(
โˆ‘."#
.",
๐”ผ{โˆ'0"%
'0"&*
๐‘“ ๐‘ '0
. {๐‘Š ๐‘ '012
โˆ’ ๐‘Š(๐‘ '0
)}}
ยจ And we know that:
ยจ ๐”ผ ๐‘Š ๐‘ .+% โˆ’ ๐‘Š ๐‘ . . [๐‘Š ๐‘ /+% โˆ’ ๐‘Š(๐‘ /)] = ๐›ฟ.,/. [๐‘ /+% โˆ’ ๐‘ /]
ยจ So we are just left with:
ยจ ๐”ผ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
%3
= lim
&โ†’(
โˆ‘)"#
)"&
โˆ4!"+
4!"%3
๐›ฟ!"#
,!"!
๐‘“ ๐‘ 4!
. ๐‘“ ๐‘ 4#
. ๐‘ 4!$%
โˆ’ ๐‘ 4!
ยจ Which similar to the isometry derivation, reduces itself to:
ยจ ๐”ผ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
&*
= lim
,โ†’(
โˆ‘."#
.", โˆ'0"%
'0"*
๐‘“ ๐‘ '0
&
. ๐‘ '012
โˆ’ ๐‘ '0
ยจ We switch the sum and the product back and we get:
ยจ ๐”ผ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
&*
= {โˆซ
!"#
!"$
๐‘“ ๐‘  &. ๐‘‘๐‘ }*
16
Luc_Faucheux_2021
Useful relationship through isometry - X
ยจ So we have:
ยจ ๐”ผ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
&*+%
= 0
ยจ ๐”ผ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
&*
= {โˆซ
!"#
!"$
๐‘“ ๐‘  &. ๐‘‘๐‘ }*
ยจ Which we plug back into:
ยจ ๐”ผ exp โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = โˆ‘'"#
'"( %
'!
. ๐”ผ{ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
'
}
ยจ ๐”ผ exp โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = โˆ‘*"#
*"( %
(&*)!
. ๐”ผ{ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
&*
}
ยจ ๐”ผ exp โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = โˆ‘*"#
*"( %
(&*)!
. {โˆซ
!"#
!"$
๐‘“ ๐‘  &. ๐‘‘๐‘ }*
17
Luc_Faucheux_2021
Useful relationship through isometry - XI
ยจ ๐”ผ exp โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = โˆ‘*"#
*"( %
(&*)!
. {โˆซ
!"#
!"$
๐‘“ ๐‘  &. ๐‘‘๐‘ }*
ยจ ๐”ผ exp โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = โˆ‘*"#
*"( %
(*)!
. {
%
&
โˆซ
!"#
!"$
๐‘“ ๐‘  &. ๐‘‘๐‘ }*
ยจ ๐”ผ exp โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = exp(
%
&
โˆซ
!"#
!"$
๐‘“ ๐‘  &. ๐‘‘๐‘ )
ยจ ๐”ผ exp โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = exp(โˆซ
!"#
!"$ %
&
๐‘“ ๐‘  &. ๐‘‘๐‘ )
ยจ This result is TRUE, however the derivation is utterly WRONG.
ยจ Gilles Franchini found all the wrong steps in less than 2 minutes.
ยจ I made it a little easier for you.
ยจ There will be a prize for those of you who find all the mistakes.
ยจ I will have the mistakes highlighted and explained in the next deck
18
Luc_Faucheux_2021
A quick note on martingale and driftless
processes
19
Luc_Faucheux_2021
Quick side note
ยจ
Dโ„š
Dโ„™
= exp[โˆ’ โˆซ
!"#
!"$
๐œ‰ ๐‘  . ๐‘‘๐‘Š ๐‘  โˆ’
%
&
โˆซ
!"#
!"$
๐œ‰ ๐‘  &. ๐‘‘๐‘ ]
ยจ ๐”ผ$
โ„™ Dโ„š
Dโ„™
|๐”‰ 0 = 1
ยจ We also have if we define :
ยจ ๐‘Œ ๐‘ก = ๐‘Œ ๐‘ก = 0 . exp(โˆซ
!"#
!"$
๐œ‰ ๐‘  . ๐‘‘๐‘Š ๐‘  โˆ’
%
&
โˆซ
!"#
!"$
๐œ‰ ๐‘  &. ๐‘‘๐‘ )
ยจ ๐”ผ$
โ„™
๐‘Œ ๐‘ก |๐”‰ 0 = ๐‘Œ(0)
ยจ So the process
ยจ ๐‘Œ ๐‘ก = ๐‘Œ 0 . exp(โˆซ
!"#
!"$
๐œ‰ ๐‘  . ๐‘‘๐‘Š ๐‘  โˆ’
%
&
โˆซ
!"#
!"$
๐œ‰ ๐‘  &. ๐‘‘๐‘ )
ยจ Is a martingale under the โ„™-measure associated with the Brownian motion ๐‘Š
20
Luc_Faucheux_2021
Quick side note - II
ยจ ๐‘Œ ๐‘ก = ๐‘Œ 0 . exp(โˆซ
!"#
!"$
๐œ‰ ๐‘  . ๐‘‘๐‘Š ๐‘  โˆ’
%
&
โˆซ
!"#
!"$
๐œ‰ ๐‘  &. ๐‘‘๐‘ )
ยจ Is a martingale under the โ„™-measure associated with the Brownian motion ๐‘Š
ยจ So ๐‘Œ ๐‘ก is driftless and can be written (maybe) as the solution of an SDE that could look like:
ยจ ๐‘‘๐‘Œ ๐‘ก = 0. ๐‘‘๐‘ก + ๐‘ ๐‘Œ ๐‘ก , ๐‘ก . ([). ๐‘‘๐‘Š(๐‘ก)
ยจ Letโ€™s use ITO lemma on:
ยจ ๐‘Œ ๐‘ก = ๐‘Œ 0 . exp(โˆซ
!"#
!"$
๐œ‰ ๐‘  . ๐‘‘๐‘Š ๐‘  โˆ’
%
&
โˆซ
!"#
!"$
๐œ‰ ๐‘  &. ๐‘‘๐‘ )
ยจ ๐‘‹ ๐‘ก = โˆซ
!"#
!"$
๐œ‰ ๐‘  . ๐‘‘๐‘Š ๐‘  โˆ’
%
&
โˆซ
!"#
!"$
๐œ‰ ๐‘  &. ๐‘‘๐‘ 
ยจ We apply ITO lemma to ๐‘Œ ๐‘ก = ๐‘Œ 0 . exp(๐‘‹ ๐‘ก )
21
Luc_Faucheux_2021
Quick side note - III
ยจ Applying Ito lemma:
ยจ ๐‘“ ๐‘‹ ๐‘กG โˆ’ ๐‘“ ๐‘‹ ๐‘กH = โˆซ
$"$H
$"$G IJ
IK
. ([). ๐‘‘๐‘‹(๐‘ก) + โˆซ
$"$H
$"$G %
&
.
I5J
IK5 . ([). (๐›ฟ๐‘‹)&
ยจ In the โ€limitโ€ of small me increments, this can be wri]en formally as the Ito lemma:
ยจ ๐›ฟ๐‘“ =
IJ
IK
. ([). ๐›ฟ๐‘‹ +
%
&
.
I5J
IK5 . (๐›ฟ๐‘‹)&
ยจ For a function of the Brownian motion ๐‘Š(๐‘ก):
ยจ ๐‘“ ๐‘Š ๐‘กG โˆ’ ๐‘“ ๐‘Š ๐‘กH = โˆซ
$"$H
$"$G IJ
IL
. ([). ๐‘‘๐‘Š(๐‘ก) + โˆซ
$"$H
$"$G %
&
.
I5J
IL5 . ([). ๐‘‘๐‘ก
22
Luc_Faucheux_2021
Quick side note - IV
ยจ ๐‘“ ๐‘‹ ๐‘กG โˆ’ ๐‘“ ๐‘‹ ๐‘กH = โˆซ
$"$H
$"$G IJ
IK
. ([). ๐‘‘๐‘‹(๐‘ก) + โˆซ
$"$H
$"$G %
&
.
I5J
IK5 . ([). (๐‘‘๐‘‹)&
ยจ ๐›ฟ๐‘“ =
IJ
IK
. ([). ๐›ฟ๐‘‹ +
%
&
.
I5J
IK5 . (๐›ฟ๐‘‹)&
ยจ ๐‘‹ ๐‘ก = โˆซ
!"#
!"$
๐œ‰ ๐‘  . ๐‘‘๐‘Š ๐‘  โˆ’
%
&
โˆซ
!"#
!"$
๐œ‰ ๐‘  &. ๐‘‘๐‘ 
ยจ ๐‘‘๐‘‹ ๐‘ก = ๐œ‰ ๐‘ก . ๐‘‘๐‘Š ๐‘ก โˆ’
%
&
๐œ‰ ๐‘ก &. ๐‘‘๐‘ก
ยจ ๐‘‘๐‘‹ & ๐‘ก = ๐œ‰ ๐‘ก &. ๐‘‘๐‘ก
ยจ ๐‘Œ ๐‘ก = ๐‘Œ 0 . exp(๐‘‹ ๐‘ก )
ยจ
IJ
IK
= ๐‘Œ 0 . exp(๐‘‹ ๐‘ก )
ยจ
I5J
IK5 = ๐‘Œ 0 . exp(๐‘‹ ๐‘ก )
23
Luc_Faucheux_2021
Quick side note - V
ยจ ๐›ฟ๐‘“ =
IJ
IK
. ([). ๐›ฟ๐‘‹ +
%
&
.
I5J
IK5 . (๐›ฟ๐‘‹)&
ยจ ๐‘‘๐‘Œ(๐‘ก) = ๐‘Œ(๐‘ก). ([). {๐œ‰ ๐‘ก . ๐‘‘๐‘Š ๐‘ก โˆ’
%
&
๐œ‰ ๐‘ก &. ๐‘‘๐‘ก} +
%
&
. ๐‘Œ ๐‘ก . {๐œ‰ ๐‘ก &. ๐‘‘๐‘ก}
ยจ ๐‘‘๐‘Œ(๐‘ก) = ๐‘Œ(๐‘ก). ([). {๐œ‰ ๐‘ก . ๐‘‘๐‘Š ๐‘ก }
ยจ ๐‘‘๐‘Œ ๐‘ก = ๐œ‰ ๐‘ก . ๐‘Œ(๐‘ก). ๐‘‘๐‘Š ๐‘ก
ยจ So we showed that the stochastic process:
ยจ ๐‘Œ ๐‘ก = ๐‘Œ 0 . exp(โˆซ
!"#
!"$
๐œ‰ ๐‘  . ๐‘‘๐‘Š ๐‘  โˆ’
%
&
โˆซ
!"#
!"$
๐œ‰ ๐‘  &. ๐‘‘๐‘ )
ยจ Is a solution (we leave to pure math people the rigorous work of showing unicity, stability,
well-behaved and all that good stuff)
ยจ Is a solution of the SDE:
ยจ ๐‘‘๐‘Œ ๐‘ก = ๐œ‰ ๐‘ก . ๐‘Œ(๐‘ก). ๐‘‘๐‘Š ๐‘ก
24
Luc_Faucheux_2021
Quick side note - VI
ยจ ๐‘Œ ๐‘ก = ๐‘Œ 0 . exp(โˆซ
!"#
!"$
๐œ‰ ๐‘  . ๐‘‘๐‘Š ๐‘  โˆ’
%
&
โˆซ
!"#
!"$
๐œ‰ ๐‘  &. ๐‘‘๐‘ )
ยจ SDE: ๐‘‘๐‘Œ ๐‘ก = ๐œ‰ ๐‘ก . ๐‘Œ(๐‘ก). ๐‘‘๐‘Š ๐‘ก
ยจ SIE: ๐‘Œ ๐‘กG โˆ’ ๐‘Œ ๐‘กH = โˆซ
$"$H
$"$G
๐œ‰ ๐‘ก . ๐‘Œ(๐‘ก). ๐‘‘๐‘Š ๐‘ก
ยจ In the regular (Newtonian) calculus,
ยจ ๐‘‘๐‘Œ ๐‘ก = ๐œ‰ ๐‘ก . ๐‘Œ(๐‘ก). ๐‘‘๐‘ก
ยจ Would yield:
ยจ ๐‘Œ ๐‘ก = ๐‘Œ 0 . exp(โˆซ
!"#
!"$
๐œ‰ ๐‘  . ๐‘‘๐‘ )
ยจ Which is the regular exponential function
25
Luc_Faucheux_2021
Quick side note - VII
ยจ In the stochastic calculus (ITO), the solution of the SDE:
ยจ ๐‘‘๐‘Œ ๐‘ก = ๐œ‰ ๐‘ก . ๐‘Œ(๐‘ก). ๐‘‘๐‘Š ๐‘ก
ยจ Is NOT the regular exponential that we are used to, but instead:
ยจ ๐‘Œ ๐‘ก = ๐‘Œ 0 . exp(โˆซ
!"#
!"$
๐œ‰ ๐‘  . ๐‘‘๐‘Š ๐‘  โˆ’
%
&
โˆซ
!"#
!"$
๐œ‰ ๐‘  &. ๐‘‘๐‘ )
ยจ Sometimes the above function is referred to the Doleans-Dade exponential in memory of
Catherine Doleans-Dade, and because is it so useful and used
ยจ โ„ฐ โˆซ
#
$
๐œ‰ ๐‘  . ๐‘‘๐‘Š ๐‘  = exp(โˆซ
#
$
๐œ‰ ๐‘  . ๐‘‘๐‘Š(๐‘ ) โˆ’ โˆซ
#
$ %
&
๐œ‰ ๐‘  &. ๐‘‘๐‘ )
ยจ ๐‘Œ ๐‘ก = ๐‘Œ 0 . exp(โˆซ
!"#
!"$
๐œ‰ ๐‘  . ๐‘‘๐‘Š ๐‘  โˆ’
%
&
โˆซ
!"#
!"$
๐œ‰ ๐‘  &. ๐‘‘๐‘ )
ยจ ๐‘Œ ๐‘ก = ๐‘Œ 0 . โ„ฐ โˆซ
#
$
๐œ‰ ๐‘  . ๐‘‘๐‘Š ๐‘ 
26
Luc_Faucheux_2021
Quick side note - VIII
ยจ Note the formal analogy:
ยจ REGULAR CALCULUS (Newtonian)
ยจ ๐‘‘๐‘Œ ๐‘ก = ๐œ‰ ๐‘ก . ๐‘Œ(๐‘ก). ๐‘‘๐‘ก
ยจ ๐‘Œ ๐‘ก = ๐‘Œ 0 . exp(โˆซ
!"#
!"$
๐œ‰ ๐‘  . ๐‘‘๐‘ )
ยจ STOCHASTIC CALCULUS (Brownian) in the ITO convention
ยจ ๐‘‘๐‘Œ ๐‘ก = ๐œ‰ ๐‘ก . ๐‘Œ ๐‘ก . [ . ๐‘‘๐‘Š ๐‘ก
ยจ ๐‘Œ ๐‘ก = ๐‘Œ 0 . โ„ฐ โˆซ
#
$
๐œ‰ ๐‘  . [ . ๐‘‘๐‘Š ๐‘ 
27
Luc_Faucheux_2021
Quick side note - IX
ยจ The interesting thing is that:
ยจ ๐‘‘๐‘Œ ๐‘ก = ๐œ‰ ๐‘ก . ๐‘Œ ๐‘ก . [ . ๐‘‘๐‘Š ๐‘ก
ยจ Is driftless, and the solution of it is:
ยจ ๐‘Œ ๐‘ก = ๐‘Œ 0 . exp(โˆซ
!"#
!"$
๐œ‰ ๐‘  . [ . ๐‘‘๐‘Š ๐‘  โˆ’
%
&
โˆซ
!"#
!"$
๐œ‰ ๐‘  &. ๐‘‘๐‘ )
ยจ Such that it is a martingale:
ยจ ๐”ผ$
โ„™ ๐‘Œ ๐‘ก |๐”‰ 0 = ๐‘Œ(0)
ยจ That would be another way to recover the useful relationship, is to use the property that a
driftless process is a martingale.
ยจ This is the end of this quick note, but I wanted to point out the nice connection between a
process that is driftless and the fact that it is a martingale, in the case where we can have an
explicit solution of the SDE
28
Luc_Faucheux_2021
Quick side note - X
ยจ There is an awful lot of complicated math to prove the equivalence, but very roughly, if the
Novikov condition is respected:
ยจ ๐”ผ$
โ„™ exp(
%
&
โˆซ
!"#
!"$
๐œ‰ ๐‘  &. ๐‘‘๐‘ )|๐”‰ 0 < โˆž
ยจ Then you have equivalence between driftless and martingale.
ยจ Just like in Mario Kart, with the evil Wario, if the Novikov condition is not respected, then
the process becomes a wartingale
29
Luc_Faucheux_2021
Quick side note - XI
30
ยจ Mario driving a martingale
Luc_Faucheux_2021
Quick side note - XII
ยจ Wario driving a wartingale
31
Luc_Faucheux_2021
Quick side note - XIII
ยจ In Mario Kart just like in stochastic processes, the crucial part is the drift.
ยจ A martingale is a driftless process.
ยจ Once you start drifting, both in Mario Kart and in your stochastic process, you could end up
in big trouble.
32
Luc_Faucheux_2021
Quick side note - XIV
ยจ The Girsanov theorem essentially โ€œtakes care of the driftโ€
33
Luc_Faucheux_2021
Quick side note - XV
ยจ In Finance you want to remove the drift (find the martingale)
ยจ In Mario Kart, you want to control the drift especially around the corners
ยจ I want to thank Gilles Franchini for pointing out how crucial the drift was in both situations
34
Luc_Faucheux_2021
From Short Rate to Affine models
35
Luc_Faucheux_2021
From short rate to Affine model
ยจ We note here that there is a strong connection between short rates models and affine
models.
ยจ This just illustrates how strong that connection is:
ยจ Suppose that we start with an SDE of the form:
ยจ ๐‘‘๐‘… ๐‘ก, ๐‘ก, ๐‘ก = ๐œƒ ๐‘ก . ๐‘‘๐‘ก โˆ’ ๐œŽ(๐‘ก). ([). ๐‘‘๐‘Š(๐‘ก)
ยจ The corresponding SIE is:
ยจ ๐‘… ๐‘ก, ๐‘ก, ๐‘ก โˆ’ ๐‘… ๐‘ , ๐‘ , ๐‘  = โˆซ
M"!
M"$
๐œƒ ๐‘ข . ๐‘‘๐‘ข โˆ’ โˆซ
M"!
M"$
๐œŽ(๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข)
ยจ In the risk-free measure โ„š(where we assumed that the Brownian motion ๐‘Š(๐‘ก) is the one
associated to this โ„š -measure)
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = ๐”ผ$0
โ„š N($)
N($0)
|๐”‰(๐‘ก) = ๐”ผ$0
โ„š
exp(โˆ’ โˆซ
!"$
!"$0
๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ )|๐”‰(๐‘ก)
36
Luc_Faucheux_2021
From short rate to Affine model - II
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = ๐”ผ$0
โ„š N($)
N($0)
|๐”‰(๐‘ก) = ๐”ผ$0
โ„š
exp(โˆ’ โˆซ
!"$
!"$0
๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ )|๐”‰(๐‘ก)
ยจ ๐‘… ๐‘ , ๐‘ , ๐‘  โˆ’ ๐‘… ๐‘ก, ๐‘ก, ๐‘ก = โˆซ
M"$
M"!
๐œƒ ๐‘ข . ๐‘‘๐‘ข โˆ’ โˆซ
M"$
M"!
๐œŽ(๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข)
ยจ ๐‘… ๐‘ , ๐‘ , ๐‘  = ๐‘… ๐‘ก, ๐‘ก, ๐‘ก + โˆซ
M"$
M"!
๐œƒ ๐‘ข . ๐‘‘๐‘ข โˆ’ โˆซ
M"$
M"!
๐œŽ(๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข)
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = ๐”ผ$0
โ„š
exp(โˆ’ โˆซ
!"$
!"$0
{๐‘… ๐‘ก, ๐‘ก, ๐‘ก + โˆซ
M"$
M"!
๐œƒ ๐‘ข . ๐‘‘๐‘ข โˆ’ โˆซ
M"$
M"!
๐œŽ(๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘ )|๐”‰(๐‘ก)
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก) = ๐”ผ$!
โ„š
exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . (๐‘ก) โˆ’ ๐‘ก) . exp(โˆ’ โˆซ
!"$
!"$!
{โˆซ
7"$
7"!
๐œƒ ๐‘ข . ๐‘‘๐‘ข โˆ’ โˆซ
7"$
7"!
๐œŽ(๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘ )|๐”‰(๐‘ก)
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก) = exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . (๐‘ก) โˆ’ ๐‘ก) . ๐”ผ$!
โ„š
exp(โˆ’ โˆซ
!"$
!"$!
{โˆซ
7"$
7"!
๐œƒ ๐‘ข . ๐‘‘๐‘ข โˆ’ โˆซ
7"$
7"!
๐œŽ(๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘ )|๐”‰(๐‘ก)
ยจ Remember that the Affine model assumption was:
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = exp(๐ด ๐‘ก, ๐‘ก. โˆ’ ๐ต ๐‘ก, ๐‘ก. . ๐‘… ๐‘ก, ๐‘ก, ๐‘ก )
37
Luc_Faucheux_2021
From short rate to Affine model - III
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก) = exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . (๐‘ก) โˆ’ ๐‘ก) . ๐”ผ$!
โ„š
exp(โˆ’ โˆซ
!"$
!"$!
{โˆซ
7"$
7"!
๐œƒ ๐‘ข . ๐‘‘๐‘ข โˆ’ โˆซ
7"$
7"!
๐œŽ(๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘ )|๐”‰(๐‘ก)
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = exp(๐ด ๐‘ก, ๐‘ก. โˆ’ ๐ต ๐‘ก, ๐‘ก. . ๐‘… ๐‘ก, ๐‘ก, ๐‘ก )
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . ๐ต ๐‘ก, ๐‘ก. . exp(๐ด ๐‘ก, ๐‘ก. )
ยจ In that formulation we see that naturally:
ยจ ๐ต ๐‘ก, ๐‘ก. = (๐‘ก. โˆ’ ๐‘ก)
ยจ exp ๐ด ๐‘ก, ๐‘ก. = ๐”ผ$0
โ„š
exp(โˆ’ โˆซ
!"$
!"$0
{โˆซ
M"$
M"!
๐œƒ ๐‘ข . ๐‘‘๐‘ข โˆ’ โˆซ
M"$
M"!
๐œŽ(๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘ )|๐”‰(๐‘ก)
ยจ Which can be super complicated
38
Luc_Faucheux_2021
From short rate to Affine model - IV
ยจ In the Ho-Lee model, we recovered:
ยจ ๐ต ๐‘ก, ๐‘ก. = (๐‘ก. โˆ’ ๐‘ก)
ยจ exp ๐ด ๐‘ก, ๐‘ก. = ๐”ผ$0
โ„š
exp(โˆ’ โˆซ
!"$
!"$0
{โˆซ
M"$
M"!
๐œƒ ๐‘ข . ๐‘‘๐‘ข โˆ’ โˆซ
M"$
M"!
๐œŽ(๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘ )|๐”‰(๐‘ก)
ยจ ๐œŽ ๐‘ข = ๐œŽ
ยจ ๐œƒ ๐‘ข =
I
IM
๐‘… 0, ๐‘ข, ๐‘ข + ๐œŽ&. ๐‘ข
ยจ Which led to:
ยจ ๐ด ๐‘ก, ๐‘ก. = โˆ’ โˆซ
!"$
!"$0
{[
IO #,M,M
IM
+ ๐œŽ&. ๐‘ข]. ๐‘ก. โˆ’ ๐‘ข โˆ’
%
&
. (๐‘ก. โˆ’ ๐‘ข)&. ๐œŽ&}. ๐‘‘๐‘ข
ยจ ๐ด ๐‘ก, ๐‘ก. = ๐‘… 0, ๐‘ก, ๐‘ก . ๐‘ก. โˆ’ ๐‘ก + ln(
PQ #,#,$0
PQ #,#,$
) โˆ’
R5
&
๐‘ก(๐‘ก. โˆ’ ๐‘ก)&
39
Luc_Faucheux_2021
From short rate to Affine model โ€“ V
ยจ Alternatively, we could also have made use of the useful relationship:
ยจ ๐”ผ exp โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = exp[โˆซ
!"#
!"$ %
&
๐‘“ ๐‘  &. ๐‘‘๐‘ ]
ยจ And apply it to:
ยจ exp ๐ด ๐‘ก, ๐‘ก. = ๐”ผ$0
โ„š
exp(โˆ’ โˆซ
!"$
!"$0
{โˆซ
M"$
M"!
๐œƒ ๐‘ข . ๐‘‘๐‘ข โˆ’ โˆซ
M"$
M"!
๐œŽ(๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘ )|๐”‰(๐‘ก)
ยจ Piterbarg (p.409) illustrates that approach on the even simpler model:
ยจ ๐‘‘๐‘… ๐‘ก, ๐‘ก, ๐‘ก = ๐œŽ. ([). ๐‘‘๐‘Š(๐‘ก)
ยจ Just for kicks, letโ€™s redo it here also:
40
Luc_Faucheux_2021
From short rate to Affine model โ€“ VI
ยจ ๐‘‘๐‘… ๐‘ก, ๐‘ก, ๐‘ก = ๐œŽ. ([). ๐‘‘๐‘Š(๐‘ก)
ยจ ๐‘… ๐‘ , ๐‘ , ๐‘  โˆ’ ๐‘… ๐‘ก, ๐‘ก, ๐‘ก = โˆ’ โˆซ
M"$
M"!
๐œŽ. ([). ๐‘‘๐‘Š(๐‘ข)
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = ๐”ผ$0
โ„š N($)
N($0)
|๐”‰(๐‘ก) = ๐”ผ$0
โ„š
exp(โˆ’ โˆซ
!"$
!"$0
๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ )|๐”‰(๐‘ก)
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = ๐”ผ$0
โ„š
exp(โˆ’ โˆซ
!"$
!"$0
{๐‘…(๐‘ก, ๐‘ก, ๐‘ก) โˆ’ โˆซ
M"$
M"!
๐œŽ. ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘ )|๐”‰(๐‘ก)
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . (๐‘ก. โˆ’ ๐‘ก) . ๐”ผ$0
โ„š
exp(โˆ’ โˆซ
!"$
!"$0
{โˆ’ โˆซ
M"$
M"!
๐œŽ. ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘ )|๐”‰(๐‘ก)
ยจ We recognize here once again our good old friend Guido Fubini so that we can change the
order of integration:
ยจ ๐‘‹ = โˆซ
!"$
!"$0
โˆซ
M"$
M"!
๐œŽ. ([). ๐‘‘๐‘Š(๐‘ข) . ๐‘‘๐‘ 
41
Luc_Faucheux_2021
From short rate to Affine model โ€“ VII
ยจ ๐‘‹ = โˆซ
!"$
!"$0
{โˆซ
M"$
M"!
๐œŽ. ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘ 
42
s
๐‘  = ๐‘ก!
u
s
u
๐‘  = ๐‘ก ๐‘  = ๐‘ก!
๐‘  = ๐‘ก
๐‘‹ = X
!"$
!"$0
{ X
M"$
M"!
๐œŽ. ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘  ๐‘‹ = X
M"$
M"$0
{ X
!"M
!"$0
๐œŽ. ๐‘‘๐‘ }. ([). ๐‘‘๐‘Š(๐‘ข)
Luc_Faucheux_2021
From short rate to Affine model โ€“ VIII
ยจ ๐‘‹ = โˆซ
!"$
!"$0
{โˆซ
M"$
M"!
๐œŽ. ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘ 
ยจ ๐‘‹ = โˆซ
M"$
M"$0
{โˆซ
!"M
!"$0
๐œŽ. ๐‘‘๐‘ }. ([). ๐‘‘๐‘Š(๐‘ข)
ยจ ๐‘‹ = โˆซ
M"$
M"$0
{๐œŽ. [๐‘ ]!"M
!"$0
}. ([). ๐‘‘๐‘Š(๐‘ข)
ยจ ๐‘‹ = โˆซ
M"$
M"$0
{๐œŽ. (๐‘ก. โˆ’ ๐‘ข)}. ([). ๐‘‘๐‘Š(๐‘ข)
ยจ ๐‘‹ = โˆซ
M"$
M"$0
๐œŽ. (๐‘ก. โˆ’ ๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข)
43
Luc_Faucheux_2021
From short rate to Affine model โ€“ IX
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . (๐‘ก. โˆ’ ๐‘ก) . ๐”ผ$0
โ„š
exp(โˆ’ โˆซ
!"$
!"$0
{โˆ’ โˆซ
M"$
M"!
๐œŽ. ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘ )|๐”‰(๐‘ก)
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . (๐‘ก. โˆ’ ๐‘ก) . ๐”ผ$0
โ„š
exp(โˆซ
!"$
!"$0
{โˆซ
M"$
M"!
๐œŽ. ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘ )|๐”‰(๐‘ก)
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . (๐‘ก. โˆ’ ๐‘ก) . ๐”ผ$0
โ„š
exp(โˆซ
M"$
M"$0
๐œŽ. (๐‘ก. โˆ’ ๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข))|๐”‰(๐‘ก)
ยจ And there we then again make good use of the useful relationship:
ยจ ๐”ผ exp โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = exp[โˆซ
!"#
!"$ %
&
๐‘“ ๐‘  &. ๐‘‘๐‘ ]
ยจ ๐”ผ exp โˆซ
M"$
M"$0
๐‘“ ๐‘ข . ๐‘‘๐‘Š ๐‘ข = exp[โˆซ
M"$
M"$0 %
&
๐‘“ ๐‘ข &. ๐‘‘๐‘ข]
ยจ On the function:
ยจ ๐‘“ ๐‘ข = ๐œŽ. (๐‘ก. โˆ’ ๐‘ข)
44
Luc_Faucheux_2021
From short rate to Affine model โ€“ X
ยจ ๐”ผ exp โˆซ
M"$
M"$0
๐‘“ ๐‘ข . ๐‘‘๐‘Š ๐‘ข = exp[โˆซ
M"$
M"$0 %
&
๐‘“ ๐‘ข &. ๐‘‘๐‘ข]
ยจ ๐‘“ ๐‘ข = ๐œŽ. (๐‘ก. โˆ’ ๐‘ข)
ยจ exp[โˆซ
M"$
M"$0 %
&
๐‘“ ๐‘ข &. ๐‘‘๐‘ข] = exp[โˆซ
M"$
M"$0 %
&
๐œŽ. ๐‘ก. โˆ’ ๐‘ข &. ๐‘‘๐‘ข] =
R5
&
. exp[โˆซ
M"$
M"$0
๐‘ก. โˆ’ ๐‘ข &. ๐‘‘๐‘ข]
ยจ exp โˆซ
M"$
M"$0 %
&
๐‘“ ๐‘ข &. ๐‘‘๐‘ข =
R5
&
. โˆ’
$0SM 8
T M"$
M"$0
=
R5
&
. โˆ’0 +
$0S$ 8
T
=
R5
U
. ๐‘ก. โˆ’ ๐‘ก T
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . (๐‘ก. โˆ’ ๐‘ก) . ๐”ผ$0
โ„š
exp(โˆซ
M"$
M"$0
๐œŽ. (๐‘ก. โˆ’ ๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข))|๐”‰(๐‘ก)
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . ๐‘ก. โˆ’ ๐‘ก . exp(
R5
U
. ๐‘ก. โˆ’ ๐‘ก T)
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = ๐”ผ$0
โ„š N($)
N($0)
|๐”‰(๐‘ก) = ๐”ผ$0
โ„š
exp(โˆ’ โˆซ
!"$
!"$0
๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ )|๐”‰(๐‘ก)
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . ๐‘ก. โˆ’ ๐‘ก +
R5
U
. ๐‘ก. โˆ’ ๐‘ก T
45
Luc_Faucheux_2021
From short rate to Affine model โ€“ XI
ยจ Alternatively, like Piterbarg does on p. 409, we can actually be a tad more general, and
instead of using the useful relationship:
ยจ ๐”ผ exp โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = exp[โˆซ
!"#
!"$ %
&
๐‘“ ๐‘  &. ๐‘‘๐‘ ]
ยจ We use the slightly more general result:
ยจ ๐”ผ exp ๐‘‹ = exp ๐”ผ ๐‘‹ . exp
%
&
๐”ผ (๐‘‹(๐‘ก)โˆ’ < ๐‘‹ >$)&
ยจ ๐”ผ exp ๐‘‹ = exp ๐‘€[๐‘‹(๐‘ก)] . exp
%
&
๐‘‰[๐‘‹(๐‘ก)]
ยจ ๐”ผ exp ๐‘‹ = exp[๐‘€] . exp
%
&
๐‘‰
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . (๐‘ก. โˆ’ ๐‘ก) . ๐”ผ$0
โ„š
exp(โˆซ
M"$
M"$0
๐œŽ. (๐‘ก. โˆ’ ๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข))|๐”‰(๐‘ก)
46
Luc_Faucheux_2021
From short rate to Affine model โ€“ X
ยจ So we will apply:
ยจ ๐”ผ exp ๐‘‹ = exp ๐‘€[๐‘‹(๐‘ก)] . exp
%
&
๐‘‰[๐‘‹(๐‘ก)]
ยจ To the process:
ยจ ๐‘Œ = โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . ๐‘ก. โˆ’ ๐‘ก + โˆซ
M"$
M"$0
๐œŽ. (๐‘ก. โˆ’ ๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข)
ยจ Since:
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = ๐”ผ$0
โ„š
exp(โˆ’ โˆซ
!"$
!"$0
{๐‘…(๐‘ก, ๐‘ก, ๐‘ก) โˆ’ โˆซ
M"$
M"!
๐œŽ. ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘ )|๐”‰(๐‘ก)
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = ๐”ผ$0
โ„š
exp(โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . ๐‘ก. โˆ’ ๐‘ก + โˆซ
!"$
!"$0
{โˆซ
M"$
M"!
๐œŽ. ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘ )|๐”‰(๐‘ก)
47
Luc_Faucheux_2021
From short rate to Affine model โ€“ XI
ยจ ๐‘Œ = โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . ๐‘ก. โˆ’ ๐‘ก + โˆซ
M"$
M"$0
๐œŽ. (๐‘ก. โˆ’ ๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข)
ยจ Since the ITO integral is a martingale:
ยจ ๐‘€ ๐‘Œ ๐‘ก = โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . ๐‘ก. โˆ’ ๐‘ก
ยจ So we then compute the variance:
ยจ ๐‘‰ ๐‘Œ ๐‘ก = ๐”ผ (๐‘Œ(๐‘ก)โˆ’ < ๐‘Œ >$)%
= ๐”ผ (๐‘Œ(๐‘ก) โˆ’ ๐‘€ ๐‘Œ ๐‘ก )%
= ๐”ผ (โˆซ
7"$
7"$!
๐œŽ. (๐‘ก) โˆ’ ๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข))%
ยจ There, we can use once again the good old isometry property of the ITO integral:
ยจ ๐”ผ (โˆซ
M"$
M"$0
๐œŽ. (๐‘ก. โˆ’ ๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข))& = โˆซ
M"$
M"$0
{๐œŽ. (๐‘ก. โˆ’ ๐‘ข)}&. ๐‘‘๐‘ข =
R5
T
. ๐‘ก. โˆ’ ๐‘ก T
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From short rate to Affine model โ€“ XII
ยจ ๐‘Œ = โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . ๐‘ก. โˆ’ ๐‘ก + โˆซ
M"$
M"$0
๐œŽ. (๐‘ก. โˆ’ ๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข)
ยจ ๐‘€ ๐‘Œ ๐‘ก = โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . ๐‘ก. โˆ’ ๐‘ก
ยจ ๐‘‰ ๐‘Œ ๐‘ก =
R5
T
. ๐‘ก. โˆ’ ๐‘ก T
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = ๐”ผ$0
โ„š
exp(๐‘Œ(๐‘ก))|๐”‰(๐‘ก) = exp ๐‘€[๐‘Œ(๐‘ก)] . exp
%
&
๐‘‰[๐‘Œ(๐‘ก)]
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = exp ๐‘€[๐‘Œ(๐‘ก)] . exp
%
&
๐‘‰[๐‘Œ(๐‘ก)] = exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . ๐‘ก. โˆ’ ๐‘ก . exp
%
&
R5
T
. ๐‘ก. โˆ’ ๐‘ก T
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . ๐‘ก. โˆ’ ๐‘ก . exp
%
&
R5
T
. ๐‘ก. โˆ’ ๐‘ก T
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . ๐‘ก. โˆ’ ๐‘ก +
R5
U
. ๐‘ก. โˆ’ ๐‘ก T
ยจ And we do recover indeed the same result for that simple model
49
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A couple of useful tools
50
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Useful tools
ยจ As you go through those slides, it is quite apparent that there are some relations or
properties that we keep using over and over again, or that are worth mentioning.
ยจ I tried to put all of them together in a quick summary section here
ยจ I still need to work on a notation section, maybe once I get my book deal
ยจ Would love to get your feedback on this section, if there are tools that you tend to use a lot
and find useful, just drop me a note and I would be happy to include those
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Useful tools โ€“ ITO LEMMA
ยจ The ITO lemma is revered in stochastic calculus.
ยจ In the somewhat misleading โ€œdifferentialโ€ form it reads:
ยจ ๐›ฟ๐‘“ =
IJ
IK
. ๐›ฟ๐‘‹ +
%
&
.
I5J
IK5 . (๐›ฟ๐‘‹)&
ยจ It should really only be expressed as:
ยจ ๐‘“ ๐‘‹ ๐‘กG โˆ’ ๐‘“ ๐‘‹ ๐‘กH = โˆซ
$"$9
$"$: IJ
IV
. ([). ๐‘‘๐‘‹(๐‘ก) + โˆซ
$"$9
$"$: %
&
.
I5J
IK5 . ([). (๐›ฟ๐‘‹)&
ยจ The ITO convention for the ITO integral is that we take the โ€œLHSโ€ (Left Hand side) in the
partition as noted by: ([)
ยจ And the definition of the integral is:
ยจ โˆซ
$"$9
$"$:
๐‘“ ๐‘‹(๐‘ก) . ๐‘‘๐‘Š ๐‘ก = lim
,โ†’(
โˆ‘."#
.",
๐‘“ ๐‘‹(๐‘ก.) . {๐‘Š ๐‘ก.+% โˆ’ ๐‘Š(๐‘ก.)}
ยจ Where we assume that we do not choose a pathological mesh and the the function is
relatively well behaved
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Useful tools โ€“ ITO LEMMA - II
ยจ Be careful that stochastic calculus in many ways has NOTHING to do with regular calculus
ยจ So it is quite dangerous to write:
ยจ ๐›ฟ๐‘“ =
IJ
IK
. ๐›ฟ๐‘‹ +
%
&
.
I5J
IK5 . (๐›ฟ๐‘‹)&
ยจ And say โ€œ oh well stochastic calculus is the same as regular calculus, it is just when I do
Taylor expansion I should really go up one more order in order to go up to all the orders that
are at least linear in timeโ€
ยจ Again, this is ONLY a formal correspondence, or a way to write down two things that are
almost completely different
ยจ Stochastic processes are NOT differentiable, so do not even think of using a โ€œTaylor
expansion on a stochastic processโ€
ยจ ALWAYS go back to the integral, always try to use the SIE format (Stochastic Integral
Equation), never the SDE format (Stochastic Differential Equation)
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Useful tools โ€“ ITO Leibniz
ยจ Again, for ease of notation, we use the โ€œdifferentialโ€ form, but by now we know better than
to trust is:
ยจ ๐›ฟ๐‘“ ๐‘‹, ๐‘Œ =
IJ
IV
. ๐›ฟ๐‘‹ +
IJ
IW
. ๐›ฟ๐‘Œ +
%
&
.
I5J
IV5 . ๐›ฟ๐‘‹& +
%
&
.
I5J
IW5 . ๐›ฟ๐‘Œ& +
I5J
IVIW
. ๐›ฟ๐‘‹. ๐›ฟ๐‘Œ
ยจ Note: should really be written as:
ยจ ๐›ฟ๐‘“ ๐‘‹, ๐‘Œ =
IJ
IK
. ๐›ฟ๐‘‹ +
IJ
IX
. ๐›ฟ๐‘Œ +
%
&
.
I5J
IK5 . ๐›ฟ๐‘‹& +
%
&
.
I5J
IX5 . ๐›ฟ๐‘Œ& +
I5J
IKIX
. ๐›ฟ๐‘‹. ๐›ฟ๐‘Œ
ยจ Lower case ๐‘ฅ is a regular variable
ยจ Upper case ๐‘‹ is a stochastic variable
ยจ ๐‘“ ๐‘‹, ๐‘Œ is really noted ๐‘“ ๐‘ฅ = ๐‘‹, ๐‘ฆ = ๐‘Œ and all the partial derivatives are for example:
ยจ
I5J
IKIX
=
I5J
IKIX
|K"V $ ,X"W($)
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Useful tools โ€“ ITO and STRATO correspondence
ยจ ITO integral is defined as LHS (Left Hand Side)
ยจ โˆซ
$"$H
$"$G
๐น ๐‘‹ ๐‘ก . ([). ๐‘‘๐‘‹(๐‘ก) = lim
Yโ†’(
{โˆ‘'"%
'"Y
๐น(๐‘‹(๐‘ก')). [๐‘‹(๐‘ก'+%) โˆ’ ๐‘‹(๐‘ก')]}
ยจ STRATO integral is defined as M (Middle)
ยจ โˆซ
$"$H
$"$G
๐น ๐‘‹ ๐‘ก . (โˆ˜). ๐‘‘๐‘‹(๐‘ก) = lim
Yโ†’(
{โˆ‘'"%
'"Y
๐น(
V($/12 +V($/)]
&
). [๐‘‹(๐‘ก'+%) โˆ’ ๐‘‹(๐‘ก')]}
ยจ For a simple Brownian motion
ยจ โˆซ
$"$H
$"$G
๐‘“ ๐‘Š ๐‘ก . (โˆ˜). ๐‘‘๐‘Š(๐‘ก) = โˆซ
$"$H
$"$G
๐‘“ ๐‘Š ๐‘ก . ([). ๐‘‘๐‘Š(๐‘ก) +
%
&
โˆซ
$"$H
$"$G IJ
IL
|L"[($). ๐‘‘๐‘ก
ยจ The integral in time โˆซ
$"$H
$"$G IJ
IL
|L"[($). ๐‘‘๐‘ก is the usual Riemann integral defined as
ยจ โˆซ
$"$H
$"$G
๐น ๐‘‹ ๐‘ก . ๐‘‘๐‘ก = lim
Yโ†’(
{โˆ‘'"%
'"Y
๐น(๐‘‹(๐œ‘[๐‘ก', ๐‘ก'+%])). [๐‘ก'+% โˆ’ ๐‘ก']}
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Useful tools โ€“ ITO and STRATO correspondence - II
ยจ Where ๐œ‘[๐‘ก', ๐‘ก'+%] is a function that takes some point within the mesh (does not matter
where, LHS, RIHS, middle, anywhere, could also varies from one bucket to the next, that is
the beauty of the Riemann integral in regular, or Newtonian, calculus, is that you do not
have all those pesky differences between ITO or Stratonovitch,โ€ฆ)
ยจ For a more complicated stochastic process
ยจ ๐‘‘๐‘‹ ๐‘ก = ๐‘Ž ๐‘ก, ๐‘‹ ๐‘ก . ๐‘‘๐‘ก + ๐‘ ๐‘ก, ๐‘‹ ๐‘ก . ๐‘‘๐‘Š
ยจ We have:
ยจ โˆซ
$"$H
$"$G
๐‘“ ๐‘‹ ๐‘ก . โˆ˜ . ๐‘‘๐‘Š ๐‘ก = โˆซ
$"$H
$"$G
๐‘“ ๐‘‹ ๐‘ก . ([). ๐‘‘๐‘Š(๐‘ก) + โˆซ
$"$H
$"$G %
&
. ๐‘ ๐‘ก, ๐‘‹ ๐‘ก .
IJ
IK
|K"V($). ๐‘‘๐‘ก
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Useful tools โ€“ ITO integral is a martingale
ยจ This is super useful
ยจ For a Brownian motion ๐‘Š ๐‘  associated to the measure
ยจ ๐”ผ{โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  } = ๐”ผ{lim
,โ†’(
โˆ‘."#
.",
๐‘“ ๐‘ . . {๐‘Š ๐‘ .+% โˆ’ ๐‘Š(๐‘ .)} }
ยจ ๐”ผ{โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  } = lim
,โ†’(
โˆ‘."#
.",
๐‘“ ๐‘ . . ๐”ผ{๐‘Š ๐‘ .+% โˆ’ ๐‘Š(๐‘ .)}
ยจ ๐”ผ{โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  } = lim
,โ†’(
โˆ‘."#
.",
๐‘“ ๐‘ . . 0 = 0
ยจ ๐”ผ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = 0
ยจ ๐”ผ โˆซ
!"#
!"$
๐‘“ ๐‘Š ๐‘  , ๐‘  . ๐‘‘๐‘Š ๐‘  = 0
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Useful tools โ€“ Isometry of the ITO integral
ยจ ๐”ผ{ โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘ 
&
} = โˆซ
!"#
!"$
๐‘“ ๐‘  &. ๐‘‘๐‘ 
ยจ ๐”ผ{ โˆซ
!"#
!"$
๐‘“ ๐‘Š ๐‘  , ๐‘  . ๐‘‘๐‘Š ๐‘ 
&
} = โˆซ
!"#
!"$
๐‘“ ๐‘Š ๐‘  , ๐‘  &. ๐‘‘๐‘ 
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Useful tools โ€“ A martingale is driftless, a driftless process is a
martingale
ยจ ๐”ผ[{๐‘‹(๐‘ก)|)|๐”‰(๐‘ )} = 0
ยจ ๐‘‘๐‘‹ ๐‘ก = ๐‘Ž ๐‘ก, ๐‘‹ ๐‘ก . ๐‘‘๐‘ก + ๐‘ ๐‘ก, ๐‘‹ ๐‘ก . ๐‘‘๐‘Š
ยจ ๐‘Ž ๐‘ก, ๐‘‹ ๐‘ก = 0
ยจ ๐‘‘๐‘‹ ๐‘ก = ๐‘ ๐‘ก, ๐‘‹ ๐‘ก . ๐‘‘๐‘Š
ยจ No advection, no drift for a martingale
ยจ ๐‘‹ ๐‘ก = โˆซ
!"#
!"$
๐‘ ๐‘ก, ๐‘‹ ๐‘ก . ๐‘‘๐‘Š ๐‘ 
ยจ Again the ITO integral is a martingale
ยจ ๐”ผ โˆซ
!"#
!"$
๐‘ ๐‘ก, ๐‘‹ ๐‘ก . ๐‘‘๐‘Š ๐‘  = 0
ยจ ๐”ผ[{๐‘‹(๐‘ก)|)|๐”‰(๐‘ )} = 0
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Useful tools โ€“ useful relationship
ยจ ๐”ผ exp โˆซ
!"#
!"$
๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = exp[โˆซ
!"#
!"$ %
&
๐‘“ ๐‘  &. ๐‘‘๐‘ ]
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Useful tools โ€“ expected value of the exponential
ยจ ๐”ผ exp ๐‘‹ = exp ๐”ผ ๐‘‹ . exp
%
&
๐”ผ (๐‘‹ โˆ’ ๐”ผ ๐‘‹ )&
ยจ ๐‘€ ๐‘‹ ๐‘ก = ๐”ผ ๐‘‹
ยจ ๐‘‰ ๐‘‹ ๐‘ก = ๐”ผ (๐‘‹(๐‘ก) โˆ’ ๐‘€ ๐‘‹ ๐‘ก )&
ยจ ๐”ผ exp ๐‘‹ = exp ๐‘€[๐‘‹(๐‘ก)] . exp
%
&
๐‘‰[๐‘‹(๐‘ก)]
ยจ ๐”ผ exp ๐‘‹ = exp[๐‘€] . exp
%
&
๐‘‰
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Useful tools - Fubini
ยจ ๐‘‹ = โˆซ
!"$
!"$0
{โˆซ
M"$
M"!
๐‘“(๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘ 
62
s
๐‘  = ๐‘ก!
u
s
u
๐‘  = ๐‘ก ๐‘  = ๐‘ก!
๐‘  = ๐‘ก
๐‘‹ = X
!"$
!"$0
{ X
M"$
M"!
๐‘“(๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘  ๐‘‹ = X
M"$
M"$0
{ X
!"M
!"$0
๐‘“(๐‘ ). ๐‘‘๐‘ }. ([). ๐‘‘๐‘Š(๐‘ข)
Luc_Faucheux_2021
Useful tools โ€“ how to always create a martingale
ยจ We use here the Tower property:
ยจ For any process ๐‘‹ ๐‘ก , we create:
ยจ ๐‘ ๐‘ก = ๐”ผ
[
{๐‘‹(๐‘‡)|๐”‰(๐‘ก)}
ยจ ๐”ผ
[
๐‘ ๐‘ก ๐”‰ ๐‘  = ๐”ผ
[
๐”ผ
[
๐‘‹ ๐‘‡ ๐”‰ ๐‘ก ๐”‰ ๐‘  = ๐”ผ
[
๐‘‹ ๐‘‡ ๐”‰ ๐‘  = ๐‘(๐‘ )
ยจ Because conditioning firstly on information back to time ๐‘ก then back to time ๐‘  is just the
same as conditioning back to time ๐‘  to start with.
ยจ ๐”ผ
[
๐‘ ๐‘ก ๐”‰ ๐‘  = ๐‘(๐‘ )
ยจ So ๐‘ ๐‘ก = ๐”ผ
[
{๐‘‹(๐‘‡)|๐”‰(๐‘ก)} is by construction a ๐‘Š-martingale
ยจ That is a neat little trick to always create a martingale process (Baxter p. 77)
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A few good measures
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A few good measures
ยจ This section is a summary of some of the measures used in Finance, and their differences /
notation
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A few good measures โ€“ The Physical measure
ยจ This is the โ€œnaturalโ€ measure
ยจ It is usually noted โ„™ (I guess the P stands for Physical)
ยจ Its characteristics (drift, variance) are usually calculated from historical data
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A few good measures โ€“ The Risk-Neutral measure
ยจ This one usually comes right after the Physical measure
ยจ It is usually noted โ„š (I guess because in the alphabet Q comes right after P)
ยจ In Finance its Numeraire is the rolling Bank account (MMN-Money Market Numeraire)
ยจ As a first approximation (especially for equity derivative), the rates are assumed to be
deterministic (non-stochastic) and even further sometimes constant in time:
ยจ It is then usually noted as follows
ยจ ๐‘‘โ„ณ ๐‘ก = ๐‘Ÿ. โ„ณ. ๐‘‘๐‘ก with: โ„ณ ๐‘ก = ๐‘’]$
ยจ As a further approximation (especially for short-dated options for which the discounting
does not matter too much, or especially nowadays where rates are not moving and are fixed
at 0 essentially due to the Central Banks Ponzi scheme, according to my good friend Bogac
Ozdemir)
ยจ ๐‘Ÿ = 0
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A few good measures โ€“ The Risk-Neutral measure - II
ยจ Essentially using the MMM as a numeraire, you just replace the drift by the risk-free rate ๐‘Ÿ
ยจ If you had for a stock:
ยจ ๐‘‘๐‘† ๐‘ก = ๐œ‡โ„™. ๐‘†. ๐‘‘๐‘ก + ๐œŽโ„™. ๐‘†. ๐‘‘๐‘Šโ„™
ยจ ๐‘‘โ„ณ ๐‘ก = ๐‘Ÿ. โ„ณ. ๐‘‘๐‘ก
ยจ Then the ratio
^
โ„ณ
will ALSO follows a geometric Brownian motion
ยจ ๐‘‘
^
โ„ณ
= ๐œ‡โ„™ โˆ’ ๐‘Ÿ .
^
โ„ณ
. ๐‘‘๐‘ก + ๐œŽโ„™.
^
โ„ณ
. ๐‘‘๐‘Šโ„™
ยจ The ratio
^
โ„ณ
is a martingale under the risk-free measure, it is thus driftless
ยจ ๐‘‘
^
โ„ณ
= 0. ๐‘‘๐‘ก + {๐‘ ๐‘œ๐‘š๐‘’๐‘กโ„Ž๐‘–๐‘›๐‘”}. ๐‘‘๐‘Šโ„š
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A few good measures โ€“ The Risk-Neutral measure - IIa
ยจ ๐‘‘
^
โ„ณ
= ๐œ‡โ„™ โˆ’ ๐‘Ÿ .
^
โ„ณ
. ๐‘‘๐‘ก + ๐œŽโ„™.
^
โ„ณ
. ๐‘‘๐‘Šโ„™
ยจ ๐‘‘
^
โ„ณ
= 0. ๐‘‘๐‘ก + {๐‘ ๐‘œ๐‘š๐‘’๐‘กโ„Ž๐‘–๐‘›๐‘”}. ๐‘‘๐‘Šโ„š
ยจ In this case, we see that we can define the โ„š-Brownian motion as:
ยจ ๐‘‘๐‘Šโ„š = ๐‘‘๐‘Šโ„™ โˆ’
`โ„™S]
Rโ„™
. ๐‘‘๐‘ก
ยจ We then get:
ยจ ๐‘‘
^
โ„ณ
= 0. ๐‘‘๐‘ก + ๐œŽโ„™.
^
โ„ณ
. ๐‘‘๐‘Šโ„š
ยจ We also see our old friend the โ€œmarket price of riskโ€, the excess return over the risk free
rate, normalized by the volatility of the asset.
ยจ ๐œ† =
`โ„™S]
Rโ„™
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A few good measures โ€“ The Risk-Neutral measure - IIb
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A few good measures โ€“ The Risk-Neutral measure - III
ยจ Things become a little more complicated once we assume that rates are stochastic
ยจ The usual notation becomes then:
ยจ ๐ต ๐‘ก = exp(โˆซ
!"#
!"$
๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ )
ยจ In the extended Zeros framework of Mercurio and Lyashenko:
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, 0 = exp โˆซ
!"#
!"$
๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘  = ๐ต ๐‘ก
ยจ The rolling Bank Account has the useful property that:
ยจ ๐ต 0 = exp(โˆซ
!"#
!"#
๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ) = exp 0 = 1
ยจ This is useful when valuing claims and derivatives
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A few good measures โ€“ The Risk-Neutral measure - IV
ยจ For example
ยจ
a #,$%,$,$
N(#)
= ๐”ผ$
โ„š a $,$%,$,$
N($)
|๐”‰(0) = ๐”ผ$
โ„š a $,$%,$,$
cde(โˆซ
<=>
<=?
O !,!,! .D!)
|๐”‰(0)
ยจ
a #,$%,$,$
N(#)
=
a #,$%,$,$
%
= ๐‘๐ถ 0,0, ๐‘ก = ๐”ผ$
โ„š a $,$%,$,$
cde(โˆซ
<=>
<=?
O !,!,! .D!)
|๐”‰(0)
ยจ ๐‘๐ถ 0,0, ๐‘ก = ๐”ผ$
โ„š a $,$%,$,$
cde(โˆซ
<=>
<=?
O !,!,! .D!)
|๐”‰(0) = ๐”ผ$
โ„š %
cde(โˆซ
<=>
<=?
O !,!,! .D!)
|๐”‰(0)
ยจ ๐‘๐ถ 0,0, ๐‘ก = ๐”ผ$
โ„š
exp(โˆ’ โˆซ
!"#
!"$
๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ )|๐”‰(0)
ยจ This is usually used when calibrating a model to the current time ๐‘ก = 0 term structure of
Zero Coupon bond prices
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A few good measures โ€“ The Risk-Neutral measure - V
ยจ Similarly as we go further in time:
ยจ ๐‘๐ถ 0,0, ๐‘ก =
PQ #,#,$
N(#)
= o
๐‘ 0,0, ๐‘ก = ๐”ผ$
โ„š
exp(โˆ’ โˆซ
!"#
!"$
๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ )|๐”‰(0)
ยจ
PQ #,#,$@
N(#)
= o
๐‘ 0,0, ๐‘ก/ = ๐”ผ$
โ„š PQ $,$,$@
N($)
|๐”‰(0) = ๐”ผ$
โ„š o
๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ |๐”‰(0)
ยจ
PQ $,$,$@
N($)
= o
๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ = ๐”ผ$0
โ„š PQ $0,$0,$@
N($0)
|๐”‰(๐‘ก) = ๐”ผ$0
โ„š o
๐‘ ๐‘ก., ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก) for: ๐‘ก < ๐‘ก.< ๐‘ก/
ยจ In particular for: ๐‘ก. = ๐‘ก/
ยจ
PQ $,$,$@
N($)
= o
๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ = ๐”ผ$@
โ„š PQ $@,$@,$@
N($@)
|๐”‰(๐‘ก) = ๐”ผ$@
โ„š o
๐‘ ๐‘ก/, ๐‘ก/, ๐‘ก/ |๐”‰(๐‘ก) = ๐”ผ$@
โ„š %
N($@)
|๐”‰(๐‘ก)
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก/ = ๐ต ๐‘ก . ๐”ผ$@
โ„š %
N($@)
|๐”‰(๐‘ก) = ๐”ผ$@
โ„š
๐ต ๐‘ก .
%
N($@)
|๐”‰(๐‘ก) = ๐”ผ$@
โ„š N $
N($@)
|๐”‰(๐‘ก)
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A few good measures โ€“ The Risk-Neutral measure - VI
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก/ = ๐”ผ$@
โ„š N $
N($@)
|๐”‰(๐‘ก)
ยจ ๐ต ๐‘ก = exp(โˆซ
!"#
!"$
๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ )
ยจ ๐ต ๐‘ก/ = exp(โˆซ
!"#
!"$@
๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ )
ยจ
A $
A($#)
= exp(โˆซ
!"#
!"$
๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ) /exp(โˆซ
!"#
!"$#
๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ) = exp(โˆซ
!"#
!"$
๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘  โˆ’ โˆซ
!"#
!"$#
๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ))
ยจ โˆซ
!"#
!"$#
๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ) = โˆซ
!"#
!"$
๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ) + โˆซ
!"$
!"$#
๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ )
ยจ
A $
A($#)
= exp(โˆ’ โˆซ
!"$
!"$#
๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ )
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก- = ๐”ผ$#
โ„š
exp[โˆ’ โˆซ
!"$
!"$#
๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ]|๐”‰(๐‘ก)
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก- = ๐”ผ$#
โ„š
๐‘’D โˆซ&'(
&'(#
F !,!,! .H!
|๐”‰(๐‘ก)
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A few good measures โ€“ The Risk-Neutral measure - VII
ยจ Similar to the SDE for stocks, the drift for a tradeable security in the Risk neutral measure is
the instantaneous short rate ๐‘… ๐‘ , ๐‘ , ๐‘ 
ยจ
DPQ $,$,$0
PQ $,$,$0
= ๐‘… ๐‘ก, ๐‘ก, ๐‘ก . ๐‘‘๐‘ก + ๐‘‰ ๐‘ก, ๐‘ก., ๐‘ก. . ([). ๐‘‘๐‘Šโ„š ๐‘ก
ยจ
D f
P $,$,$0
f
P $,$,$0
= ๐‘‰ ๐‘ก, ๐‘ก., ๐‘ก. . ([). ๐‘‘๐‘Šโ„š ๐‘ก
ยจ This is a driftless process. In particular:
ยจ ๐‘‘ o
๐‘ ๐‘ก, ๐‘ก, ๐‘ก. = o
๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐‘‰ ๐‘ก, ๐‘ก., ๐‘ก. . ([). ๐‘‘๐‘Šโ„š ๐‘ก
ยจ This is also a driftless process, hence the deflated Zeros are martingale under the risk-
neutral โ„š measure
ยจ ๐‘‘๐ต ๐‘ก = ๐‘… ๐‘ก, ๐‘ก, ๐‘ก . ๐ต ๐‘ก . ๐‘‘๐‘ก or ๐ต ๐‘ก = exp(โˆซ
!"#
!"$
๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ )
ยจ p
๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = o
๐‘ ๐‘ก, ๐‘ก, ๐‘ก. =
PQ $,$,$0
N($)
=
P $,$,$0
N($)
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A few good measures โ€“ The Terminal/Forward measure
ยจ This is the measure where the numeraire is the Zero ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก/
ยจ Also referred to in textbooks as the ๐‘ก/-Terminal or sometimes ๐‘‡/-terminal
ยจ It is also called the Forward measure because under this measure the Forward rate (simply
compounded, not every forward rate!) spanning a period [๐‘ก., ๐‘ก/] is a martingale.
ยจ Not super easy to convince yourself of, so worth looking at it again (it was a while since we
did it, was in deck II and III)
ยจ Also worth redoing it with the full notation that we have slowly developed as we went along
ยจ This hopefully will be rigorous enough to stand the test of reading it back later on.
ยจ Just to be on the safe side, we will write it:
ยจ ๐”ผ$@
โ„ค($@)
ยจ It is usually when the claims get PAID (the early one is when the claim gets SET or FIXED)
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A few good measures โ€“ The Terminal/Forward measure - II
ยจ We have to do a little refresher on the notation (because remember unlike in Physics, what
matters really in Finance is WHEN you get paid, not when you observe/fix/set the payment)
ยจ ๐‘‰(๐‘ก) = ๐‘‰ ๐‘ก, $๐ป(๐‘ก), ๐‘ก., ๐‘ก/
79
๐‘ƒ๐‘Ž๐‘–๐‘‘ ๐‘Ž๐‘ก ๐‘ก๐‘–๐‘š๐‘’ ๐‘ก/
๐น๐‘–๐‘ฅ๐‘’๐‘‘ ๐‘œ๐‘Ÿ ๐‘ ๐‘’๐‘ก ๐‘Ž๐‘ก ๐‘ก๐‘–๐‘š๐‘’ ๐‘ก.
๐บ๐‘’๐‘›๐‘’๐‘Ÿ๐‘Ž๐‘™ ๐‘ƒ๐‘Ž๐‘ฆ๐‘œ๐‘“๐‘“ ๐ป ๐‘ก ๐‘–๐‘› ๐‘๐‘ข๐‘Ÿ๐‘Ÿ๐‘’๐‘›๐‘๐‘ฆ $
๐‘‰๐‘Ž๐‘™๐‘ข๐‘’ ๐‘œ๐‘“ ๐‘กโ„Ž๐‘’ ๐‘๐‘Ž๐‘ฆ๐‘œ๐‘“๐‘“ ๐‘œ๐‘๐‘ ๐‘’๐‘Ÿ๐‘ฃ๐‘’๐‘‘ ๐‘Ž๐‘ก ๐‘ก๐‘–๐‘š๐‘’ ๐‘ก
Luc_Faucheux_2021
A few good measures โ€“ The Terminal/Forward measure - III
ยจ ๐‘‰(๐‘ก) = ๐‘‰ ๐‘ก, $๐ป(๐‘ก), ๐‘ก., ๐‘ก/
ยจ Of paramount importance is the payoff that ALWAYS pays $1
ยจ $๐ป ๐‘ก = $1
ยจ
a $,$h($),$0,$@
P($,$,$@)
is a martingale under ๐”ผ$@
โ„ค($@)
ยจ
a $,$h($),$0,$@
P($,$,$@)
= ๐”ผ$@
โ„ค($@) a $@,$h $@ ,$0,$@
P $@,$@,$@
๐”‰ ๐‘ก = ๐”ผ$@
โ„ค($@)
๐‘‰ ๐‘ก/, $๐ป ๐‘ก/ , ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก
ยจ Because in a very convenient fashion:
ยจ ๐‘ ๐‘ก/, ๐‘ก/, ๐‘ก/ = 1
ยจ For $๐ป ๐‘ก = $1
ยจ
a $,$%,$0,$@
P($,$,$@)
= ๐”ผ$@
โ„ค($@) a $@,$%,$0,$@
P $@,$@,$@
๐”‰ ๐‘ก = ๐”ผ$@
โ„ค($@)
๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = $1
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A few good measures โ€“ The Terminal/Forward measure - IV
ยจ
a $,$%,$0,$@
P($,$,$@)
= ๐”ผ$@
โ„ค($@) a $@,$%,$0,$@
P $@,$@,$@
๐”‰ ๐‘ก = ๐”ผ$@
โ„ค($@)
๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = $1
ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก, ๐‘ก/)
ยจ Note that this result is independent of the measure, and validates our intuition about the
Zeros
ยจ All right, letโ€™s nest those bad boys so that we recover the bootstrap relations
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A few good measures โ€“ The Terminal/Forward measure - V
ยจ
a $,$%,$0,$@
P($,$,$@)
= ๐”ผ$@
โ„ค($@) a $@,$%,$0,$@
P $@,$@,$@
๐”‰ ๐‘ก so: ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก, ๐‘ก/)
ยจ
a $,$%,$0,$0
P($,$,$0)
= ๐”ผ$0
โ„ค($0) a $0,$%,$0,$0
P $0,$0,$0
๐”‰ ๐‘ก so: ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก. = ๐‘(๐‘ก, ๐‘ก, ๐‘ก.)
ยจ Note that the โ€œsettingโ€ or โ€œfixingโ€ time for a claim that always pays $1 is irrelevant, but letโ€™s
keep it for now, we did not go through a couple of thousand slides of building a rigorous
formalism to throw it all away now.
ยจ
a $,$%,$0,$@
P($,$,$0)
= ๐”ผ$0
โ„ค($0) a $0,$%,$0,$@
P $0,$0,$0
๐”‰ ๐‘ก
ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐”ผ$0
โ„ค($0) a $0,$%,$0,$@
P $0,$0,$0
๐”‰ ๐‘ก = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก
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A few good measures โ€“ The Terminal/Forward measure - VI
ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก
ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก, ๐‘ก/)
ยจ From the bootstrap relations we have by definition:
ยจ ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐‘(๐‘ก, ๐‘ก., ๐‘ก/)
ยจ So we have:
ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก
ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐‘(๐‘ก, ๐‘ก., ๐‘ก/)
ยจ Which leads us quite naturally to:
ยจ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก
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A few good measures โ€“ The Terminal/Forward measure - VII
ยจ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก
ยจ That is where the nesting comes into the game:
ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก, ๐‘ก/)
ยจ And so by just changing the variables:
ยจ ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก., ๐‘ก., ๐‘ก/)
ยจ Again when you read it aloud it makes sense: The value at time ๐‘ก. of a claim that will pay $1
no matter what at time ๐‘ก/ is equal to the value of the Zero Coupon Bond at time ๐‘ก. that pays
$1 at time ๐‘ก/.
ยจ That is almost a tautology, but we can nest that one into the first equation:
ยจ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = ๐”ผ$0
โ„ค($0)
๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก
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A few good measures โ€“ The Terminal/Forward measure - VIII
ยจ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = ๐”ผ$0
โ„ค($0)
๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก
ยจ I know that we are mixing a little the Terminal/Forward measure ๐”ผ$@
โ„ค($@)
and the
early/discount measure ๐”ผ$0
โ„ค($0)
but they are exactly the same just with a different end time
for the expectation and the numeraire, but the above relationship is quite cool.
ยจ In the early/discount measure ๐”ผ$0
โ„ค($0)
, the Zeros are a martingale. And by Zeros we mean the
๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ , so not every Zeros, careful about that.
ยจ So the process of those guys will be driftless:
ยจ ๐‘‘๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = 0. ๐‘‘๐‘ก + ๐‘ ๐‘œ๐‘š๐‘’๐‘กโ„Ž๐‘–๐‘›๐‘” . ๐‘‘๐‘Š
$0
โ„ค $0
(๐‘ก)
ยจ Where ๐‘Š
$0
โ„ค $0
(๐‘ก) is the Brownian motion associated to the early measure ๐”ผ$0
โ„ค($0)
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A few good measures โ€“ The Terminal/Forward measure - IX
ยจ OK, so we are like a fifth of the way there, so grab a coke and some popcorn
ยจ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = ๐”ผ$0
โ„ค($0)
๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก
ยจ Now we did define the simply compounded Forward Rates as:
ยจ ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ =
%
i $,$0,$@
. [
%
PQ $,$0,$@
โˆ’ 1]
ยจ Where ๐œ ๐‘ก, ๐‘ก., ๐‘ก/ is the daycount fraction that for sake of simplicity we will note ๐œ in this
section.
ยจ ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ =
%
i
. [
%
PQ $,$0,$@
โˆ’ 1]
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ =
%
%+i.j $,$0,$@
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A few good measures โ€“ The Terminal/Forward measure - X
ยจ Remember that we could have defined a number of other rates:
ยจ Continuously compounded FORWARD : ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ = exp โˆ’๐‘… ๐‘ก, ๐‘ก., ๐‘ก/ . ๐œ ๐‘ก, ๐‘ก., ๐‘ก/
ยจ Simply compounded FORWARD: ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ =
%
%+i $,$0,$@ .j $,$0,$@
ยจ Annually compounded FORWARD : ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ =
%
(%+W $,$0,$@ )
I ?,?0,?@
ยจ ๐‘ž-times per year compounded FOWARD: ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ =
%
(%+
2
J
.WJ $,$0,$@ )
J.I ?,?0,?@
ยจ The function ๐œ ๐‘ก, ๐‘ก., ๐‘ก/ is the daycount fraction, will usually depends on what convention
(ACT/ACT, ACT/360, 30/360,โ€ฆ) you will choose, and potentially adjustment for holidays and
what holiday center
87
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A few good measures โ€“ The Terminal/Forward measure - XI
ยจ We are looking at the usual graph we had in section II and III
88
๐‘ก
๐‘ก๐‘–๐‘š๐‘’
๐‘™ ๐‘ก, ๐‘ก., ๐‘ก/ . ๐œ ๐‘ก, ๐‘ก., ๐‘ก/
๐‘ก!
=
๐‘ก
๐‘ก๐‘–๐‘š๐‘’
๐‘ก!
๐‘ก"
$1
$1
๐‘ก"
Luc_Faucheux_2021
A few good measures โ€“ The Terminal/Forward measure - XII
ยจ Letโ€™s construct a claim on a portfolio that pays $1 at time ๐‘ก. and pays back $1 at time ๐‘ก/
ยจ Note, by now that specific portfolio should not come as a surprise
ยจ Letโ€™s note ฮ (๐‘ก) the value at time ๐‘ก of this portfolio
ยจ ฮ  ๐‘ก = ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก. โˆ’ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/
ยจ ฮ  ๐‘ก = ๐‘(๐‘ก, ๐‘ก, ๐‘ก.) โˆ’ ๐‘(๐‘ก, ๐‘ก, ๐‘ก/)
ยจ ฮ  ๐‘ก = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ . {
P $,$,$0
P($,$,$@)
โˆ’ 1}
ยจ ฮ  ๐‘ก = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ . {
%
P($,$0,$@)
โˆ’ 1}
ยจ ฮ  ๐‘ก = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ . 1 + ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ โˆ’ 1 = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ . ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/
ยจ OK, keep that on the back of your minds for just a couple of slides
89
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A few good measures โ€“ The Terminal/Forward measure - XIII
ยจ Because that claim is tradeable, the ratio of it to the numeraire is a martingale in the
terminal measure:
ยจ
k $
P($,$,$@)
= ๐”ผ$@
โ„ค($@) k $@
P $@,$@,$@
๐”‰ ๐‘ก = ๐”ผ$@
โ„ค($@)
ฮ  ๐‘ก/ ๐”‰ ๐‘ก
ยจ ฮ  ๐‘ก = ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก. โˆ’ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/
ยจ ฮ  ๐‘ก/ = ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. โˆ’ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก/
ยจ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก/ is the value at time ๐‘ก/ of a claim that pays $1 at time ๐‘ก/
ยจ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก/ = 1
ยจ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. is the value at time ๐‘ก/ of a claim that PAID $1 at time ๐‘ก.
ยจ ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) is the value at time ๐‘ก. of a Zero coupon bond that pays $1 at time ๐‘ก/
90
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A few good measures โ€“ The Terminal/Forward measure - XIV
ยจ Letโ€™s make sure that we get this right:
ยจ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. is the value at time ๐‘ก/ of a claim that PAID $1 at time ๐‘ก.
ยจ ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) is the value at time ๐‘ก. of a Zero coupon bond that pays $1 at time ๐‘ก/
ยจ Letโ€™s say it another way
ยจ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. is the value at time ๐‘ก/ of a claim that PAID $1 at time ๐‘ก.
ยจ If you invested $1 at time ๐‘ก. you will receive ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. at time ๐‘ก/
ยจ ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) is the value at time ๐‘ก. of a Zero coupon bond that pays $1 at time ๐‘ก/
ยจ If you invested ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) at time ๐‘ก. you will receive $1 at time ๐‘ก/
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A few good measures โ€“ The Terminal/Forward measure - XV
ยจ If you invested $1 at time ๐‘ก. you will receive ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. at time ๐‘ก/
ยจ If you invested ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) at time ๐‘ก. you will receive $1 at time ๐‘ก/
ยจ And so:
ยจ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. =
%
P($0,$0,$@)
ยจ Really make sure that you are 100% convinced on that one.
ยจ Just to be sure, letโ€™s break it down in the next slide:
92
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A few good measures โ€“ The Terminal/Forward measure - XVI
ยจ If you invested $1 at time ๐‘ก. you will receive ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. at time ๐‘ก/
ยจ If you invested ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) at time ๐‘ก. you will receive $1 at time ๐‘ก/
ยจ If you invested ๐‘ ๐‘ก., ๐‘ก., ๐‘ก/ . {
%
P $0,$0,$@
} at time ๐‘ก. you will receive $1. {
%
P $0,$0,$@
} at time ๐‘ก/
ยจ If you invested {
P $0,$0,$@
P $0,$0,$@
} at time ๐‘ก. you will receive $1. {
%
P $0,$0,$@
} at time ๐‘ก/
ยจ If you invested {$1} at time ๐‘ก. you will receive $1. {
%
P $0,$0,$@
} at time ๐‘ก/
ยจ But we also have the first relation:
ยจ If you invested $1 at time ๐‘ก. you will receive ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. at time ๐‘ก/
ยจ And so:
ยจ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. =
%
P $0,$0,$@
93
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A few good measures โ€“ The Terminal/Forward measure - XVII
ยจ Ok, we are 3 fifths of the way there
ยจ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. =
%
P $0,$0,$@
ยจ But remember ! This is really saying:
ยจ If you invested {$1} at time ๐‘ก. you will receive $1. {
%
P $0,$0,$@
} at time ๐’•๐’‹
ยจ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. = ๐‘‰(๐‘ก/,
%
P $0,$0,$@
, ๐‘ก., ๐‘ก/)
ยจ The value at time ๐‘ก/ of receiving $1 that was PAID at time ๐‘ก., is equal to the value at time ๐‘ก/
of receiving $
%
P $0,$0,$@
that was set at time ๐‘ก. and paid at time ๐‘ก/
94
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A few good measures โ€“ The Terminal/Forward measure - XVIII
ยจ Which we now plug back into:
ยจ
k $
P($,$,$@)
= ๐”ผ$@
โ„ค($@) k $@
P $@,$@,$@
๐”‰ ๐‘ก = ๐”ผ$@
โ„ค($@)
ฮ  ๐‘ก/ ๐”‰ ๐‘ก
ยจ ฮ  ๐‘ก = ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก. โˆ’ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/
ยจ ฮ  ๐‘ก/ = ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. โˆ’ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก/
ยจ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก/ = $1
ยจ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. = ๐‘‰(๐‘ก/,
%
P $0,$0,$@
, ๐‘ก., ๐‘ก/)
95
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A few good measures โ€“ The Terminal/Forward measure - XIX
ยจ We now go back to the bootstrap definition
ยจ
%
P $0,$0,$@
= 1 + ๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/
ยจ ฮ  ๐‘ก/ = ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. โˆ’ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก/
ยจ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก/ = $1
ยจ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. = ๐‘‰(๐‘ก/, $
%
P $0,$0,$@
, ๐‘ก., ๐‘ก/)
ยจ ฮ  ๐‘ก/ = ๐‘‰ ๐‘ก/, $
%
P $0,$0,$@
, ๐‘ก., ๐‘ก/ โˆ’ $1
ยจ ฮ  ๐‘ก/ = ๐‘‰(๐‘ก/, $
%
P $0,$0,$@
โˆ’ $1, ๐‘ก., ๐‘ก/)
96
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A few good measures โ€“ The Terminal/Forward measure - XX
ยจ ฮ  ๐‘ก/ = ๐‘‰(๐‘ก/, $
%
P $0,$0,$@
โˆ’ $1, ๐‘ก., ๐‘ก/)
ยจ ฮ  ๐‘ก/ = ๐‘‰(๐‘ก/, $(1 + ๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ ) โˆ’ $1, ๐‘ก., ๐‘ก/)
ยจ ฮ  ๐‘ก/ = ๐‘‰(๐‘ก/, $(1 + ๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ โˆ’ 1), ๐‘ก., ๐‘ก/)
ยจ ฮ  ๐‘ก/ = ๐‘‰(๐‘ก/, $(๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ ), ๐‘ก., ๐‘ก/)
ยจ ฮ  ๐‘ก/ = ๐‘‰(๐‘ก/, $๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/)
ยจ ฮ  ๐‘ก/ = ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. โˆ’ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก/
ยจ That is essentially the intuition that we had built and illustrated in the graph
97
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A few good measures โ€“ The Terminal/Forward measure โ€“ XXa
ยจ ฮ  ๐‘ก = ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก. โˆ’ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/
ยจ ฮ  ๐‘ก = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ . ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/
ยจ ฮ  ๐‘ก/ = ๐‘‰(๐‘ก/, $๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/)
ยจ ฮ  ๐‘ก/ = ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. โˆ’ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก/
ยจ ฮ  ๐‘ก = ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก. โˆ’ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/
ยจ ฮ  ๐‘ก = ๐‘‰(๐‘ก, $๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/)
98
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A few good measures โ€“ The Terminal/Forward measure โ€“ XXb
ยจ ฮ  ๐‘ก = ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก. โˆ’ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/
99
๐‘ก
๐‘ก๐‘–๐‘š๐‘’
๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ . ๐œ ๐‘ก, ๐‘ก., ๐‘ก/
๐‘ก!
=
๐‘ก
๐‘ก๐‘–๐‘š๐‘’
๐‘ก!
๐‘ก"
$1
$1
๐‘ก"
Luc_Faucheux_2021
A few good measures โ€“ The Terminal/Forward measure โ€“ XXc
ยจ ฮ  ๐‘ก = ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก. โˆ’ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/
ยจ ฮ  ๐‘ก = ๐‘‰(๐‘ก, $๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/)
ยจ Receiving $1 at time ๐‘ก. and paying it back at time ๐‘ก/ is equivalent to:
ยจ Receiving at time ๐‘ก/ the simply compounded forward rate ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ covering the period
[๐‘ก., ๐‘ก/], multiplied by the appropriate daycount fraction. This forward rate sets at time ๐‘ก..
ยจ Simply compounded FORWARD: ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ =
%
%+i $,$0,$@ .j $,$0,$@
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ =
%
%+i.j $,$0,$@
ยจ ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ =
%
PQ $,$0,$@
โˆ’ 1
100
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A few good measures โ€“ The Terminal/Forward measure โ€“ XXd
ยจ ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ =
%
PQ $,$0,$@
โˆ’ 1 before it sets for all time ๐‘ก < ๐‘ก.
ยจ ๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ =
%
PQ $0,$0,$@
โˆ’ 1 when it sets at time ๐‘ก.
ยจ ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ for all time after it did set ๐‘ก โ‰ฅ ๐‘ก.
101
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A few good measures โ€“ The Terminal/Forward measure - XXI
ยจ ฮ  ๐‘ก/ = ๐‘‰(๐‘ก/, $๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/)
ยจ ฮ  ๐‘ก/ = ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. โˆ’ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก/
ยจ The value at time ๐‘ก/ of a portfolio that consists of receiving $1 at time ๐‘ก. and paying it back at
time ๐‘ก/ is the same value at time ๐‘ก/ of a portfolio paying at time ๐‘ก/ the simply compounded rate
(times the daycount fraction), set at time ๐‘ก., and spanning the period [๐‘ก., ๐‘ก/]
ยจ All right, past the halfway point:
ยจ ฮ  ๐‘ก/ = ๐‘‰(๐‘ก/, $๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/)
ยจ We also know that:
ยจ
k $
P($,$,$@)
= ๐”ผ$@
โ„ค($@) k $@
P $@,$@,$@
๐”‰ ๐‘ก = ๐”ผ$@
โ„ค($@)
ฮ  ๐‘ก/ ๐”‰ ๐‘ก
ยจ ฮ  ๐‘ก = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ . 1 + ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ โˆ’ 1 = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ . ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/
102
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A few good measures โ€“ The Terminal/Forward measure - XXII
ยจ Letโ€™s put those together:
ยจ
k $
P($,$,$@)
= ๐”ผ$@
โ„ค($@) k $@
P $@,$@,$@
๐”‰ ๐‘ก = ๐”ผ$@
โ„ค($@)
ฮ  ๐‘ก/ ๐”‰ ๐‘ก
ยจ ฮ  ๐‘ก = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ . 1 + ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ โˆ’ 1 = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ . ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/
ยจ
k $
P($,$,$@)
=
P $,$,$@ .i.j $,$0,$@
P($,$,$@)
= ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/
ยจ ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$@
โ„ค($@)
ฮ  ๐‘ก/ ๐”‰ ๐‘ก
ยจ ฮ  ๐‘ก/ = ๐‘‰(๐‘ก/, $๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/)
ยจ ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$@
โ„ค($@)
๐‘‰(๐‘ก/, $๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก
103
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A few good measures โ€“ The Terminal/Forward measure - XXIII
ยจ ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$@
โ„ค($@)
๐‘‰(๐‘ก/, $๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก
ยจ Letโ€™s drop the daycount fraction
ยจ ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$@
โ„ค($@)
๐‘‰(๐‘ก/, $๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก
ยจ We are almost tempted to say that the simply compounded forward rate is a martingale
under the terminal measure
ยจ Not quite
ยจ What the above says is that the expectation under the ๐‘ก/-terminal measure of a claim that
pays at time ๐‘ก/ the value of the simply compounded forward ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ set at time ๐‘ก. is the
current value of the simply compounded forward spanning the period [๐‘ก., ๐‘ก/]
ยจ We almost there, but before we perform the final step, letโ€™s take a small detour through
swaps valuation
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A few good measures โ€“ The Terminal/Forward measure - XXIV
ยจ ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$@
โ„ค($@)
๐‘‰(๐‘ก/, $๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก
ยจ ๐”ผ$@
โ„ค($@)
๐‘‰(๐‘ก/, $๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก = ๐”ผ$@
โ„ค($@) a($@,$i.j $0,$0,$@ ,$0,$@)
P($@,$@,$@)
๐”‰ ๐‘ก
ยจ ๐”ผ$@
โ„ค($@) a($@,$i.j $0,$0,$@ ,$0,$@)
P($@,$@,$@)
๐”‰ ๐‘ก =
a($,$i.j $0,$0,$@ ,$0,$@)
P($,$,$@)
= ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/
ยจ ๐‘‰ ๐‘ก, $๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ . {๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ }
ยจ This is why we can value swaps without a volatility curve just using the yield curve and
nothing else (remember for REGULAR swaps)
ยจ A swaplet pays at time ๐‘ก/ (end of the period) a Libor rate set at time ๐‘ก. (beginning of the
period) times the appropriate daycount fraction? Boom, the current value of that swaplet is
the current value of that simply compounded forward rate ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ times the daycount
fraction time the discount factor observed in the current discount curve between now (time
๐‘ก) and the payment date ๐‘ก/, that discount factor is ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/
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A few good measures โ€“ The Terminal/Forward measure - XXV
ยจ So this is really what underpins the valuation of all swaps (REGULAR, no funny business
about paying and setting at different dates that the ones wee just talked about!), fixed cash
flows of course and all that.
ยจ Fairly cool right ?
ยจ Almost there about why this is called the forward measure.
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ยจ ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$@
โ„ค($@)
๐‘‰(๐‘ก/, $๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก
ยจ We know take the limit ๐‘ก/ โ†’ ๐‘ก. in order to recover the usual instantaneous forward
ยจ Just for sake of completeness letโ€™s refresh our knowledge from deck V-a
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Another summary - XVII
ยจ From the variable ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ , we are absolutely free to define a bunch of other variables,
and we certainly did not deprive ourselves of doing so:
ยจ Continuously compounded FORWARD : ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ = exp โˆ’๐‘… ๐‘ก, ๐‘ก., ๐‘ก/ . ๐œ ๐‘ก, ๐‘ก., ๐‘ก/
ยจ Simply compounded FORWARD: ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ =
%
%+i $,$0,$@ .j $,$0,$@
ยจ Annually compounded FORWARD : ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ =
%
(%+W $,$0,$@ )
I ?,?0,?@
ยจ ๐‘ž-times per year compounded FOWARD: ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ =
%
(%+
2
J
.WJ $,$0,$@ )
J.I ?,?0,?@
ยจ The function ๐œ ๐‘ก, ๐‘ก., ๐‘ก/ is the daycount fraction, will usually depends on what convention
(ACT/ACT, ACT/360, 30/360,โ€ฆ) you will choose, and potentially adjustment for holidays and
what holiday center
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Another summary - XVIII
ยจ In the small ๐œ ๐‘ก, ๐‘ก., ๐‘ก/ โ†’ 0 limit (also if the rates themselves are such that they are <<1)
ยจ In bootstrap form which is the intuitive way:
ยจ Continuously compounded spot: ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ = 1 โˆ’ ๐‘… ๐‘ก, ๐‘ก., ๐‘ก/ . ๐œ + ๐’ช(๐œ&. ๐‘…&)
ยจ Simply compounded spot: ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ = 1 โˆ’ ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ . ๐œ + ๐’ช(๐œ&. ๐‘™&)
ยจ Annually compounded spot: ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ = 1 โˆ’ ๐‘Œ ๐‘ก, ๐‘ก., ๐‘ก/ . ๐œ + ๐’ช(๐œ&. ๐‘ฆ&)
ยจ ๐‘ž-times per year compounded spot ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ = 1 โˆ’ ๐‘Œm ๐‘ก, ๐‘ก., ๐‘ก/ . ๐œ + ๐’ช(๐œ&. ๐‘ฆm
&)
ยจ So in the limit of small ๐œ ๐‘ก, ๐‘ก., ๐‘ก/ , (and also small rates), in particular when: ๐‘ก/ โ†’ ๐‘ก., all rates
converge to the same limit we call
ยจ ๐ฟ๐‘–๐‘š ๐‘ก/ โ†’ ๐‘ก. = lim
$@โ†’$0
(
%SPQ $,$0,$@
i $,$0,$@
) that we will note Instantaneous Forward Rate
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Another summary - XIX
ยจ ๐ฟ๐‘–๐‘š ๐‘ก/ โ†’ ๐‘ก. = lim
$@โ†’$0
(
%SPQ $,$0,$@
i $,$0,$@
) that we will note Instantaneous Forward Rate
ยจ ๐ผ๐น๐‘ค๐‘… ๐‘ก, ๐‘ก., ๐‘ก/ = ๐ผ๐น๐‘ค๐‘… ๐‘ก, ๐‘ก., ๐‘ก.+ = ๐ผ๐น๐‘ค๐‘… ๐‘ก, ๐‘ก. = lim
$@โ†’$0
(
%SPQ $,$0,$@
i $,$0,$@
)
ยจ In the small ๐œ ๐‘ก, ๐‘ก., ๐‘ก/ limit, (and also small rates) since really what matters is how small the
product of the defined rate by the daycount fraction, ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ is close to 1.
ยจ ๐ฟ๐‘–๐‘š ๐‘ก/ โ†’ ๐‘ก. = lim
$@โ†’$0
(
%SPQ $,$0,$@
i $,$0,$@
) = lim
$@โ†’$0
(
Sno(PQ $,$0,$@ )
i $,$0,$@
)
ยจ Usually most textbooks will assume without explicitly telling you that in that limit we will also
have:
ยจ lim
$@โ†’$0
(๐œ ๐‘ก, ๐‘ก., ๐‘ก/ ) = (๐‘ก/ โˆ’ ๐‘ก.), so that ๐ฟ๐‘–๐‘š ๐‘ก/ โ†’ ๐‘ก. = lim
$@โ†’$0
S no PQ $,$0,$@
i $,$0,$@
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Another summary - XX
ยจ ๐ผ๐น๐‘ค๐‘… ๐‘ก, ๐‘ก., ๐‘ก/ = ๐ผ๐น๐‘ค๐‘… ๐‘ก, ๐‘ก., ๐‘ก.+ = ๐ผ๐น๐‘ค๐‘… ๐‘ก, ๐‘ก. = lim
$@โ†’$0
(
%SPQ $,$0,$@
i $,$0,$@
)
ยจ ๐ผ๐น๐‘ค๐‘… ๐‘ก, ๐‘ก. = lim
$@โ†’$0
(
%SPQ $,$0,$@
i $,$0,$@
) = lim
$@โ†’$0
(๐‘… ๐‘ก, ๐‘ก., ๐‘ก/ ) = ๐‘… ๐‘ก, ๐‘ก., ๐‘ก.
ยจ ๐ผ๐น๐‘ค๐‘… ๐‘ก, ๐‘ก. = lim
$@โ†’$0
(
%SPQ $,$0,$@
i $,$0,$@
) = lim
$@โ†’$0
(๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ ) = ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก.
ยจ ๐ผ๐น๐‘ค๐‘… ๐‘ก, ๐‘ก. = lim
$@โ†’$0
(
%SPQ $,$0,$@
i $,$0,$@
) = lim
$@โ†’$0
(๐‘Œ ๐‘ก, ๐‘ก., ๐‘ก/ ) = ๐‘Œ ๐‘ก, ๐‘ก., ๐‘ก.
ยจ ๐ผ๐น๐‘ค๐‘… ๐‘ก, ๐‘ก. = lim
$@โ†’$0
(
%SPQ $,$0,$@
i $,$0,$@
) = lim
$@โ†’$0
(๐‘Œm ๐‘ก, ๐‘ก., ๐‘ก/ ) = ๐‘Œm ๐‘ก, ๐‘ก., ๐‘ก.
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ยจ ๐‘… ๐‘ก, ๐‘ก., ๐‘ก. = ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก. = ๐‘Œ ๐‘ก, ๐‘ก., ๐‘ก. = ๐‘Œm ๐‘ก, ๐‘ก., ๐‘ก. = ๐‘“(๐‘ก, ๐‘ก.) as per the notation in most
textbooks
ยจ lim
$@โ†’$0
(
%SPQ $,$0,$@
i $,$0,$@
) = lim
$@โ†’$0
(
Sno(PQ $,$0,$@ )
i $,$0,$@
)
ยจ From bootstrap:
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก/ /๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก.
ยจ ln(๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ = ln(๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก/ โˆ’ ln(๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก.
ยจ lim
$@โ†’$0
(
Sno(PQ $,$0,$@ )
i $,$0,$@
) = โˆ’ lim
$@โ†’$0
no(PQ $,$,$@ Sno(PQ $,$,$0
i $,$0,$@
= โˆ’ lim
$@โ†’$0
(
no(PQ $,$,$@ Sno(PQ $,$,$0
$@ S $0
)
ยจ lim
$@โ†’$0
(
Sno(PQ $,$0,$@ )
i $,$0,$@
) = โˆ’
Ino(PQ $,$,$0
I$0
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Another summary - XXII
ยจ ๐‘… ๐‘ก, ๐‘ก., ๐‘ก. = ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก. = ๐‘Œ ๐‘ก, ๐‘ก., ๐‘ก. = ๐‘Œm ๐‘ก, ๐‘ก., ๐‘ก. = ๐‘“ ๐‘ก, ๐‘ก. = โˆ’
Ino(PQ $,$,$0
I$0
ยจ A lot of models loooove to use the Instantaneous Forward Rate (HJM)
ยจ We can also take another limit, the Instantaneous Short Rate defined as:
ยจ ๐ผ๐‘†โ„Ž๐‘… ๐‘ก, ๐‘ก., ๐‘ก/ = ๐ผ๐‘†โ„Ž๐‘… ๐‘ก, ๐‘ก+, ๐‘ก + = ๐ผ๐‘†โ„Ž๐‘… ๐‘ก = lim
$@โ†’$0,$@โ†’$
(
%SPQ $,$0,$@
i $,$0,$@
)
ยจ ๐ผ๐‘†โ„Ž๐‘… ๐‘ก = lim
$0โ†’$
๐ผ๐น๐‘ค๐‘… ๐‘ก, ๐‘ก. = ๐‘… ๐‘ก, ๐‘ก, ๐‘ก = ๐ฟ ๐‘ก, ๐‘ก, ๐‘ก = ๐‘Œ ๐‘ก, ๐‘ก, ๐‘ก = ๐‘Œm ๐‘ก, ๐‘ก, ๐‘ก = ๐‘“ ๐‘ก, ๐‘ก = ๐‘Ÿ(๐‘ก)
ยจ Most of the early models were built on the short rate, and then a lot of models were โ€œaffine
modelsโ€ meaning that there were assumptions of linearity for a lot of the functions.
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ยจ ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$@
โ„ค($@)
๐‘‰(๐‘ก/, $๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก
ยจ lim
$@โ†’$0
๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐‘… ๐‘ก, ๐‘ก., ๐‘ก. sometimes noted (Mercurio) ๐‘… ๐‘ก, ๐‘ก., ๐‘ก. +
ยจ lim
$@โ†’$0
[๐”ผ$@
โ„ค($@)
๐‘‰(๐‘ก/, $๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก ] = ๐”ผ$0
โ„ค($0)
๐‘‰(๐‘ก., $๐ฟ ๐‘ก., ๐‘ก., ๐‘ก. , ๐‘ก., ๐‘ก.) ๐”‰ ๐‘ก
ยจ This is also true, as you can visualize essentially squeezing the period [๐‘ก., ๐‘ก/] to [๐‘ก., ๐‘ก.]
ยจ Payment date ๐‘ก/ goes to ๐‘ก.
ยจ Because ๐‘‰(๐‘ก., $๐ฟ ๐‘ก., ๐‘ก., ๐‘ก. , ๐‘ก., ๐‘ก.) is the value at time ๐‘ก. of a claim that pays , $๐ฟ ๐‘ก., ๐‘ก., ๐‘ก. , set
at time ๐‘ก. and paid at time ๐‘ก., all the dates are the same and thus it is legitimate to write:
ยจ ๐‘‰ ๐‘ก., $๐ฟ ๐‘ก., ๐‘ก., ๐‘ก. , ๐‘ก., ๐‘ก. = ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก. = ๐‘… ๐‘ก., ๐‘ก., ๐‘ก.
ยจ This is reminding us of Physics where variables are observed, defined and โ€œpaidโ€ at the same
time so we do not have to go through that notation.
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A few good measures โ€“ The Terminal/Forward measure - XXVIII
ยจ ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$@
โ„ค($@)
๐‘‰(๐‘ก/, $๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก
ยจ lim
$@โ†’$0
๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐‘… ๐‘ก, ๐‘ก., ๐‘ก.
ยจ lim
$@โ†’$0
[๐”ผ$@
โ„ค($@)
๐‘‰(๐‘ก/, $๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก ] = ๐”ผ$0
โ„ค($0)
๐‘‰(๐‘ก., $๐ฟ ๐‘ก., ๐‘ก., ๐‘ก. , ๐‘ก., ๐‘ก.) ๐”‰ ๐‘ก
ยจ ๐‘… ๐‘ก, ๐‘ก., ๐‘ก. = ๐”ผ$0
โ„ค($0)
๐‘‰(๐‘ก., $๐ฟ ๐‘ก., ๐‘ก., ๐‘ก. , ๐‘ก., ๐‘ก.) ๐”‰ ๐‘ก = ๐”ผ$0
โ„ค($0)
๐‘…(๐‘ก., ๐‘ก., ๐‘ก.) ๐”‰ ๐‘ก
ยจ This is why is it called the forward measure
ยจ Essentially in textbooks you will see it as (for example Mercurio p. 34) โ€œthe expected value of
any future instantaneous spot interest rate, under the corresponding measure, is equal to
the related instantaneous forward rateโ€
ยจ We almost there where we recognize our good old friend the instantaneous forward:
ยจ ๐‘… ๐‘ก, ๐‘ก., ๐‘ก. = โˆ’
Ino(PQ $,$,$0
I$0
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ยจ ๐‘… ๐‘ก, ๐‘ก., ๐‘ก. = โˆ’
Ino(PQ $,$,$0
I$0
ยจ Just to make it simpler
ยจ ๐‘… ๐‘ก, ๐‘ข, ๐‘ข = โˆ’
Ino(PQ $,$,M
IM
ยจ โˆซ
M"$
M"$0
๐‘… ๐‘ก, ๐‘ข, ๐‘ข . ๐‘‘๐‘ข = โˆซ
M"$
M"$0
โˆ’
Ino(PQ $,$,M
IM
. ๐‘‘๐‘ข = [โˆ’ln(๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ข )]M"$
M"$0
ยจ โˆซ
M"$
M"$0
๐‘… ๐‘ก, ๐‘ข, ๐‘ข . ๐‘‘๐‘ข = [โˆ’ln(๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. ) + ln ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก ) = โˆ’ln(๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. )
ยจ โˆ’ ln ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = โˆซ
M"$
M"$0
๐‘… ๐‘ก, ๐‘ข, ๐‘ข . ๐‘‘๐‘ข
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. = exp(โˆ’ โˆซ
M"$
M"$0
๐‘… ๐‘ก, ๐‘ข, ๐‘ข . ๐‘‘๐‘ข)
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ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. = exp(โˆ’ โˆซ
M"$
M"$0
๐‘… ๐‘ก, ๐‘ข, ๐‘ข . ๐‘‘๐‘ข)
ยจ In particular for ๐‘ก = 0
ยจ ๐‘๐ถ 0,0, ๐‘ก. = ๐‘ 0,0, ๐‘ก. = exp(โˆ’ โˆซ
M"#
M"$0
๐‘… 0, ๐‘ข, ๐‘ข . ๐‘‘๐‘ข)
ยจ Almost there
ยจ ๐‘… ๐‘ก, ๐‘ก., ๐‘ก. = ๐”ผ$0
โ„ค($0)
๐‘‰(๐‘ก., $๐ฟ ๐‘ก., ๐‘ก., ๐‘ก. , ๐‘ก., ๐‘ก.) ๐”‰ ๐‘ก = ๐”ผ$0
โ„ค($0)
๐‘…(๐‘ก., ๐‘ก., ๐‘ก.) ๐”‰ ๐‘ก
ยจ ๐‘… ๐‘ก, ๐‘ข, ๐‘ข = ๐”ผM
โ„ค(M)
๐‘…(๐‘ข, ๐‘ข, ๐‘ข) ๐”‰ ๐‘ก so ๐‘… 0, ๐‘ข, ๐‘ข = ๐”ผM
โ„ค(M)
๐‘…(๐‘ข, ๐‘ข, ๐‘ข) ๐”‰ 0
ยจ ๐‘ 0,0, ๐‘ก. = exp(โˆ’ โˆซ
M"#
M"$0
๐”ผM
โ„ค(M)
๐‘…(๐‘ข, ๐‘ข, ๐‘ข) ๐”‰ 0 . ๐‘‘๐‘ข)
ยจ And remember we had in the Risk free measure
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก/ = ๐”ผ$@
โ„š
๐‘’S โˆซ
<=?
<=?@
O !,!,! .D!
|๐”‰(๐‘ก)
117
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A few good measures โ€“ The Terminal/Forward measure - XXXI
ยจ ๐‘ 0,0, ๐‘ก. = exp(โˆ’ โˆซ
M"#
M"$0
๐”ผM
โ„ค(M)
๐‘…(๐‘ข, ๐‘ข, ๐‘ข) ๐”‰ 0 . ๐‘‘๐‘ข)
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก/ = ๐”ผ$@
โ„š
exp[โˆ’ โˆซ
!"$
!"$@
๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ]|๐”‰(๐‘ก)
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = ๐”ผ$0
โ„š
exp[โˆ’ โˆซ
!"$
!"$0
๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ]|๐”‰(๐‘ก)
ยจ ๐‘ 0,0, ๐‘ก. = exp(โˆ’ โˆซ
M"#
M"$0
๐”ผM
โ„ค(M)
๐‘…(๐‘ข, ๐‘ข, ๐‘ข) ๐”‰ 0 . ๐‘‘๐‘ข)
ยจ ๐‘ 0,0, ๐‘ก = exp(โˆ’ โˆซ
M"#
M"$
๐”ผM
โ„ค(M)
๐‘…(๐‘ข, ๐‘ข, ๐‘ข) ๐”‰ 0 . ๐‘‘๐‘ข)
ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = ๐”ผ$0
โ„š
exp[โˆ’ โˆซ
!"$
!"$0
๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ]|๐”‰(๐‘ก)
ยจ ๐‘๐ถ 0,0, ๐‘ก = ๐”ผ$
โ„š
exp[โˆ’ โˆซ
!"#
!"$
๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ]|๐”‰(0)
118
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A few good measures โ€“ The Terminal/Forward measure - XXXII
ยจ Those are the famous relations that look almost the same but are quite different:
ยจ ๐‘ 0,0, ๐‘ก = exp(โˆ’ โˆซ
!"#
!"$
๐”ผ!
โ„ค(!)
๐‘…(๐‘ , ๐‘ , ๐‘ ) ๐”‰ 0 . ๐‘‘๐‘ )
ยจ ๐‘๐ถ 0,0, ๐‘ก = ๐”ผ$
โ„š
exp[โˆ’ โˆซ
!"#
!"$
๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ]|๐”‰(0)
ยจ To quote Rebonato (p. 33): โ€œconsiderable grief has come to erstwhile happy individuals and
their families by confusing the two equations aboveโ€
119
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A few good measures โ€“ The Terminal/Forward measure - XXXIII
ยจ ๐‘ 0,0, ๐‘ก = exp(โˆ’ โˆซ
!"#
!"$
๐”ผ!
โ„ค(!)
๐‘…(๐‘ , ๐‘ , ๐‘ ) ๐”‰ 0 . ๐‘‘๐‘ )
ยจ ๐‘๐ถ 0,0, ๐‘ก = ๐”ผ$
โ„š
exp[โˆ’ โˆซ
!"#
!"$
๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ]|๐”‰(0)
ยจ So you are in good company if by now you are confused about the equations above as it
looks like we switched the Expectation, the integral and the exponential in just a different
order.
ยจ Remember that those are NOT the same measures
ยจ In fact in the first equation, we integrate over an infinite number of different measures
ยจ Also just to make it clear remember to write the first one as:
ยจ ๐‘ 0,0, ๐‘ก = exp(โˆ’ โˆซ
!"#
!"$
๐”ผ!
โ„ค(!)
๐‘‰(๐‘ , $๐‘… ๐‘ , ๐‘ , ๐‘  , ๐‘ , ๐‘ ) ๐”‰ 0 . ๐‘‘๐‘ )
120
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A few good measures โ€“ The Terminal/Forward measure - XXXIV
ยจ So juts one more time (hopefully the last time)
ยจ ๐‘ 0,0, ๐‘ก = exp(โˆ’ โˆซ
!"#
!"$
๐”ผ!
โ„ค(!)
๐‘…(๐‘ , ๐‘ , ๐‘ ) ๐”‰ 0 . ๐‘‘๐‘ )
ยจ ๐‘๐ถ 0,0, ๐‘ก = ๐”ผ$
โ„š
exp[โˆ’ โˆซ
!"#
!"$
๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ]|๐”‰(0)
121
Luc_Faucheux_2021
A few good measures โ€“ The early/discount measure
ยจ This is the measure where the numeraire is the Zero ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก.
ยจ Also referred to in textbooks as the ๐‘ก.-Terminal or sometimes ๐‘‡.-terminal
ยจ ๐”ผ$0
โ„ค($0)
ยจ It is usually when the claims get SET at the beginning of the period (the forward one is when
the claim gets PAID at the end of the period)
ยจ We sort of did it as we went over the terminal measure but worth noting that number of
relations
ยจ So letโ€™s redo a couple of the early slides with the early (or ๐‘ก.-Terminal) measure in mind
122
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A few good measures โ€“ The early/discount measure - II
ยจ
a $,$%,$0,$@
P($,$,$@)
= ๐”ผ$@
โ„ค($@) a $@,$%,$0,$@
P $@,$@,$@
๐”‰ ๐‘ก so: ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก, ๐‘ก/)
ยจ
a $,$%,$0,$0
P($,$,$0)
= ๐”ผ$0
โ„ค($0) a $0,$%,$0,$0
P $0,$0,$0
๐”‰ ๐‘ก so: ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก. = ๐‘(๐‘ก, ๐‘ก, ๐‘ก.)
ยจ Note that the โ€œsettingโ€ or โ€œfixingโ€ time for a claim that always pays $1 is irrelevant, but letโ€™s
keep it for now, we did not go through a couple of thousand slides of building a rigorous
formalism to throw it all away now.
ยจ
a $,$%,$0,$@
P($,$,$0)
= ๐”ผ$0
โ„ค($0) a $0,$%,$0,$@
P $0,$0,$0
๐”‰ ๐‘ก
ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐”ผ$0
โ„ค($0) a $0,$%,$0,$@
P $0,$0,$0
๐”‰ ๐‘ก = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก
ยจ To be super specific, we should really if we want write:
ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘‰ ๐‘ก, $๐‘(๐‘ก, ๐‘ก, ๐‘ก/), ๐‘ก, ๐‘ก = $๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก, ๐‘ก/)
123
Luc_Faucheux_2021
A few good measures โ€“ The early/discount measure - III
ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก
ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก, ๐‘ก/)
ยจ From the bootstrap relations we have by definition:
ยจ ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐‘(๐‘ก, ๐‘ก., ๐‘ก/)
ยจ So we have:
ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก
ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐‘(๐‘ก, ๐‘ก., ๐‘ก/)
ยจ Which leads us quite naturally to:
ยจ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก
124
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A few good measures โ€“ The early/discount measure - IV
ยจ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก
ยจ That is where the nesting comes into the game:
ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก, ๐‘ก/)
ยจ And so by just changing the variables:
ยจ ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก., ๐‘ก., ๐‘ก/)
ยจ Again when you read it aloud it makes sense: The value at time ๐‘ก. of a claim that will pay $1
no matter what at time ๐‘ก/ is equal to the value of the Zero Coupon Bond at time ๐‘ก. that pays
$1 at time ๐‘ก/.
ยจ That is almost a tautology, but we can nest that one into the first equation:
ยจ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = ๐”ผ$0
โ„ค($0)
๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก
125
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A few good measures โ€“ The early/discount measure - V
ยจ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = ๐”ผ$0
โ„ค($0)
๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก
ยจ I know that we are mixing a little the Terminal/Forward measure ๐”ผ$@
โ„ค($@)
and the
early/discount measure ๐”ผ$0
โ„ค($0)
but they are exactly the same just with a different end time
for the expectation and the numeraire, but the above relationship is quite cool.
ยจ In the early/discount measure ๐”ผ$0
โ„ค($0)
, the Zeros are a martingale. And by Zeros we mean the
๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ , so not every Zeros, careful about that.
ยจ So the process of those guys will be driftless:
ยจ ๐‘‘๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = 0. ๐‘‘๐‘ก + ๐‘ ๐‘œ๐‘š๐‘’๐‘กโ„Ž๐‘–๐‘›๐‘” . ๐‘‘๐‘Š
$0
โ„ค $0
(๐‘ก)
ยจ Where ๐‘Š
$0
โ„ค $0
(๐‘ก) is the Brownian motion associated to the early measure ๐”ผ$0
โ„ค($0)
126
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A few good measures โ€“ The early/discount measure - VI
ยจ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = ๐”ผ$0
โ„ค($0)
๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก
ยจ Simply compounded FORWARD at time ๐‘ก: ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ =
%
%+i $,$0,$@ .j $,$0,$@
ยจ Simply compounded FORWARD at time ๐‘ก. : ๐‘๐ถ ๐‘ก., ๐‘ก., ๐‘ก/ =
%
%+i $0,$0,$@ .j $0,$0,$@
ยจ With for sake of simplicity: ๐œ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐œ
ยจ ๐”ผ$0
โ„ค($0) %
%+i.j $0,$0,$@
๐”‰ ๐‘ก =
%
%+i.j $,$0,$@
ยจ ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $
%
%+i.j $,$0,$@
, ๐‘ก., ๐‘ก. ๐”‰ ๐‘ก =
%
%+i.j $,$0,$@
127
Luc_Faucheux_2021
A few good measures โ€“ The early/discount measure - VII
ยจ ๐”ผ$0
โ„ค($0) %
%+i.j $0,$0,$@
๐”‰ ๐‘ก =
%
%+i.j $,$0,$@
ยจ We use the magic equation: ๐‘‹ โˆ’ ๐‘‹ = 0 which is true whatever the value of ๐‘‹
ยจ
%
%+i.j $0,$0,$@
=
%+i.j $0,$0,$@ Si.j $0,$0,$@
%+i.j $0,$0,$@
= 1 โˆ’
i.j $0,$0,$@
%+i.j $0,$0,$@
ยจ Same on the right hand side at time ๐‘ก
ยจ
%
%+i.j $,$0,$@
=
%+i.j $,$0,$@ Si.j $,$0,$@
%+i.j $,$0,$@
= 1 โˆ’
i.j $,$0,$@
%+i.j $,$0,$@
ยจ ๐”ผ$0
โ„ค($0)
1 โˆ’
i.j $0,$0,$@
%+i.j $0,$0,$@
๐”‰ ๐‘ก = 1 โˆ’ ๐”ผ$0
โ„ค $0 i.j $0,$0,$@
%+i.j $0,$0,$@
๐”‰ ๐‘ก = 1 โˆ’
i.j $,$0,$@
%+i.j $,$0,$@
ยจ ๐”ผ$0
โ„ค $0 i.j $0,$0,$@
%+i.j $0,$0,$@
๐”‰ ๐‘ก =
i.j $,$0,$@
%+i.j $,$0,$@
128
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A few good measures โ€“ The early/discount measure - VIII
ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ = ๐‘‰ ๐‘ก, $๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ , ๐‘ก, ๐‘ก
ยจ The value of a claim at time ๐‘ก that pays $1 at time ๐‘ก/ is ๐‘(๐‘ก, ๐‘ก, ๐‘ก/)
ยจ If you pay $๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ at time ๐‘ก, you will receive $1 at time ๐‘ก/
ยจ If you pay $1 = $๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ . {
%
P $,$,$@
} at time ๐‘ก, you will receive $1.
%
P $,$,$@
at time ๐‘ก/
129
๐‘ก"
๐‘ก!
๐‘ก
๐‘ก๐‘–๐‘š๐‘’
$1
{
1
๐‘(๐‘ก, ๐‘ก, ๐‘ก/)
}
{
1
๐‘(๐‘ก, ๐‘ก, ๐‘ก.)
} {? }
Luc_Faucheux_2021
A few good measures โ€“ The early/discount measure - IX
ยจ If you pay $1 = $๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ . {
%
P $,$,$@
} at time ๐‘ก, you will receive $1.
%
P $,$,$@
at time ๐‘ก/
ยจ If you pay $1 at time ๐‘ก, you will receive $
%
P $,$,$@
at time ๐‘ก/
ยจ If you pay $1 at time ๐‘ก, you will receive $
%
P $,$,$0
at time ๐‘ก.
130
๐‘ก๐‘–๐‘š๐‘’
$1
{
1
๐‘(๐‘ก, ๐‘ก, ๐‘ก/)
}
{
1
๐‘(๐‘ก, ๐‘ก, ๐‘ก.)
} {? }
๐‘ก"
๐‘ก!
๐‘ก
Luc_Faucheux_2021
A few good measures โ€“ The early/discount measure - X
ยจ If you pay $1 at time ๐‘ก, you will receive $
%
P $,$,$@
at time ๐‘ก/
ยจ If you pay $1 at time ๐‘ก, you will receive $
%
P $,$,$0
at time ๐‘ก.
ยจ From the bootstrap definition: ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐‘(๐‘ก, ๐‘ก., ๐‘ก/)
131
๐‘ก๐‘–๐‘š๐‘’
$1
{
1
๐‘(๐‘ก, ๐‘ก, ๐‘ก/)
}
{
1
๐‘(๐‘ก, ๐‘ก, ๐‘ก.)
} {? }
๐‘ก"
๐‘ก!
๐‘ก
Luc_Faucheux_2021
A few good measures โ€“ The early/discount measure - XI
ยจ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = ๐”ผ$0
โ„ค($0)
๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก
ยจ If you pay $1 at time ๐‘ก, you will receive $
%
P $,$,$0
at time ๐‘ก.
ยจ If you then re-invest that amount at time ๐‘ก. until time ๐‘ก/, what would you expect to receive
then?
132
๐‘ก๐‘–๐‘š๐‘’
$1
{
1
๐‘(๐‘ก, ๐‘ก, ๐‘ก/)
}
{
1
๐‘(๐‘ก, ๐‘ก, ๐‘ก.)
} {? }
๐‘ก"
๐‘ก!
๐‘ก
Luc_Faucheux_2021
ยจ
a $,$%,$0,$@
P($,$,$@)
= ๐”ผ$@
โ„ค($@) a $@,$%,$0,$@
P $@,$@,$@
๐”‰ ๐‘ก = ๐”ผ$@
โ„ค($@) %
%
๐”‰ ๐‘ก = ๐”ผ$@
โ„ค($@)
1 ๐”‰ ๐‘ก = 1
ยจ so: ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก, ๐‘ก/)
ยจ
a $,$%,$0,$@
P($,$,$@)
= ๐”ผ$@
โ„ค($@) a $@,$%,$0,$@
P $@,$@,$@
๐”‰ ๐‘ก so: ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก, ๐‘ก/)
133
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A few good measures โ€“ The early/discount measure - XI
134
๐‘ก๐‘–๐‘š๐‘’
$1
{
1
๐‘(๐‘ก, ๐‘ก, ๐‘ก/)
}
{
1
๐‘(๐‘ก, ๐‘ก, ๐‘ก.)
}
{? }
๐‘ก"
๐‘ก!
๐‘ก
Luc_Faucheux_2021
ยจ
a $,$%,$0,$0
P($,$,$0)
= ๐”ผ$0
โ„ค($0) a $0,$%,$0,$0
P $0,$0,$0
๐”‰ ๐‘ก = ๐”ผ$0
โ„ค($0) %
%
๐”‰ ๐‘ก = ๐”ผ$0
โ„ค($0)
1 ๐”‰ ๐‘ก = 1
ยจ so: ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก. = ๐‘(๐‘ก, ๐‘ก, ๐‘ก.)
ยจ So: ๐‘‰ ๐‘ก, $
%
P($,$,$0)
, ๐‘ก., ๐‘ก. =
%
P $,$,$0
. ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก. =
%
P $,$,$0
. ๐‘(๐‘ก, ๐‘ก, ๐‘ก.) = $1
A few good measures โ€“ The early/discount measure - XI
135
๐‘ก๐‘–๐‘š๐‘’
$1
{
1
๐‘(๐‘ก, ๐‘ก, ๐‘ก.)
}
๐‘ก"
๐‘ก!
๐‘ก
Luc_Faucheux_2021
ยจ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = ๐”ผ$0
โ„ค($0)
๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก
A few good measures โ€“ The early/discount measure - XI
136
๐‘ก๐‘–๐‘š๐‘’
๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ ๐”ผ$0
โ„ค($0)
๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก
๐‘ก"
๐‘ก!
๐‘ก
Luc_Faucheux_2021
A few good measures โ€“ The early/discount measure - XII
ยจ First of all, it is always useful from time to time to go back to the โ€œZero volatility worldโ€ or
deterministic, or โ€œyield curve worldโ€ where nothing is assumed to be stochastic, and all
functions are deterministic
ยจ This is nice because then we do not have to worry about ITO and STRATO and all the
strangeness and alienness of stochastic calculus
ยจ It is also nice because it is a nice check of our understanding and intuition
ยจ The really cool thing about 0 volatility is that there is no convexity adjustment
ยจ ๐”ผ
%
V
=
%
๐”ผ p
ยจ So you cannot really mess up anything thereโ€ฆ.
137
Luc_Faucheux_2021
ยจ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = ๐”ผ$0
โ„ค($0)
๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก
ยจ At Zero volatility, ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = ๐‘‰# ๐‘ก., $1, ๐‘ก., ๐‘ก/ = ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/
A few good measures โ€“ The early/discount measure - XIII
138
๐‘ก๐‘–๐‘š๐‘’
$1
{
1
๐‘ ๐‘ก, ๐‘ก, ๐‘ก.
}
๐‘ก"
๐‘ก!
๐‘ก
Luc_Faucheux_2021
ยจ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = ๐”ผ$0
โ„ค($0)
๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก
ยจ At Zero volatility,๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = ๐‘# ๐‘ก., ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก., ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก., ๐‘ก/)
A few good measures โ€“ The early/discount measure - XIV
139
๐‘ก๐‘–๐‘š๐‘’
๐‘ ๐‘ก, ๐‘ก., ๐‘ก/
๐”ผ$0
โ„ค($0)
๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก =
๐‘ ๐‘ก., ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก., ๐‘ก/)
๐‘ก"
๐‘ก!
๐‘ก
Luc_Faucheux_2021
ยจ At Zero volatility ? =
%
P $,$0,$@
=
%
P $0,$0,$@
ยจ At Zero volatility, ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = ๐‘# ๐‘ก., ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก., ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก., ๐‘ก/)
ยจ At Zero volatility, ๐”ผ$0
โ„ค($0) %
a $0,$%,$0,$@
๐”‰ ๐‘ก =
%
P> $0,$0,$@
=
%
P $0,$0,$@
=
%
P $,$0,$@
A few good measures โ€“ The early/discount measure - XV
140
๐‘ก๐‘–๐‘š๐‘’
$1
{
1
๐‘(๐‘ก, ๐‘ก, ๐‘ก/)
}
{
1
๐‘(๐‘ก, ๐‘ก, ๐‘ก.)
}
๐‘ก"
๐‘ก!
๐‘ก
{
1
๐‘(๐‘ก, ๐‘ก., ๐‘ก/)
}
Luc_Faucheux_2021
A few good measures โ€“ The early/discount measure - XVI
ยจ For a non-zero volatility (outside the pure yield curve world), we cannot have a unique value
for {?}. There is a distribution to the possible middle points at time ๐‘ก. as we explained in
section III
ยจ So the graph below is misleading, we really have a distribution at time ๐‘ก. and also at time ๐‘ก/
141
๐‘ก๐‘–๐‘š๐‘’
$1
{
1
๐‘(๐‘ก, ๐‘ก, ๐‘ก/)
}
{
1
๐‘(๐‘ก, ๐‘ก, ๐‘ก.)
} {? }
๐‘ก"
๐‘ก!
๐‘ก
Luc_Faucheux_2021
A few good measures โ€“ The early/discount measure -
Deferred premium
ยจ Suppose that we have a general claim , $๐ป ๐‘ก that only depends on thins that happen before
the time it finally โ€œsetsโ€ at time ๐‘ก.
ยจ I wish that there was a way to express the sentence above.
ยจ Oh wait there is actually, the filtration ๐”‰(๐‘ก.)
ยจ So we can write that claim as $๐ป ๐‘ก = $๐ป ๐”‰(๐‘ก.)
ยจ Suppose now that this claims is paid at a time ๐‘ก/ > ๐‘ก.
ยจ The deferred premium formula (part III) can be expressed as:
ยจ ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $๐ป ๐”‰(๐‘ก.) , ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก) = ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $๐ป ๐”‰(๐‘ก.) . ๐‘๐ถ(๐‘ก., ๐‘ก., ๐‘ก/), ๐‘ก., ๐‘ก. |๐”‰(๐‘ก)
ยจ ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $๐ป ๐”‰(๐‘ก.) , ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก) = ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $๐ป ๐”‰(๐‘ก.) . ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก. |๐”‰(๐‘ก)
ยจ Note the nesting in the function ๐‘‰
142
Luc_Faucheux_2021
A few good measures โ€“ The early/discount measure -
Deferred premium - II
ยจ We have reduced the calculation of the claim to an expectation at time ๐‘ก. under the early
(discount) measure.
ยจ This is super useful when building trees and discounting back on the tree (in order to price
callable for example), that way when discounting back you pick up the value of the claim at
the time ๐‘ก., as opposed to picking it up at time ๐‘ก/ (where you would not necessiraly know
how to spread it on the tree nodes because you would have to forward propagate it from ๐‘ก.
to ๐‘ก/in the first place, which is the thing that you are trying to avoid)
ยจ This is why in 99% of the tree valuation models out there, the measure being used is the
โ€œearly/discountโ€, which maybe we should just rename the โ€œtree measureโ€
143
Luc_Faucheux_2021
A few good measures โ€“ The early/discount measure -
Deferred premium - III
144
๐‘ก"
๐‘ก!
๐‘ก๐‘–๐‘š๐‘’
๐‘ก
Luc_Faucheux_2021
A few good measures โ€“ The early/discount measure -
Deferred premium - IV
145
๐‘ก"
๐‘ก!
๐‘ก๐‘–๐‘š๐‘’
๐ป(๐”‰ ๐‘ก) ) sets at time ๐‘ก) and depends only on ๐”‰ ๐‘ก) , what happens before ๐‘ก)
๐‘‰ ๐‘ก., $๐ป(๐”‰ ๐‘ก. ), ๐‘ก., ๐‘ก/
Luc_Faucheux_2021
A few good measures โ€“ The early/discount measure -
Deferred premium - V
ยจ ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $๐ป ๐”‰(๐‘ก.) , ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก) = ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $๐ป ๐”‰(๐‘ก.) . ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก. |๐”‰(๐‘ก)
ยจ In particular if we set : $๐ป ๐”‰(๐‘ก.) = $1
ยจ ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก) = ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1. ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก. |๐”‰(๐‘ก)
ยจ ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก) = ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก. |๐”‰(๐‘ก)
ยจ ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก) = ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $๐‘ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก. |๐”‰(๐‘ก)
ยจ ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก) = ๐”ผ$0
โ„ค($0)
๐‘ ๐‘ก., ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก)
ยจ ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก) = ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/
ยจ
%
P($,$,$0)
. ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก) =
%
P($,$,$0)
. ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ =
P $,$0,$@
P($,$,$0)
146
Luc_Faucheux_2021
A few good measures โ€“ The early/discount measure -
Deferred premium - VI
ยจ
%
P($,$,$0)
. ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก) =
%
P($,$,$0)
. ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ =
P $,$0,$@
P($,$,$0)
ยจ ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $
%
P($,$,$0)
, ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก) =
%
P($,$,$0)
. ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ =
P $,$0,$@
P($,$,$0)
ยจ Which is equivalent to setting: $๐ป ๐”‰(๐‘ก.) = $
%
P($,$,$0)
147
Luc_Faucheux_2021
A few good measures โ€“ The early/discount measure -
Deferred premium - VII
ยจ
a $,$%,$0,$@
P($,$,$@)
= ๐”ผ$@
โ„ค($@) a $@,$%,$0,$@
P $@,$@,$@
๐”‰ ๐‘ก = ๐”ผ$@
โ„ค($@) %
%
๐”‰ ๐‘ก = ๐”ผ$@
โ„ค($@)
1 ๐”‰ ๐‘ก = 1
ยจ so: ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก, ๐‘ก/)
ยจ So: ๐‘‰ ๐‘ก, $
%
P($,$,$@)
, ๐‘ก., ๐‘ก/ =
%
P $,$,$@
. ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ =
%
P $,$,$@
. ๐‘(๐‘ก, ๐‘ก, ๐‘ก/) = $1
ยจ If you invest $1 today until time ๐‘ก/, on average you expect to get back
%
P $,$,$@
ยจ
a $,$%,$0,$@
P($,$,$0)
= ๐”ผ$0
โ„ค($0) a $0,$%,$0,$@
P $0,$0,$0
๐”‰ ๐‘ก = ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก)
ยจ
a $,$%,$0,$@
P($,$,$0)
= ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1. ๐‘(๐‘ก., ๐‘ก., ๐‘ก/), ๐‘ก., ๐‘ก. |๐”‰(๐‘ก) = ๐”ผ$0
โ„ค($0)
๐‘(๐‘ก., ๐‘ก., ๐‘ก/)|๐”‰(๐‘ก) = ๐‘(๐‘ก, ๐‘ก., ๐‘ก/)
ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/
148
Luc_Faucheux_2021
A few good measures โ€“ The early/discount measure -
Deferred premium - VIII
ยจ
a $,$%,$0,$@
P($,$,$0)
= ๐”ผ$0
โ„ค($0) a $0,$%,$0,$@
P $0,$0,$0
๐”‰ ๐‘ก = ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก)
ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐”ผ$0
โ„ค($0)
๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก
ยจ ๐‘‰ ๐‘ก, $
%
P $,$,$0
, ๐‘ก., ๐‘ก/ = ๐”ผ$0
โ„ค($0)
๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก = ๐‘(๐‘ก, ๐‘ก., ๐‘ก/)
ยจ ๐‘‰ ๐‘ก, $
%
P $,$,$0
, ๐‘ก., ๐‘ก. = $1
ยจ If you invest $1 today until time ๐‘ก., on average you will expect
%
P $,$,$0
ยจ If you invest $1 today until time ๐‘ก/, on average you expect to get back
%
P $,$,$@
149
Luc_Faucheux_2021
A few good measures โ€“ The early/discount measure -
Deferred premium - IX
ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก)
ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $๐‘(๐‘ก., ๐‘ก., ๐‘ก/), ๐‘ก., ๐‘ก. ๐”‰ ๐‘ก
ยจ ๐‘‰ ๐‘ก, $
%
P $,$,$@
, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $
%
P $,$,$@
. ๐‘(๐‘ก., ๐‘ก., ๐‘ก/), ๐‘ก., ๐‘ก. ๐”‰ ๐‘ก
ยจ ๐‘‰ ๐‘ก, $
%
P $,$,$@
, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. .
%
P $,$,$@
. ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1. ๐‘(๐‘ก., ๐‘ก., ๐‘ก/), ๐‘ก., ๐‘ก. ๐”‰ ๐‘ก
ยจ ๐‘‰ ๐‘ก, $
%
P $,$,$@
, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. .
%
P $,$,$@
. ๐”ผ$0
โ„ค($0)
๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก
ยจ ๐‘‰ ๐‘ก, $
%
P $,$,$@
, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. .
%
P $,$,$@
. ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. .
%
P $,$,$0 โˆ—P $,$0,$@
. ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = $1
ยจ If you invest $1 at time ๐‘ก, invest it until time ๐‘ก. THEN re-invest back util time ๐‘ก/, you will STILL expect
to receive on average
%
P $,$,$@
150
Luc_Faucheux_2021
A few good measures โ€“ The early/discount measure -
Deferred premium - X
ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก, ๐‘ก/)
ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก. = ๐‘(๐‘ก, ๐‘ก, ๐‘ก.)
ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $๐‘(๐‘ก., ๐‘ก., ๐‘ก/), ๐‘ก., ๐‘ก. ๐”‰ ๐‘ก
ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐”ผ$0
โ„ค($0)
๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก
ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก. . ๐”ผ$0
โ„ค($0)
๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก
ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก. . ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก
ยจ ๐”ผ$0
โ„ค($0)
๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก =
a $,$%,$0,$@
a $,$%,$0,$0
151
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Lf 2021 rates_vi

  • 1. Luc_Faucheux_2021 THE RATES WORLD โ€“ Part VI Cleaning up a bunch of loose ends 1
  • 2. Luc_Faucheux_2021 That deck 2 ยจ After a bunch of decks, we take here a breather to revisit some of the assumptions/results, and finish up a number of sections that we had left unfinished ยจ Something to say about the notation / progression of those decks. ยจ I tried very hard to do it in a progressive manner, and so the formalism and notations became more complicated but also more complete as we went on. ยจ So in many ways the โ€simpleโ€ notation that I used at the beginning were potentially confusing. Many apologies for that, but that was intended in order to demonstrate as we go along the need for more complicated notation, as opposed to just dump it at the beginning in a very formal manner ยจ Hopefully you will have found the journey interesting and enlightning, and maybe more alive than a formal class, which again this is not. This is merely a bunch of notes that I put down in a Powerpoint in a selfish purpose so that I can more easily find them and retrieve them, and hopefully this helps you reading and understanding real serious and formal textbooks on the subject.
  • 3. Luc_Faucheux_2021 Letโ€™s play a game. Letโ€™s see if you can spot the mistakes in the next section (*) Gilles Franchini found them under 2 minutes 3
  • 4. Luc_Faucheux_2021 Rules of the game ยจ The result is correct ยจ The derivation is wrong ยจ There are a bunch of mistakes ยจ I will highlight and explain the mistakes in the next deck ยจ There is a prize for the first one to point out all the mistakes ยจ Gilles Franchini is disqualified from this contest as he found them under 2 minutes 4
  • 5. Luc_Faucheux_2021 Redoing the useful relationship with the Isometry property of the ITO integral 5
  • 6. Luc_Faucheux_2021 Useful relationship through isometry ยจ ๐”ผ exp โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = exp[โˆซ !"# !"$ % & ๐‘“ ๐‘  &. ๐‘‘๐‘ ] ยจ We know that this is true, we are trying to re-derive it in another manner, by using the regular Taylor expansion: exp(๐‘‹) = โˆ‘'"# '"( % '! . ๐‘‹' ยจ ๐”ผ exp โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = ๐”ผ{โˆ‘'"# '"( % '! . โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  ' } ยจ ๐”ผ exp โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = โˆ‘'"# '"( % '! . ๐”ผ{ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  ' } ยจ We surmise that because of the Isometry property of the ITO integral, all the odd terms in ๐‘˜ equal 0 and all the even terms in ๐‘˜ = 2๐‘ are equal to: ยจ ๐”ผ{ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  &*+% } = 0 ยจ ๐”ผ{ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  &* } = โˆซ !"# !"$ ๐‘“& ๐‘  . ๐‘‘๐‘  * 6
  • 7. Luc_Faucheux_2021 Useful relationship through isometry - II ยจ To convince ourselves of this, it pays to expand the integral as the usual limit of a sum. ยจ Remember, since we are using ITO calculus, we are using the LHS (Left Hand Side) for the value of the function in the interval/mesh/bucketing for the function to be evaluated ยจ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = โˆซ !"# !"$ ๐‘“ ๐‘  . ([). ๐‘‘๐‘Š ๐‘  ยจ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = lim ,โ†’( โˆ‘."# .", ๐‘“ ๐‘ . . {๐‘Š ๐‘ .+% โˆ’ ๐‘Š(๐‘ .)} ยจ ๐”ผ{โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  } = ๐”ผ{lim ,โ†’( โˆ‘."# .", ๐‘“ ๐‘ . . {๐‘Š ๐‘ .+% โˆ’ ๐‘Š(๐‘ .)} } ยจ ๐”ผ{โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  } = lim ,โ†’( โˆ‘."# .", ๐‘“ ๐‘ . . ๐”ผ{๐‘Š ๐‘ .+% โˆ’ ๐‘Š(๐‘ .)} ยจ ๐”ผ{โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  } = lim ,โ†’( โˆ‘."# .", ๐‘“ ๐‘ . . 0 = 0 7
  • 8. Luc_Faucheux_2021 Useful relationship through isometry - III ยจ [โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  ]&= [โˆซ !"# !"$ ๐‘“ ๐‘  . ([). ๐‘‘๐‘Š ๐‘  ]& ยจ [โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  ]&= [lim ,โ†’( โˆ‘."# .", ๐‘“ ๐‘ . . {๐‘Š ๐‘ .+% โˆ’ ๐‘Š(๐‘ .)} ]& ยจ [โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  ]% = lim &โ†’( โˆ‘)"# )"& ๐‘“ ๐‘ ) . ๐‘Š ๐‘ )*+ โˆ’ ๐‘Š ๐‘ ) . [lim ,โ†’( โˆ‘-"# -", ๐‘“ ๐‘ - . {๐‘Š ๐‘ -*+ โˆ’ ๐‘Š(๐‘ -)} ] ยจ [โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  ]% = lim ,โ†’( โˆ‘-"# -", lim &โ†’( โˆ‘)"# )"& ๐‘“ ๐‘ ) . ๐‘“ ๐‘ - . ๐‘Š ๐‘ )*+ โˆ’ ๐‘Š ๐‘ ) . {๐‘Š ๐‘ -*+ โˆ’ ๐‘Š(๐‘ -)} ยจ ๐”ผ ๐‘Š ๐‘ .+% โˆ’ ๐‘Š ๐‘ . = 0 ยจ ๐”ผ ๐‘Š ๐‘ .+% โˆ’ ๐‘Š ๐‘ . . [๐‘Š ๐‘ /+% โˆ’ ๐‘Š(๐‘ /)] = ๐›ฟ.,/. [๐‘ /+% โˆ’ ๐‘ /] ยจ Where we are using the usual Kronecker notation: ยจ ๐›ฟ.,/ = 1 if ๐‘– = ๐‘— ยจ ๐›ฟ.,/ = 0 if ๐‘– โ‰  ๐‘— 8
  • 9. Luc_Faucheux_2021 Useful relationship through isometry - IV ยจ ๐”ผ ๐‘Š ๐‘ .+% โˆ’ ๐‘Š ๐‘ . . [๐‘Š ๐‘ /+% โˆ’ ๐‘Š(๐‘ /)] = ๐›ฟ.,/. [๐‘ /+% โˆ’ ๐‘ /] ยจ [โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  ]% = lim ,โ†’( โˆ‘-"# -", lim &โ†’( โˆ‘)"# )"& ๐‘“ ๐‘ ) . ๐‘“ ๐‘ - . ๐‘Š ๐‘ )*+ โˆ’ ๐‘Š ๐‘ ) . {๐‘Š ๐‘ -*+ โˆ’ ๐‘Š(๐‘ -)} ยจ ๐”ผ{ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  % } = ๐”ผ{ lim ,โ†’( โˆ‘-"# -", lim &โ†’( โˆ‘)"# )"& ๐‘“ ๐‘ ) . ๐‘“ ๐‘ - . ๐‘Š ๐‘ )*+ โˆ’ ๐‘Š ๐‘ ) . {๐‘Š ๐‘ -*+ โˆ’ ๐‘Š(๐‘ -)} } ยจ ๐”ผ{ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  % } = lim ,โ†’( โˆ‘-"# -", lim &โ†’( โˆ‘)"# )"& ๐‘“ ๐‘ ) . ๐‘“ ๐‘ - . ๐”ผ ๐‘Š ๐‘ )*+ โˆ’ ๐‘Š ๐‘ ) . [๐‘Š ๐‘ -*+ โˆ’ ๐‘Š(๐‘ -)] ยจ ๐”ผ{ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  % } = lim ,โ†’( โˆ‘-"# -", lim &โ†’( โˆ‘)"# )"& ๐‘“ ๐‘ ) . ๐‘“ ๐‘ - . ๐›ฟ),-. [๐‘ -*+ โˆ’ ๐‘ -] ยจ ๐”ผ{ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  % } = lim ,โ†’( โˆ‘-"# -", ๐‘“ ๐‘ - . ๐‘“ ๐‘ - . [๐‘ -*+ โˆ’ ๐‘ -] ยจ ๐”ผ{ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  % } = โˆซ !"# !"$ ๐‘“ ๐‘  % . ๐‘‘๐‘  9
  • 10. Luc_Faucheux_2021 Useful relationship through isometry - V ยจ ๐”ผ{ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  & } = โˆซ !"# !"$ ๐‘“ ๐‘  &. ๐‘‘๐‘  ยจ This is the usual Isometry property of the ITO integral ยจ Note that this would NOT apply in Stratonovitch calculus ยจ So really we should write it as: ยจ ๐”ผ{ โˆซ !"# !"$ ๐‘“ ๐‘  . ([). ๐‘‘๐‘Š ๐‘  & } = โˆซ !"# !"$ ๐‘“ ๐‘  &. ๐‘‘๐‘  ยจ We can try to generalize to the higher orders: ๐”ผ{ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  ' } ยจ In particular can we write the following ? ยจ ๐”ผ{ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  &* } = โˆซ !"# !"$ ๐‘“& ๐‘  . ๐‘‘๐‘  * 10
  • 11. Luc_Faucheux_2021 Useful relationship through isometry - VI ยจ ๐”ผ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  ' = ๐”ผ{ lim ,โ†’( โˆ‘."# .", ๐‘“ ๐‘ . . {๐‘Š ๐‘ .+% โˆ’ ๐‘Š(๐‘ .)} ' } ยจ In the case where ๐‘˜ is odd, we have ๐‘˜ = 2๐‘ + 1 ยจ ๐”ผ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  &*+% = ๐”ผ{ lim ,โ†’( โˆ‘."# .", ๐‘“ ๐‘ . . {๐‘Š ๐‘ .+% โˆ’ ๐‘Š(๐‘ .)} &*+% } ยจ That leaves us with a product of (2๐‘ + 1) limits of sums that looks something like that: ยจ ๐”ผ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  &*+% = ๐”ผ{โˆ'"% '"&*+% lim ,/โ†’( โˆ‘./"# ./",/ ๐‘“ ๐‘ ./ . {๐‘Š ๐‘ ./+% โˆ’ ๐‘Š(๐‘ ./ )} } ยจ That is a product of sums ยจ We can switch to the sum of products, even though the notation gets a little ugly: 11
  • 12. Luc_Faucheux_2021 Useful relationship through isometry - VII ยจ ๐”ผ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  &*+% = ๐”ผ{โˆ'"% '"&*+% lim ,/โ†’( โˆ‘./"# ./",/ ๐‘“ ๐‘ ./ . {๐‘Š ๐‘ ./+% โˆ’ ๐‘Š(๐‘ ./ )} } ยจ So instead of indexing the elements of the sum by ๐‘–' we will now index the elements of the products by ๐‘˜. ยจ ๐”ผ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  &*+% = ๐”ผ{lim ,โ†’( โˆ‘."# .", โˆ'0"% '0"&*+% ๐‘“ ๐‘ '0 . {๐‘Š ๐‘ '012 โˆ’ ๐‘Š(๐‘ '0 )} } ยจ Note that the index for the stochastic jump went from ๐‘ ./+% to ๐‘ '012 ยจ We can now have the expectation enter into the sum: ยจ ๐”ผ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  &*+% = lim ,โ†’( โˆ‘."# .", ๐”ผ{โˆ'0"% '0"&*+% ๐‘“ ๐‘ '0 . {๐‘Š ๐‘ '012 โˆ’ ๐‘Š(๐‘ '0 )}} ยจ This is where the properties of Brownian motion will greatly simplify the product, just like in the previous slide for the simple square case (isometry property) ยจ Note: we need to also spend some time to explain why it is called โ€œisometryโ€ 12
  • 13. Luc_Faucheux_2021 Useful relationship through isometry - VIII ยจ ๐”ผ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  &*+% = lim ,โ†’( โˆ‘."# .", ๐”ผ{โˆ'0"% '0"&*+% ๐‘“ ๐‘ '0 . {๐‘Š ๐‘ '012 โˆ’ ๐‘Š(๐‘ '0 )}} ยจ And we know that: ยจ ๐”ผ ๐‘Š ๐‘ .+% โˆ’ ๐‘Š ๐‘ . . [๐‘Š ๐‘ /+% โˆ’ ๐‘Š(๐‘ /)] = ๐›ฟ.,/. [๐‘ /+% โˆ’ ๐‘ /] ยจ So we are just left with: ยจ ๐”ผ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  %&'( = lim )โ†’+ โˆ‘,"# ,") ๐”ผ{โˆ-!"( -!"%&'( ๐›ฟ!"# ,!"! ๐‘“ ๐‘ -! . ๐‘“ ๐‘ -# . ๐‘ -!$% โˆ’ ๐‘ -! & . {๐‘Š ๐‘ -!$% โˆ’ ๐‘Š(๐‘ -! )}} ยจ Since: ยจ ๐”ผ ๐‘Š ๐‘ .+% โˆ’ ๐‘Š ๐‘ . = 0 ยจ For the odd term we are left with: ยจ ๐”ผ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  &*+% = 0 13
  • 14. Luc_Faucheux_2021 Useful relationship through isometry - IX ยจ In the case where ๐‘˜ is even, we have ๐‘˜ = 2๐‘ ยจ ๐”ผ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  &* = ๐”ผ{ lim ,โ†’( โˆ‘."# .", ๐‘“ ๐‘ . . {๐‘Š ๐‘ .+% โˆ’ ๐‘Š(๐‘ .)} &* } ยจ That leaves us with a product of (2๐‘ + 1) limits of sums that looks something like that: ยจ ๐”ผ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  &* = ๐”ผ{โˆ'"% '"&* lim ,/โ†’( โˆ‘./"# ./",/ ๐‘“ ๐‘ ./ . {๐‘Š ๐‘ ./+% โˆ’ ๐‘Š(๐‘ ./ )} } ยจ That is a product of sums ยจ We can switch to the sum of products, even though the notation gets a little ugly: 14
  • 15. Luc_Faucheux_2021 Useful relationship through isometry - X ยจ ๐”ผ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  &* = ๐”ผ{โˆ'"% '"&* lim ,/โ†’( โˆ‘./"# ./",/ ๐‘“ ๐‘ ./ . {๐‘Š ๐‘ ./+% โˆ’ ๐‘Š(๐‘ ./ )} } ยจ So instead of indexing the elements of the sum by ๐‘–' we will now index the elements of the products by ๐‘˜. ยจ ๐”ผ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  &* = ๐”ผ{lim ,โ†’( โˆ‘."# .", โˆ'0"% '0"&* ๐‘“ ๐‘ '0 . {๐‘Š ๐‘ '012 โˆ’ ๐‘Š(๐‘ '0 )} } ยจ Note that the index for the stochastic jump went from ๐‘ ./+% to ๐‘ '012 ยจ We can now have the expectation enter into the sum: ยจ ๐”ผ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  &* = lim ,โ†’( โˆ‘."# .", ๐”ผ{โˆ'0"% '0"&* ๐‘“ ๐‘ '0 . {๐‘Š ๐‘ '012 โˆ’ ๐‘Š(๐‘ '0 )}} 15
  • 16. Luc_Faucheux_2021 Useful relationship through isometry - XI ยจ ๐”ผ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  &* = lim ,โ†’( โˆ‘."# .", ๐”ผ{โˆ'0"% '0"&* ๐‘“ ๐‘ '0 . {๐‘Š ๐‘ '012 โˆ’ ๐‘Š(๐‘ '0 )}} ยจ And we know that: ยจ ๐”ผ ๐‘Š ๐‘ .+% โˆ’ ๐‘Š ๐‘ . . [๐‘Š ๐‘ /+% โˆ’ ๐‘Š(๐‘ /)] = ๐›ฟ.,/. [๐‘ /+% โˆ’ ๐‘ /] ยจ So we are just left with: ยจ ๐”ผ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  %3 = lim &โ†’( โˆ‘)"# )"& โˆ4!"+ 4!"%3 ๐›ฟ!"# ,!"! ๐‘“ ๐‘ 4! . ๐‘“ ๐‘ 4# . ๐‘ 4!$% โˆ’ ๐‘ 4! ยจ Which similar to the isometry derivation, reduces itself to: ยจ ๐”ผ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  &* = lim ,โ†’( โˆ‘."# .", โˆ'0"% '0"* ๐‘“ ๐‘ '0 & . ๐‘ '012 โˆ’ ๐‘ '0 ยจ We switch the sum and the product back and we get: ยจ ๐”ผ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  &* = {โˆซ !"# !"$ ๐‘“ ๐‘  &. ๐‘‘๐‘ }* 16
  • 17. Luc_Faucheux_2021 Useful relationship through isometry - X ยจ So we have: ยจ ๐”ผ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  &*+% = 0 ยจ ๐”ผ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  &* = {โˆซ !"# !"$ ๐‘“ ๐‘  &. ๐‘‘๐‘ }* ยจ Which we plug back into: ยจ ๐”ผ exp โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = โˆ‘'"# '"( % '! . ๐”ผ{ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  ' } ยจ ๐”ผ exp โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = โˆ‘*"# *"( % (&*)! . ๐”ผ{ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  &* } ยจ ๐”ผ exp โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = โˆ‘*"# *"( % (&*)! . {โˆซ !"# !"$ ๐‘“ ๐‘  &. ๐‘‘๐‘ }* 17
  • 18. Luc_Faucheux_2021 Useful relationship through isometry - XI ยจ ๐”ผ exp โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = โˆ‘*"# *"( % (&*)! . {โˆซ !"# !"$ ๐‘“ ๐‘  &. ๐‘‘๐‘ }* ยจ ๐”ผ exp โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = โˆ‘*"# *"( % (*)! . { % & โˆซ !"# !"$ ๐‘“ ๐‘  &. ๐‘‘๐‘ }* ยจ ๐”ผ exp โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = exp( % & โˆซ !"# !"$ ๐‘“ ๐‘  &. ๐‘‘๐‘ ) ยจ ๐”ผ exp โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = exp(โˆซ !"# !"$ % & ๐‘“ ๐‘  &. ๐‘‘๐‘ ) ยจ This result is TRUE, however the derivation is utterly WRONG. ยจ Gilles Franchini found all the wrong steps in less than 2 minutes. ยจ I made it a little easier for you. ยจ There will be a prize for those of you who find all the mistakes. ยจ I will have the mistakes highlighted and explained in the next deck 18
  • 19. Luc_Faucheux_2021 A quick note on martingale and driftless processes 19
  • 20. Luc_Faucheux_2021 Quick side note ยจ Dโ„š Dโ„™ = exp[โˆ’ โˆซ !"# !"$ ๐œ‰ ๐‘  . ๐‘‘๐‘Š ๐‘  โˆ’ % & โˆซ !"# !"$ ๐œ‰ ๐‘  &. ๐‘‘๐‘ ] ยจ ๐”ผ$ โ„™ Dโ„š Dโ„™ |๐”‰ 0 = 1 ยจ We also have if we define : ยจ ๐‘Œ ๐‘ก = ๐‘Œ ๐‘ก = 0 . exp(โˆซ !"# !"$ ๐œ‰ ๐‘  . ๐‘‘๐‘Š ๐‘  โˆ’ % & โˆซ !"# !"$ ๐œ‰ ๐‘  &. ๐‘‘๐‘ ) ยจ ๐”ผ$ โ„™ ๐‘Œ ๐‘ก |๐”‰ 0 = ๐‘Œ(0) ยจ So the process ยจ ๐‘Œ ๐‘ก = ๐‘Œ 0 . exp(โˆซ !"# !"$ ๐œ‰ ๐‘  . ๐‘‘๐‘Š ๐‘  โˆ’ % & โˆซ !"# !"$ ๐œ‰ ๐‘  &. ๐‘‘๐‘ ) ยจ Is a martingale under the โ„™-measure associated with the Brownian motion ๐‘Š 20
  • 21. Luc_Faucheux_2021 Quick side note - II ยจ ๐‘Œ ๐‘ก = ๐‘Œ 0 . exp(โˆซ !"# !"$ ๐œ‰ ๐‘  . ๐‘‘๐‘Š ๐‘  โˆ’ % & โˆซ !"# !"$ ๐œ‰ ๐‘  &. ๐‘‘๐‘ ) ยจ Is a martingale under the โ„™-measure associated with the Brownian motion ๐‘Š ยจ So ๐‘Œ ๐‘ก is driftless and can be written (maybe) as the solution of an SDE that could look like: ยจ ๐‘‘๐‘Œ ๐‘ก = 0. ๐‘‘๐‘ก + ๐‘ ๐‘Œ ๐‘ก , ๐‘ก . ([). ๐‘‘๐‘Š(๐‘ก) ยจ Letโ€™s use ITO lemma on: ยจ ๐‘Œ ๐‘ก = ๐‘Œ 0 . exp(โˆซ !"# !"$ ๐œ‰ ๐‘  . ๐‘‘๐‘Š ๐‘  โˆ’ % & โˆซ !"# !"$ ๐œ‰ ๐‘  &. ๐‘‘๐‘ ) ยจ ๐‘‹ ๐‘ก = โˆซ !"# !"$ ๐œ‰ ๐‘  . ๐‘‘๐‘Š ๐‘  โˆ’ % & โˆซ !"# !"$ ๐œ‰ ๐‘  &. ๐‘‘๐‘  ยจ We apply ITO lemma to ๐‘Œ ๐‘ก = ๐‘Œ 0 . exp(๐‘‹ ๐‘ก ) 21
  • 22. Luc_Faucheux_2021 Quick side note - III ยจ Applying Ito lemma: ยจ ๐‘“ ๐‘‹ ๐‘กG โˆ’ ๐‘“ ๐‘‹ ๐‘กH = โˆซ $"$H $"$G IJ IK . ([). ๐‘‘๐‘‹(๐‘ก) + โˆซ $"$H $"$G % & . I5J IK5 . ([). (๐›ฟ๐‘‹)& ยจ In the โ€limitโ€ of small me increments, this can be wri]en formally as the Ito lemma: ยจ ๐›ฟ๐‘“ = IJ IK . ([). ๐›ฟ๐‘‹ + % & . I5J IK5 . (๐›ฟ๐‘‹)& ยจ For a function of the Brownian motion ๐‘Š(๐‘ก): ยจ ๐‘“ ๐‘Š ๐‘กG โˆ’ ๐‘“ ๐‘Š ๐‘กH = โˆซ $"$H $"$G IJ IL . ([). ๐‘‘๐‘Š(๐‘ก) + โˆซ $"$H $"$G % & . I5J IL5 . ([). ๐‘‘๐‘ก 22
  • 23. Luc_Faucheux_2021 Quick side note - IV ยจ ๐‘“ ๐‘‹ ๐‘กG โˆ’ ๐‘“ ๐‘‹ ๐‘กH = โˆซ $"$H $"$G IJ IK . ([). ๐‘‘๐‘‹(๐‘ก) + โˆซ $"$H $"$G % & . I5J IK5 . ([). (๐‘‘๐‘‹)& ยจ ๐›ฟ๐‘“ = IJ IK . ([). ๐›ฟ๐‘‹ + % & . I5J IK5 . (๐›ฟ๐‘‹)& ยจ ๐‘‹ ๐‘ก = โˆซ !"# !"$ ๐œ‰ ๐‘  . ๐‘‘๐‘Š ๐‘  โˆ’ % & โˆซ !"# !"$ ๐œ‰ ๐‘  &. ๐‘‘๐‘  ยจ ๐‘‘๐‘‹ ๐‘ก = ๐œ‰ ๐‘ก . ๐‘‘๐‘Š ๐‘ก โˆ’ % & ๐œ‰ ๐‘ก &. ๐‘‘๐‘ก ยจ ๐‘‘๐‘‹ & ๐‘ก = ๐œ‰ ๐‘ก &. ๐‘‘๐‘ก ยจ ๐‘Œ ๐‘ก = ๐‘Œ 0 . exp(๐‘‹ ๐‘ก ) ยจ IJ IK = ๐‘Œ 0 . exp(๐‘‹ ๐‘ก ) ยจ I5J IK5 = ๐‘Œ 0 . exp(๐‘‹ ๐‘ก ) 23
  • 24. Luc_Faucheux_2021 Quick side note - V ยจ ๐›ฟ๐‘“ = IJ IK . ([). ๐›ฟ๐‘‹ + % & . I5J IK5 . (๐›ฟ๐‘‹)& ยจ ๐‘‘๐‘Œ(๐‘ก) = ๐‘Œ(๐‘ก). ([). {๐œ‰ ๐‘ก . ๐‘‘๐‘Š ๐‘ก โˆ’ % & ๐œ‰ ๐‘ก &. ๐‘‘๐‘ก} + % & . ๐‘Œ ๐‘ก . {๐œ‰ ๐‘ก &. ๐‘‘๐‘ก} ยจ ๐‘‘๐‘Œ(๐‘ก) = ๐‘Œ(๐‘ก). ([). {๐œ‰ ๐‘ก . ๐‘‘๐‘Š ๐‘ก } ยจ ๐‘‘๐‘Œ ๐‘ก = ๐œ‰ ๐‘ก . ๐‘Œ(๐‘ก). ๐‘‘๐‘Š ๐‘ก ยจ So we showed that the stochastic process: ยจ ๐‘Œ ๐‘ก = ๐‘Œ 0 . exp(โˆซ !"# !"$ ๐œ‰ ๐‘  . ๐‘‘๐‘Š ๐‘  โˆ’ % & โˆซ !"# !"$ ๐œ‰ ๐‘  &. ๐‘‘๐‘ ) ยจ Is a solution (we leave to pure math people the rigorous work of showing unicity, stability, well-behaved and all that good stuff) ยจ Is a solution of the SDE: ยจ ๐‘‘๐‘Œ ๐‘ก = ๐œ‰ ๐‘ก . ๐‘Œ(๐‘ก). ๐‘‘๐‘Š ๐‘ก 24
  • 25. Luc_Faucheux_2021 Quick side note - VI ยจ ๐‘Œ ๐‘ก = ๐‘Œ 0 . exp(โˆซ !"# !"$ ๐œ‰ ๐‘  . ๐‘‘๐‘Š ๐‘  โˆ’ % & โˆซ !"# !"$ ๐œ‰ ๐‘  &. ๐‘‘๐‘ ) ยจ SDE: ๐‘‘๐‘Œ ๐‘ก = ๐œ‰ ๐‘ก . ๐‘Œ(๐‘ก). ๐‘‘๐‘Š ๐‘ก ยจ SIE: ๐‘Œ ๐‘กG โˆ’ ๐‘Œ ๐‘กH = โˆซ $"$H $"$G ๐œ‰ ๐‘ก . ๐‘Œ(๐‘ก). ๐‘‘๐‘Š ๐‘ก ยจ In the regular (Newtonian) calculus, ยจ ๐‘‘๐‘Œ ๐‘ก = ๐œ‰ ๐‘ก . ๐‘Œ(๐‘ก). ๐‘‘๐‘ก ยจ Would yield: ยจ ๐‘Œ ๐‘ก = ๐‘Œ 0 . exp(โˆซ !"# !"$ ๐œ‰ ๐‘  . ๐‘‘๐‘ ) ยจ Which is the regular exponential function 25
  • 26. Luc_Faucheux_2021 Quick side note - VII ยจ In the stochastic calculus (ITO), the solution of the SDE: ยจ ๐‘‘๐‘Œ ๐‘ก = ๐œ‰ ๐‘ก . ๐‘Œ(๐‘ก). ๐‘‘๐‘Š ๐‘ก ยจ Is NOT the regular exponential that we are used to, but instead: ยจ ๐‘Œ ๐‘ก = ๐‘Œ 0 . exp(โˆซ !"# !"$ ๐œ‰ ๐‘  . ๐‘‘๐‘Š ๐‘  โˆ’ % & โˆซ !"# !"$ ๐œ‰ ๐‘  &. ๐‘‘๐‘ ) ยจ Sometimes the above function is referred to the Doleans-Dade exponential in memory of Catherine Doleans-Dade, and because is it so useful and used ยจ โ„ฐ โˆซ # $ ๐œ‰ ๐‘  . ๐‘‘๐‘Š ๐‘  = exp(โˆซ # $ ๐œ‰ ๐‘  . ๐‘‘๐‘Š(๐‘ ) โˆ’ โˆซ # $ % & ๐œ‰ ๐‘  &. ๐‘‘๐‘ ) ยจ ๐‘Œ ๐‘ก = ๐‘Œ 0 . exp(โˆซ !"# !"$ ๐œ‰ ๐‘  . ๐‘‘๐‘Š ๐‘  โˆ’ % & โˆซ !"# !"$ ๐œ‰ ๐‘  &. ๐‘‘๐‘ ) ยจ ๐‘Œ ๐‘ก = ๐‘Œ 0 . โ„ฐ โˆซ # $ ๐œ‰ ๐‘  . ๐‘‘๐‘Š ๐‘  26
  • 27. Luc_Faucheux_2021 Quick side note - VIII ยจ Note the formal analogy: ยจ REGULAR CALCULUS (Newtonian) ยจ ๐‘‘๐‘Œ ๐‘ก = ๐œ‰ ๐‘ก . ๐‘Œ(๐‘ก). ๐‘‘๐‘ก ยจ ๐‘Œ ๐‘ก = ๐‘Œ 0 . exp(โˆซ !"# !"$ ๐œ‰ ๐‘  . ๐‘‘๐‘ ) ยจ STOCHASTIC CALCULUS (Brownian) in the ITO convention ยจ ๐‘‘๐‘Œ ๐‘ก = ๐œ‰ ๐‘ก . ๐‘Œ ๐‘ก . [ . ๐‘‘๐‘Š ๐‘ก ยจ ๐‘Œ ๐‘ก = ๐‘Œ 0 . โ„ฐ โˆซ # $ ๐œ‰ ๐‘  . [ . ๐‘‘๐‘Š ๐‘  27
  • 28. Luc_Faucheux_2021 Quick side note - IX ยจ The interesting thing is that: ยจ ๐‘‘๐‘Œ ๐‘ก = ๐œ‰ ๐‘ก . ๐‘Œ ๐‘ก . [ . ๐‘‘๐‘Š ๐‘ก ยจ Is driftless, and the solution of it is: ยจ ๐‘Œ ๐‘ก = ๐‘Œ 0 . exp(โˆซ !"# !"$ ๐œ‰ ๐‘  . [ . ๐‘‘๐‘Š ๐‘  โˆ’ % & โˆซ !"# !"$ ๐œ‰ ๐‘  &. ๐‘‘๐‘ ) ยจ Such that it is a martingale: ยจ ๐”ผ$ โ„™ ๐‘Œ ๐‘ก |๐”‰ 0 = ๐‘Œ(0) ยจ That would be another way to recover the useful relationship, is to use the property that a driftless process is a martingale. ยจ This is the end of this quick note, but I wanted to point out the nice connection between a process that is driftless and the fact that it is a martingale, in the case where we can have an explicit solution of the SDE 28
  • 29. Luc_Faucheux_2021 Quick side note - X ยจ There is an awful lot of complicated math to prove the equivalence, but very roughly, if the Novikov condition is respected: ยจ ๐”ผ$ โ„™ exp( % & โˆซ !"# !"$ ๐œ‰ ๐‘  &. ๐‘‘๐‘ )|๐”‰ 0 < โˆž ยจ Then you have equivalence between driftless and martingale. ยจ Just like in Mario Kart, with the evil Wario, if the Novikov condition is not respected, then the process becomes a wartingale 29
  • 30. Luc_Faucheux_2021 Quick side note - XI 30 ยจ Mario driving a martingale
  • 31. Luc_Faucheux_2021 Quick side note - XII ยจ Wario driving a wartingale 31
  • 32. Luc_Faucheux_2021 Quick side note - XIII ยจ In Mario Kart just like in stochastic processes, the crucial part is the drift. ยจ A martingale is a driftless process. ยจ Once you start drifting, both in Mario Kart and in your stochastic process, you could end up in big trouble. 32
  • 33. Luc_Faucheux_2021 Quick side note - XIV ยจ The Girsanov theorem essentially โ€œtakes care of the driftโ€ 33
  • 34. Luc_Faucheux_2021 Quick side note - XV ยจ In Finance you want to remove the drift (find the martingale) ยจ In Mario Kart, you want to control the drift especially around the corners ยจ I want to thank Gilles Franchini for pointing out how crucial the drift was in both situations 34
  • 35. Luc_Faucheux_2021 From Short Rate to Affine models 35
  • 36. Luc_Faucheux_2021 From short rate to Affine model ยจ We note here that there is a strong connection between short rates models and affine models. ยจ This just illustrates how strong that connection is: ยจ Suppose that we start with an SDE of the form: ยจ ๐‘‘๐‘… ๐‘ก, ๐‘ก, ๐‘ก = ๐œƒ ๐‘ก . ๐‘‘๐‘ก โˆ’ ๐œŽ(๐‘ก). ([). ๐‘‘๐‘Š(๐‘ก) ยจ The corresponding SIE is: ยจ ๐‘… ๐‘ก, ๐‘ก, ๐‘ก โˆ’ ๐‘… ๐‘ , ๐‘ , ๐‘  = โˆซ M"! M"$ ๐œƒ ๐‘ข . ๐‘‘๐‘ข โˆ’ โˆซ M"! M"$ ๐œŽ(๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข) ยจ In the risk-free measure โ„š(where we assumed that the Brownian motion ๐‘Š(๐‘ก) is the one associated to this โ„š -measure) ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = ๐”ผ$0 โ„š N($) N($0) |๐”‰(๐‘ก) = ๐”ผ$0 โ„š exp(โˆ’ โˆซ !"$ !"$0 ๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ )|๐”‰(๐‘ก) 36
  • 37. Luc_Faucheux_2021 From short rate to Affine model - II ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = ๐”ผ$0 โ„š N($) N($0) |๐”‰(๐‘ก) = ๐”ผ$0 โ„š exp(โˆ’ โˆซ !"$ !"$0 ๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ )|๐”‰(๐‘ก) ยจ ๐‘… ๐‘ , ๐‘ , ๐‘  โˆ’ ๐‘… ๐‘ก, ๐‘ก, ๐‘ก = โˆซ M"$ M"! ๐œƒ ๐‘ข . ๐‘‘๐‘ข โˆ’ โˆซ M"$ M"! ๐œŽ(๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข) ยจ ๐‘… ๐‘ , ๐‘ , ๐‘  = ๐‘… ๐‘ก, ๐‘ก, ๐‘ก + โˆซ M"$ M"! ๐œƒ ๐‘ข . ๐‘‘๐‘ข โˆ’ โˆซ M"$ M"! ๐œŽ(๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข) ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = ๐”ผ$0 โ„š exp(โˆ’ โˆซ !"$ !"$0 {๐‘… ๐‘ก, ๐‘ก, ๐‘ก + โˆซ M"$ M"! ๐œƒ ๐‘ข . ๐‘‘๐‘ข โˆ’ โˆซ M"$ M"! ๐œŽ(๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘ )|๐”‰(๐‘ก) ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก) = ๐”ผ$! โ„š exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . (๐‘ก) โˆ’ ๐‘ก) . exp(โˆ’ โˆซ !"$ !"$! {โˆซ 7"$ 7"! ๐œƒ ๐‘ข . ๐‘‘๐‘ข โˆ’ โˆซ 7"$ 7"! ๐œŽ(๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘ )|๐”‰(๐‘ก) ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก) = exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . (๐‘ก) โˆ’ ๐‘ก) . ๐”ผ$! โ„š exp(โˆ’ โˆซ !"$ !"$! {โˆซ 7"$ 7"! ๐œƒ ๐‘ข . ๐‘‘๐‘ข โˆ’ โˆซ 7"$ 7"! ๐œŽ(๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘ )|๐”‰(๐‘ก) ยจ Remember that the Affine model assumption was: ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = exp(๐ด ๐‘ก, ๐‘ก. โˆ’ ๐ต ๐‘ก, ๐‘ก. . ๐‘… ๐‘ก, ๐‘ก, ๐‘ก ) 37
  • 38. Luc_Faucheux_2021 From short rate to Affine model - III ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก) = exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . (๐‘ก) โˆ’ ๐‘ก) . ๐”ผ$! โ„š exp(โˆ’ โˆซ !"$ !"$! {โˆซ 7"$ 7"! ๐œƒ ๐‘ข . ๐‘‘๐‘ข โˆ’ โˆซ 7"$ 7"! ๐œŽ(๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘ )|๐”‰(๐‘ก) ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = exp(๐ด ๐‘ก, ๐‘ก. โˆ’ ๐ต ๐‘ก, ๐‘ก. . ๐‘… ๐‘ก, ๐‘ก, ๐‘ก ) ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . ๐ต ๐‘ก, ๐‘ก. . exp(๐ด ๐‘ก, ๐‘ก. ) ยจ In that formulation we see that naturally: ยจ ๐ต ๐‘ก, ๐‘ก. = (๐‘ก. โˆ’ ๐‘ก) ยจ exp ๐ด ๐‘ก, ๐‘ก. = ๐”ผ$0 โ„š exp(โˆ’ โˆซ !"$ !"$0 {โˆซ M"$ M"! ๐œƒ ๐‘ข . ๐‘‘๐‘ข โˆ’ โˆซ M"$ M"! ๐œŽ(๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘ )|๐”‰(๐‘ก) ยจ Which can be super complicated 38
  • 39. Luc_Faucheux_2021 From short rate to Affine model - IV ยจ In the Ho-Lee model, we recovered: ยจ ๐ต ๐‘ก, ๐‘ก. = (๐‘ก. โˆ’ ๐‘ก) ยจ exp ๐ด ๐‘ก, ๐‘ก. = ๐”ผ$0 โ„š exp(โˆ’ โˆซ !"$ !"$0 {โˆซ M"$ M"! ๐œƒ ๐‘ข . ๐‘‘๐‘ข โˆ’ โˆซ M"$ M"! ๐œŽ(๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘ )|๐”‰(๐‘ก) ยจ ๐œŽ ๐‘ข = ๐œŽ ยจ ๐œƒ ๐‘ข = I IM ๐‘… 0, ๐‘ข, ๐‘ข + ๐œŽ&. ๐‘ข ยจ Which led to: ยจ ๐ด ๐‘ก, ๐‘ก. = โˆ’ โˆซ !"$ !"$0 {[ IO #,M,M IM + ๐œŽ&. ๐‘ข]. ๐‘ก. โˆ’ ๐‘ข โˆ’ % & . (๐‘ก. โˆ’ ๐‘ข)&. ๐œŽ&}. ๐‘‘๐‘ข ยจ ๐ด ๐‘ก, ๐‘ก. = ๐‘… 0, ๐‘ก, ๐‘ก . ๐‘ก. โˆ’ ๐‘ก + ln( PQ #,#,$0 PQ #,#,$ ) โˆ’ R5 & ๐‘ก(๐‘ก. โˆ’ ๐‘ก)& 39
  • 40. Luc_Faucheux_2021 From short rate to Affine model โ€“ V ยจ Alternatively, we could also have made use of the useful relationship: ยจ ๐”ผ exp โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = exp[โˆซ !"# !"$ % & ๐‘“ ๐‘  &. ๐‘‘๐‘ ] ยจ And apply it to: ยจ exp ๐ด ๐‘ก, ๐‘ก. = ๐”ผ$0 โ„š exp(โˆ’ โˆซ !"$ !"$0 {โˆซ M"$ M"! ๐œƒ ๐‘ข . ๐‘‘๐‘ข โˆ’ โˆซ M"$ M"! ๐œŽ(๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘ )|๐”‰(๐‘ก) ยจ Piterbarg (p.409) illustrates that approach on the even simpler model: ยจ ๐‘‘๐‘… ๐‘ก, ๐‘ก, ๐‘ก = ๐œŽ. ([). ๐‘‘๐‘Š(๐‘ก) ยจ Just for kicks, letโ€™s redo it here also: 40
  • 41. Luc_Faucheux_2021 From short rate to Affine model โ€“ VI ยจ ๐‘‘๐‘… ๐‘ก, ๐‘ก, ๐‘ก = ๐œŽ. ([). ๐‘‘๐‘Š(๐‘ก) ยจ ๐‘… ๐‘ , ๐‘ , ๐‘  โˆ’ ๐‘… ๐‘ก, ๐‘ก, ๐‘ก = โˆ’ โˆซ M"$ M"! ๐œŽ. ([). ๐‘‘๐‘Š(๐‘ข) ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = ๐”ผ$0 โ„š N($) N($0) |๐”‰(๐‘ก) = ๐”ผ$0 โ„š exp(โˆ’ โˆซ !"$ !"$0 ๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ )|๐”‰(๐‘ก) ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = ๐”ผ$0 โ„š exp(โˆ’ โˆซ !"$ !"$0 {๐‘…(๐‘ก, ๐‘ก, ๐‘ก) โˆ’ โˆซ M"$ M"! ๐œŽ. ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘ )|๐”‰(๐‘ก) ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . (๐‘ก. โˆ’ ๐‘ก) . ๐”ผ$0 โ„š exp(โˆ’ โˆซ !"$ !"$0 {โˆ’ โˆซ M"$ M"! ๐œŽ. ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘ )|๐”‰(๐‘ก) ยจ We recognize here once again our good old friend Guido Fubini so that we can change the order of integration: ยจ ๐‘‹ = โˆซ !"$ !"$0 โˆซ M"$ M"! ๐œŽ. ([). ๐‘‘๐‘Š(๐‘ข) . ๐‘‘๐‘  41
  • 42. Luc_Faucheux_2021 From short rate to Affine model โ€“ VII ยจ ๐‘‹ = โˆซ !"$ !"$0 {โˆซ M"$ M"! ๐œŽ. ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘  42 s ๐‘  = ๐‘ก! u s u ๐‘  = ๐‘ก ๐‘  = ๐‘ก! ๐‘  = ๐‘ก ๐‘‹ = X !"$ !"$0 { X M"$ M"! ๐œŽ. ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘  ๐‘‹ = X M"$ M"$0 { X !"M !"$0 ๐œŽ. ๐‘‘๐‘ }. ([). ๐‘‘๐‘Š(๐‘ข)
  • 43. Luc_Faucheux_2021 From short rate to Affine model โ€“ VIII ยจ ๐‘‹ = โˆซ !"$ !"$0 {โˆซ M"$ M"! ๐œŽ. ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘  ยจ ๐‘‹ = โˆซ M"$ M"$0 {โˆซ !"M !"$0 ๐œŽ. ๐‘‘๐‘ }. ([). ๐‘‘๐‘Š(๐‘ข) ยจ ๐‘‹ = โˆซ M"$ M"$0 {๐œŽ. [๐‘ ]!"M !"$0 }. ([). ๐‘‘๐‘Š(๐‘ข) ยจ ๐‘‹ = โˆซ M"$ M"$0 {๐œŽ. (๐‘ก. โˆ’ ๐‘ข)}. ([). ๐‘‘๐‘Š(๐‘ข) ยจ ๐‘‹ = โˆซ M"$ M"$0 ๐œŽ. (๐‘ก. โˆ’ ๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข) 43
  • 44. Luc_Faucheux_2021 From short rate to Affine model โ€“ IX ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . (๐‘ก. โˆ’ ๐‘ก) . ๐”ผ$0 โ„š exp(โˆ’ โˆซ !"$ !"$0 {โˆ’ โˆซ M"$ M"! ๐œŽ. ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘ )|๐”‰(๐‘ก) ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . (๐‘ก. โˆ’ ๐‘ก) . ๐”ผ$0 โ„š exp(โˆซ !"$ !"$0 {โˆซ M"$ M"! ๐œŽ. ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘ )|๐”‰(๐‘ก) ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . (๐‘ก. โˆ’ ๐‘ก) . ๐”ผ$0 โ„š exp(โˆซ M"$ M"$0 ๐œŽ. (๐‘ก. โˆ’ ๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข))|๐”‰(๐‘ก) ยจ And there we then again make good use of the useful relationship: ยจ ๐”ผ exp โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = exp[โˆซ !"# !"$ % & ๐‘“ ๐‘  &. ๐‘‘๐‘ ] ยจ ๐”ผ exp โˆซ M"$ M"$0 ๐‘“ ๐‘ข . ๐‘‘๐‘Š ๐‘ข = exp[โˆซ M"$ M"$0 % & ๐‘“ ๐‘ข &. ๐‘‘๐‘ข] ยจ On the function: ยจ ๐‘“ ๐‘ข = ๐œŽ. (๐‘ก. โˆ’ ๐‘ข) 44
  • 45. Luc_Faucheux_2021 From short rate to Affine model โ€“ X ยจ ๐”ผ exp โˆซ M"$ M"$0 ๐‘“ ๐‘ข . ๐‘‘๐‘Š ๐‘ข = exp[โˆซ M"$ M"$0 % & ๐‘“ ๐‘ข &. ๐‘‘๐‘ข] ยจ ๐‘“ ๐‘ข = ๐œŽ. (๐‘ก. โˆ’ ๐‘ข) ยจ exp[โˆซ M"$ M"$0 % & ๐‘“ ๐‘ข &. ๐‘‘๐‘ข] = exp[โˆซ M"$ M"$0 % & ๐œŽ. ๐‘ก. โˆ’ ๐‘ข &. ๐‘‘๐‘ข] = R5 & . exp[โˆซ M"$ M"$0 ๐‘ก. โˆ’ ๐‘ข &. ๐‘‘๐‘ข] ยจ exp โˆซ M"$ M"$0 % & ๐‘“ ๐‘ข &. ๐‘‘๐‘ข = R5 & . โˆ’ $0SM 8 T M"$ M"$0 = R5 & . โˆ’0 + $0S$ 8 T = R5 U . ๐‘ก. โˆ’ ๐‘ก T ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . (๐‘ก. โˆ’ ๐‘ก) . ๐”ผ$0 โ„š exp(โˆซ M"$ M"$0 ๐œŽ. (๐‘ก. โˆ’ ๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข))|๐”‰(๐‘ก) ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . ๐‘ก. โˆ’ ๐‘ก . exp( R5 U . ๐‘ก. โˆ’ ๐‘ก T) ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = ๐”ผ$0 โ„š N($) N($0) |๐”‰(๐‘ก) = ๐”ผ$0 โ„š exp(โˆ’ โˆซ !"$ !"$0 ๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ )|๐”‰(๐‘ก) ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . ๐‘ก. โˆ’ ๐‘ก + R5 U . ๐‘ก. โˆ’ ๐‘ก T 45
  • 46. Luc_Faucheux_2021 From short rate to Affine model โ€“ XI ยจ Alternatively, like Piterbarg does on p. 409, we can actually be a tad more general, and instead of using the useful relationship: ยจ ๐”ผ exp โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = exp[โˆซ !"# !"$ % & ๐‘“ ๐‘  &. ๐‘‘๐‘ ] ยจ We use the slightly more general result: ยจ ๐”ผ exp ๐‘‹ = exp ๐”ผ ๐‘‹ . exp % & ๐”ผ (๐‘‹(๐‘ก)โˆ’ < ๐‘‹ >$)& ยจ ๐”ผ exp ๐‘‹ = exp ๐‘€[๐‘‹(๐‘ก)] . exp % & ๐‘‰[๐‘‹(๐‘ก)] ยจ ๐”ผ exp ๐‘‹ = exp[๐‘€] . exp % & ๐‘‰ ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . (๐‘ก. โˆ’ ๐‘ก) . ๐”ผ$0 โ„š exp(โˆซ M"$ M"$0 ๐œŽ. (๐‘ก. โˆ’ ๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข))|๐”‰(๐‘ก) 46
  • 47. Luc_Faucheux_2021 From short rate to Affine model โ€“ X ยจ So we will apply: ยจ ๐”ผ exp ๐‘‹ = exp ๐‘€[๐‘‹(๐‘ก)] . exp % & ๐‘‰[๐‘‹(๐‘ก)] ยจ To the process: ยจ ๐‘Œ = โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . ๐‘ก. โˆ’ ๐‘ก + โˆซ M"$ M"$0 ๐œŽ. (๐‘ก. โˆ’ ๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข) ยจ Since: ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = ๐”ผ$0 โ„š exp(โˆ’ โˆซ !"$ !"$0 {๐‘…(๐‘ก, ๐‘ก, ๐‘ก) โˆ’ โˆซ M"$ M"! ๐œŽ. ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘ )|๐”‰(๐‘ก) ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = ๐”ผ$0 โ„š exp(โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . ๐‘ก. โˆ’ ๐‘ก + โˆซ !"$ !"$0 {โˆซ M"$ M"! ๐œŽ. ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘ )|๐”‰(๐‘ก) 47
  • 48. Luc_Faucheux_2021 From short rate to Affine model โ€“ XI ยจ ๐‘Œ = โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . ๐‘ก. โˆ’ ๐‘ก + โˆซ M"$ M"$0 ๐œŽ. (๐‘ก. โˆ’ ๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข) ยจ Since the ITO integral is a martingale: ยจ ๐‘€ ๐‘Œ ๐‘ก = โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . ๐‘ก. โˆ’ ๐‘ก ยจ So we then compute the variance: ยจ ๐‘‰ ๐‘Œ ๐‘ก = ๐”ผ (๐‘Œ(๐‘ก)โˆ’ < ๐‘Œ >$)% = ๐”ผ (๐‘Œ(๐‘ก) โˆ’ ๐‘€ ๐‘Œ ๐‘ก )% = ๐”ผ (โˆซ 7"$ 7"$! ๐œŽ. (๐‘ก) โˆ’ ๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข))% ยจ There, we can use once again the good old isometry property of the ITO integral: ยจ ๐”ผ (โˆซ M"$ M"$0 ๐œŽ. (๐‘ก. โˆ’ ๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข))& = โˆซ M"$ M"$0 {๐œŽ. (๐‘ก. โˆ’ ๐‘ข)}&. ๐‘‘๐‘ข = R5 T . ๐‘ก. โˆ’ ๐‘ก T 48
  • 49. Luc_Faucheux_2021 From short rate to Affine model โ€“ XII ยจ ๐‘Œ = โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . ๐‘ก. โˆ’ ๐‘ก + โˆซ M"$ M"$0 ๐œŽ. (๐‘ก. โˆ’ ๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข) ยจ ๐‘€ ๐‘Œ ๐‘ก = โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . ๐‘ก. โˆ’ ๐‘ก ยจ ๐‘‰ ๐‘Œ ๐‘ก = R5 T . ๐‘ก. โˆ’ ๐‘ก T ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = ๐”ผ$0 โ„š exp(๐‘Œ(๐‘ก))|๐”‰(๐‘ก) = exp ๐‘€[๐‘Œ(๐‘ก)] . exp % & ๐‘‰[๐‘Œ(๐‘ก)] ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = exp ๐‘€[๐‘Œ(๐‘ก)] . exp % & ๐‘‰[๐‘Œ(๐‘ก)] = exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . ๐‘ก. โˆ’ ๐‘ก . exp % & R5 T . ๐‘ก. โˆ’ ๐‘ก T ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . ๐‘ก. โˆ’ ๐‘ก . exp % & R5 T . ๐‘ก. โˆ’ ๐‘ก T ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = exp โˆ’๐‘… ๐‘ก, ๐‘ก, ๐‘ก . ๐‘ก. โˆ’ ๐‘ก + R5 U . ๐‘ก. โˆ’ ๐‘ก T ยจ And we do recover indeed the same result for that simple model 49
  • 50. Luc_Faucheux_2021 A couple of useful tools 50
  • 51. Luc_Faucheux_2021 Useful tools ยจ As you go through those slides, it is quite apparent that there are some relations or properties that we keep using over and over again, or that are worth mentioning. ยจ I tried to put all of them together in a quick summary section here ยจ I still need to work on a notation section, maybe once I get my book deal ยจ Would love to get your feedback on this section, if there are tools that you tend to use a lot and find useful, just drop me a note and I would be happy to include those 51
  • 52. Luc_Faucheux_2021 Useful tools โ€“ ITO LEMMA ยจ The ITO lemma is revered in stochastic calculus. ยจ In the somewhat misleading โ€œdifferentialโ€ form it reads: ยจ ๐›ฟ๐‘“ = IJ IK . ๐›ฟ๐‘‹ + % & . I5J IK5 . (๐›ฟ๐‘‹)& ยจ It should really only be expressed as: ยจ ๐‘“ ๐‘‹ ๐‘กG โˆ’ ๐‘“ ๐‘‹ ๐‘กH = โˆซ $"$9 $"$: IJ IV . ([). ๐‘‘๐‘‹(๐‘ก) + โˆซ $"$9 $"$: % & . I5J IK5 . ([). (๐›ฟ๐‘‹)& ยจ The ITO convention for the ITO integral is that we take the โ€œLHSโ€ (Left Hand side) in the partition as noted by: ([) ยจ And the definition of the integral is: ยจ โˆซ $"$9 $"$: ๐‘“ ๐‘‹(๐‘ก) . ๐‘‘๐‘Š ๐‘ก = lim ,โ†’( โˆ‘."# .", ๐‘“ ๐‘‹(๐‘ก.) . {๐‘Š ๐‘ก.+% โˆ’ ๐‘Š(๐‘ก.)} ยจ Where we assume that we do not choose a pathological mesh and the the function is relatively well behaved 52
  • 53. Luc_Faucheux_2021 Useful tools โ€“ ITO LEMMA - II ยจ Be careful that stochastic calculus in many ways has NOTHING to do with regular calculus ยจ So it is quite dangerous to write: ยจ ๐›ฟ๐‘“ = IJ IK . ๐›ฟ๐‘‹ + % & . I5J IK5 . (๐›ฟ๐‘‹)& ยจ And say โ€œ oh well stochastic calculus is the same as regular calculus, it is just when I do Taylor expansion I should really go up one more order in order to go up to all the orders that are at least linear in timeโ€ ยจ Again, this is ONLY a formal correspondence, or a way to write down two things that are almost completely different ยจ Stochastic processes are NOT differentiable, so do not even think of using a โ€œTaylor expansion on a stochastic processโ€ ยจ ALWAYS go back to the integral, always try to use the SIE format (Stochastic Integral Equation), never the SDE format (Stochastic Differential Equation) 53
  • 54. Luc_Faucheux_2021 Useful tools โ€“ ITO Leibniz ยจ Again, for ease of notation, we use the โ€œdifferentialโ€ form, but by now we know better than to trust is: ยจ ๐›ฟ๐‘“ ๐‘‹, ๐‘Œ = IJ IV . ๐›ฟ๐‘‹ + IJ IW . ๐›ฟ๐‘Œ + % & . I5J IV5 . ๐›ฟ๐‘‹& + % & . I5J IW5 . ๐›ฟ๐‘Œ& + I5J IVIW . ๐›ฟ๐‘‹. ๐›ฟ๐‘Œ ยจ Note: should really be written as: ยจ ๐›ฟ๐‘“ ๐‘‹, ๐‘Œ = IJ IK . ๐›ฟ๐‘‹ + IJ IX . ๐›ฟ๐‘Œ + % & . I5J IK5 . ๐›ฟ๐‘‹& + % & . I5J IX5 . ๐›ฟ๐‘Œ& + I5J IKIX . ๐›ฟ๐‘‹. ๐›ฟ๐‘Œ ยจ Lower case ๐‘ฅ is a regular variable ยจ Upper case ๐‘‹ is a stochastic variable ยจ ๐‘“ ๐‘‹, ๐‘Œ is really noted ๐‘“ ๐‘ฅ = ๐‘‹, ๐‘ฆ = ๐‘Œ and all the partial derivatives are for example: ยจ I5J IKIX = I5J IKIX |K"V $ ,X"W($) 54
  • 55. Luc_Faucheux_2021 Useful tools โ€“ ITO and STRATO correspondence ยจ ITO integral is defined as LHS (Left Hand Side) ยจ โˆซ $"$H $"$G ๐น ๐‘‹ ๐‘ก . ([). ๐‘‘๐‘‹(๐‘ก) = lim Yโ†’( {โˆ‘'"% '"Y ๐น(๐‘‹(๐‘ก')). [๐‘‹(๐‘ก'+%) โˆ’ ๐‘‹(๐‘ก')]} ยจ STRATO integral is defined as M (Middle) ยจ โˆซ $"$H $"$G ๐น ๐‘‹ ๐‘ก . (โˆ˜). ๐‘‘๐‘‹(๐‘ก) = lim Yโ†’( {โˆ‘'"% '"Y ๐น( V($/12 +V($/)] & ). [๐‘‹(๐‘ก'+%) โˆ’ ๐‘‹(๐‘ก')]} ยจ For a simple Brownian motion ยจ โˆซ $"$H $"$G ๐‘“ ๐‘Š ๐‘ก . (โˆ˜). ๐‘‘๐‘Š(๐‘ก) = โˆซ $"$H $"$G ๐‘“ ๐‘Š ๐‘ก . ([). ๐‘‘๐‘Š(๐‘ก) + % & โˆซ $"$H $"$G IJ IL |L"[($). ๐‘‘๐‘ก ยจ The integral in time โˆซ $"$H $"$G IJ IL |L"[($). ๐‘‘๐‘ก is the usual Riemann integral defined as ยจ โˆซ $"$H $"$G ๐น ๐‘‹ ๐‘ก . ๐‘‘๐‘ก = lim Yโ†’( {โˆ‘'"% '"Y ๐น(๐‘‹(๐œ‘[๐‘ก', ๐‘ก'+%])). [๐‘ก'+% โˆ’ ๐‘ก']} 55
  • 56. Luc_Faucheux_2021 Useful tools โ€“ ITO and STRATO correspondence - II ยจ Where ๐œ‘[๐‘ก', ๐‘ก'+%] is a function that takes some point within the mesh (does not matter where, LHS, RIHS, middle, anywhere, could also varies from one bucket to the next, that is the beauty of the Riemann integral in regular, or Newtonian, calculus, is that you do not have all those pesky differences between ITO or Stratonovitch,โ€ฆ) ยจ For a more complicated stochastic process ยจ ๐‘‘๐‘‹ ๐‘ก = ๐‘Ž ๐‘ก, ๐‘‹ ๐‘ก . ๐‘‘๐‘ก + ๐‘ ๐‘ก, ๐‘‹ ๐‘ก . ๐‘‘๐‘Š ยจ We have: ยจ โˆซ $"$H $"$G ๐‘“ ๐‘‹ ๐‘ก . โˆ˜ . ๐‘‘๐‘Š ๐‘ก = โˆซ $"$H $"$G ๐‘“ ๐‘‹ ๐‘ก . ([). ๐‘‘๐‘Š(๐‘ก) + โˆซ $"$H $"$G % & . ๐‘ ๐‘ก, ๐‘‹ ๐‘ก . IJ IK |K"V($). ๐‘‘๐‘ก 56
  • 57. Luc_Faucheux_2021 Useful tools โ€“ ITO integral is a martingale ยจ This is super useful ยจ For a Brownian motion ๐‘Š ๐‘  associated to the measure ยจ ๐”ผ{โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  } = ๐”ผ{lim ,โ†’( โˆ‘."# .", ๐‘“ ๐‘ . . {๐‘Š ๐‘ .+% โˆ’ ๐‘Š(๐‘ .)} } ยจ ๐”ผ{โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  } = lim ,โ†’( โˆ‘."# .", ๐‘“ ๐‘ . . ๐”ผ{๐‘Š ๐‘ .+% โˆ’ ๐‘Š(๐‘ .)} ยจ ๐”ผ{โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  } = lim ,โ†’( โˆ‘."# .", ๐‘“ ๐‘ . . 0 = 0 ยจ ๐”ผ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = 0 ยจ ๐”ผ โˆซ !"# !"$ ๐‘“ ๐‘Š ๐‘  , ๐‘  . ๐‘‘๐‘Š ๐‘  = 0 57
  • 58. Luc_Faucheux_2021 Useful tools โ€“ Isometry of the ITO integral ยจ ๐”ผ{ โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  & } = โˆซ !"# !"$ ๐‘“ ๐‘  &. ๐‘‘๐‘  ยจ ๐”ผ{ โˆซ !"# !"$ ๐‘“ ๐‘Š ๐‘  , ๐‘  . ๐‘‘๐‘Š ๐‘  & } = โˆซ !"# !"$ ๐‘“ ๐‘Š ๐‘  , ๐‘  &. ๐‘‘๐‘  58
  • 59. Luc_Faucheux_2021 Useful tools โ€“ A martingale is driftless, a driftless process is a martingale ยจ ๐”ผ[{๐‘‹(๐‘ก)|)|๐”‰(๐‘ )} = 0 ยจ ๐‘‘๐‘‹ ๐‘ก = ๐‘Ž ๐‘ก, ๐‘‹ ๐‘ก . ๐‘‘๐‘ก + ๐‘ ๐‘ก, ๐‘‹ ๐‘ก . ๐‘‘๐‘Š ยจ ๐‘Ž ๐‘ก, ๐‘‹ ๐‘ก = 0 ยจ ๐‘‘๐‘‹ ๐‘ก = ๐‘ ๐‘ก, ๐‘‹ ๐‘ก . ๐‘‘๐‘Š ยจ No advection, no drift for a martingale ยจ ๐‘‹ ๐‘ก = โˆซ !"# !"$ ๐‘ ๐‘ก, ๐‘‹ ๐‘ก . ๐‘‘๐‘Š ๐‘  ยจ Again the ITO integral is a martingale ยจ ๐”ผ โˆซ !"# !"$ ๐‘ ๐‘ก, ๐‘‹ ๐‘ก . ๐‘‘๐‘Š ๐‘  = 0 ยจ ๐”ผ[{๐‘‹(๐‘ก)|)|๐”‰(๐‘ )} = 0 59
  • 60. Luc_Faucheux_2021 Useful tools โ€“ useful relationship ยจ ๐”ผ exp โˆซ !"# !"$ ๐‘“ ๐‘  . ๐‘‘๐‘Š ๐‘  = exp[โˆซ !"# !"$ % & ๐‘“ ๐‘  &. ๐‘‘๐‘ ] 60
  • 61. Luc_Faucheux_2021 Useful tools โ€“ expected value of the exponential ยจ ๐”ผ exp ๐‘‹ = exp ๐”ผ ๐‘‹ . exp % & ๐”ผ (๐‘‹ โˆ’ ๐”ผ ๐‘‹ )& ยจ ๐‘€ ๐‘‹ ๐‘ก = ๐”ผ ๐‘‹ ยจ ๐‘‰ ๐‘‹ ๐‘ก = ๐”ผ (๐‘‹(๐‘ก) โˆ’ ๐‘€ ๐‘‹ ๐‘ก )& ยจ ๐”ผ exp ๐‘‹ = exp ๐‘€[๐‘‹(๐‘ก)] . exp % & ๐‘‰[๐‘‹(๐‘ก)] ยจ ๐”ผ exp ๐‘‹ = exp[๐‘€] . exp % & ๐‘‰ 61
  • 62. Luc_Faucheux_2021 Useful tools - Fubini ยจ ๐‘‹ = โˆซ !"$ !"$0 {โˆซ M"$ M"! ๐‘“(๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘  62 s ๐‘  = ๐‘ก! u s u ๐‘  = ๐‘ก ๐‘  = ๐‘ก! ๐‘  = ๐‘ก ๐‘‹ = X !"$ !"$0 { X M"$ M"! ๐‘“(๐‘ข). ([). ๐‘‘๐‘Š(๐‘ข)}. ๐‘‘๐‘  ๐‘‹ = X M"$ M"$0 { X !"M !"$0 ๐‘“(๐‘ ). ๐‘‘๐‘ }. ([). ๐‘‘๐‘Š(๐‘ข)
  • 63. Luc_Faucheux_2021 Useful tools โ€“ how to always create a martingale ยจ We use here the Tower property: ยจ For any process ๐‘‹ ๐‘ก , we create: ยจ ๐‘ ๐‘ก = ๐”ผ [ {๐‘‹(๐‘‡)|๐”‰(๐‘ก)} ยจ ๐”ผ [ ๐‘ ๐‘ก ๐”‰ ๐‘  = ๐”ผ [ ๐”ผ [ ๐‘‹ ๐‘‡ ๐”‰ ๐‘ก ๐”‰ ๐‘  = ๐”ผ [ ๐‘‹ ๐‘‡ ๐”‰ ๐‘  = ๐‘(๐‘ ) ยจ Because conditioning firstly on information back to time ๐‘ก then back to time ๐‘  is just the same as conditioning back to time ๐‘  to start with. ยจ ๐”ผ [ ๐‘ ๐‘ก ๐”‰ ๐‘  = ๐‘(๐‘ ) ยจ So ๐‘ ๐‘ก = ๐”ผ [ {๐‘‹(๐‘‡)|๐”‰(๐‘ก)} is by construction a ๐‘Š-martingale ยจ That is a neat little trick to always create a martingale process (Baxter p. 77) 63
  • 67. Luc_Faucheux_2021 A few good measures ยจ This section is a summary of some of the measures used in Finance, and their differences / notation 67
  • 68. Luc_Faucheux_2021 A few good measures โ€“ The Physical measure ยจ This is the โ€œnaturalโ€ measure ยจ It is usually noted โ„™ (I guess the P stands for Physical) ยจ Its characteristics (drift, variance) are usually calculated from historical data 68
  • 69. Luc_Faucheux_2021 A few good measures โ€“ The Risk-Neutral measure ยจ This one usually comes right after the Physical measure ยจ It is usually noted โ„š (I guess because in the alphabet Q comes right after P) ยจ In Finance its Numeraire is the rolling Bank account (MMN-Money Market Numeraire) ยจ As a first approximation (especially for equity derivative), the rates are assumed to be deterministic (non-stochastic) and even further sometimes constant in time: ยจ It is then usually noted as follows ยจ ๐‘‘โ„ณ ๐‘ก = ๐‘Ÿ. โ„ณ. ๐‘‘๐‘ก with: โ„ณ ๐‘ก = ๐‘’]$ ยจ As a further approximation (especially for short-dated options for which the discounting does not matter too much, or especially nowadays where rates are not moving and are fixed at 0 essentially due to the Central Banks Ponzi scheme, according to my good friend Bogac Ozdemir) ยจ ๐‘Ÿ = 0 69
  • 70. Luc_Faucheux_2021 A few good measures โ€“ The Risk-Neutral measure - II ยจ Essentially using the MMM as a numeraire, you just replace the drift by the risk-free rate ๐‘Ÿ ยจ If you had for a stock: ยจ ๐‘‘๐‘† ๐‘ก = ๐œ‡โ„™. ๐‘†. ๐‘‘๐‘ก + ๐œŽโ„™. ๐‘†. ๐‘‘๐‘Šโ„™ ยจ ๐‘‘โ„ณ ๐‘ก = ๐‘Ÿ. โ„ณ. ๐‘‘๐‘ก ยจ Then the ratio ^ โ„ณ will ALSO follows a geometric Brownian motion ยจ ๐‘‘ ^ โ„ณ = ๐œ‡โ„™ โˆ’ ๐‘Ÿ . ^ โ„ณ . ๐‘‘๐‘ก + ๐œŽโ„™. ^ โ„ณ . ๐‘‘๐‘Šโ„™ ยจ The ratio ^ โ„ณ is a martingale under the risk-free measure, it is thus driftless ยจ ๐‘‘ ^ โ„ณ = 0. ๐‘‘๐‘ก + {๐‘ ๐‘œ๐‘š๐‘’๐‘กโ„Ž๐‘–๐‘›๐‘”}. ๐‘‘๐‘Šโ„š 70
  • 71. Luc_Faucheux_2021 A few good measures โ€“ The Risk-Neutral measure - IIa ยจ ๐‘‘ ^ โ„ณ = ๐œ‡โ„™ โˆ’ ๐‘Ÿ . ^ โ„ณ . ๐‘‘๐‘ก + ๐œŽโ„™. ^ โ„ณ . ๐‘‘๐‘Šโ„™ ยจ ๐‘‘ ^ โ„ณ = 0. ๐‘‘๐‘ก + {๐‘ ๐‘œ๐‘š๐‘’๐‘กโ„Ž๐‘–๐‘›๐‘”}. ๐‘‘๐‘Šโ„š ยจ In this case, we see that we can define the โ„š-Brownian motion as: ยจ ๐‘‘๐‘Šโ„š = ๐‘‘๐‘Šโ„™ โˆ’ `โ„™S] Rโ„™ . ๐‘‘๐‘ก ยจ We then get: ยจ ๐‘‘ ^ โ„ณ = 0. ๐‘‘๐‘ก + ๐œŽโ„™. ^ โ„ณ . ๐‘‘๐‘Šโ„š ยจ We also see our old friend the โ€œmarket price of riskโ€, the excess return over the risk free rate, normalized by the volatility of the asset. ยจ ๐œ† = `โ„™S] Rโ„™ 71
  • 72. Luc_Faucheux_2021 A few good measures โ€“ The Risk-Neutral measure - IIb 72
  • 73. Luc_Faucheux_2021 A few good measures โ€“ The Risk-Neutral measure - III ยจ Things become a little more complicated once we assume that rates are stochastic ยจ The usual notation becomes then: ยจ ๐ต ๐‘ก = exp(โˆซ !"# !"$ ๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ) ยจ In the extended Zeros framework of Mercurio and Lyashenko: ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, 0 = exp โˆซ !"# !"$ ๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘  = ๐ต ๐‘ก ยจ The rolling Bank Account has the useful property that: ยจ ๐ต 0 = exp(โˆซ !"# !"# ๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ) = exp 0 = 1 ยจ This is useful when valuing claims and derivatives 73
  • 74. Luc_Faucheux_2021 A few good measures โ€“ The Risk-Neutral measure - IV ยจ For example ยจ a #,$%,$,$ N(#) = ๐”ผ$ โ„š a $,$%,$,$ N($) |๐”‰(0) = ๐”ผ$ โ„š a $,$%,$,$ cde(โˆซ <=> <=? O !,!,! .D!) |๐”‰(0) ยจ a #,$%,$,$ N(#) = a #,$%,$,$ % = ๐‘๐ถ 0,0, ๐‘ก = ๐”ผ$ โ„š a $,$%,$,$ cde(โˆซ <=> <=? O !,!,! .D!) |๐”‰(0) ยจ ๐‘๐ถ 0,0, ๐‘ก = ๐”ผ$ โ„š a $,$%,$,$ cde(โˆซ <=> <=? O !,!,! .D!) |๐”‰(0) = ๐”ผ$ โ„š % cde(โˆซ <=> <=? O !,!,! .D!) |๐”‰(0) ยจ ๐‘๐ถ 0,0, ๐‘ก = ๐”ผ$ โ„š exp(โˆ’ โˆซ !"# !"$ ๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ )|๐”‰(0) ยจ This is usually used when calibrating a model to the current time ๐‘ก = 0 term structure of Zero Coupon bond prices 74
  • 75. Luc_Faucheux_2021 A few good measures โ€“ The Risk-Neutral measure - V ยจ Similarly as we go further in time: ยจ ๐‘๐ถ 0,0, ๐‘ก = PQ #,#,$ N(#) = o ๐‘ 0,0, ๐‘ก = ๐”ผ$ โ„š exp(โˆ’ โˆซ !"# !"$ ๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ )|๐”‰(0) ยจ PQ #,#,$@ N(#) = o ๐‘ 0,0, ๐‘ก/ = ๐”ผ$ โ„š PQ $,$,$@ N($) |๐”‰(0) = ๐”ผ$ โ„š o ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ |๐”‰(0) ยจ PQ $,$,$@ N($) = o ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ = ๐”ผ$0 โ„š PQ $0,$0,$@ N($0) |๐”‰(๐‘ก) = ๐”ผ$0 โ„š o ๐‘ ๐‘ก., ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก) for: ๐‘ก < ๐‘ก.< ๐‘ก/ ยจ In particular for: ๐‘ก. = ๐‘ก/ ยจ PQ $,$,$@ N($) = o ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ = ๐”ผ$@ โ„š PQ $@,$@,$@ N($@) |๐”‰(๐‘ก) = ๐”ผ$@ โ„š o ๐‘ ๐‘ก/, ๐‘ก/, ๐‘ก/ |๐”‰(๐‘ก) = ๐”ผ$@ โ„š % N($@) |๐”‰(๐‘ก) ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก/ = ๐ต ๐‘ก . ๐”ผ$@ โ„š % N($@) |๐”‰(๐‘ก) = ๐”ผ$@ โ„š ๐ต ๐‘ก . % N($@) |๐”‰(๐‘ก) = ๐”ผ$@ โ„š N $ N($@) |๐”‰(๐‘ก) 75
  • 76. Luc_Faucheux_2021 A few good measures โ€“ The Risk-Neutral measure - VI ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก/ = ๐”ผ$@ โ„š N $ N($@) |๐”‰(๐‘ก) ยจ ๐ต ๐‘ก = exp(โˆซ !"# !"$ ๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ) ยจ ๐ต ๐‘ก/ = exp(โˆซ !"# !"$@ ๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ) ยจ A $ A($#) = exp(โˆซ !"# !"$ ๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ) /exp(โˆซ !"# !"$# ๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ) = exp(โˆซ !"# !"$ ๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘  โˆ’ โˆซ !"# !"$# ๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ )) ยจ โˆซ !"# !"$# ๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ) = โˆซ !"# !"$ ๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ) + โˆซ !"$ !"$# ๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ) ยจ A $ A($#) = exp(โˆ’ โˆซ !"$ !"$# ๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ) ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก- = ๐”ผ$# โ„š exp[โˆ’ โˆซ !"$ !"$# ๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ]|๐”‰(๐‘ก) ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก- = ๐”ผ$# โ„š ๐‘’D โˆซ&'( &'(# F !,!,! .H! |๐”‰(๐‘ก) 76
  • 77. Luc_Faucheux_2021 A few good measures โ€“ The Risk-Neutral measure - VII ยจ Similar to the SDE for stocks, the drift for a tradeable security in the Risk neutral measure is the instantaneous short rate ๐‘… ๐‘ , ๐‘ , ๐‘  ยจ DPQ $,$,$0 PQ $,$,$0 = ๐‘… ๐‘ก, ๐‘ก, ๐‘ก . ๐‘‘๐‘ก + ๐‘‰ ๐‘ก, ๐‘ก., ๐‘ก. . ([). ๐‘‘๐‘Šโ„š ๐‘ก ยจ D f P $,$,$0 f P $,$,$0 = ๐‘‰ ๐‘ก, ๐‘ก., ๐‘ก. . ([). ๐‘‘๐‘Šโ„š ๐‘ก ยจ This is a driftless process. In particular: ยจ ๐‘‘ o ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. = o ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐‘‰ ๐‘ก, ๐‘ก., ๐‘ก. . ([). ๐‘‘๐‘Šโ„š ๐‘ก ยจ This is also a driftless process, hence the deflated Zeros are martingale under the risk- neutral โ„š measure ยจ ๐‘‘๐ต ๐‘ก = ๐‘… ๐‘ก, ๐‘ก, ๐‘ก . ๐ต ๐‘ก . ๐‘‘๐‘ก or ๐ต ๐‘ก = exp(โˆซ !"# !"$ ๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ) ยจ p ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = o ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. = PQ $,$,$0 N($) = P $,$,$0 N($) 77
  • 78. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure ยจ This is the measure where the numeraire is the Zero ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก/ ยจ Also referred to in textbooks as the ๐‘ก/-Terminal or sometimes ๐‘‡/-terminal ยจ It is also called the Forward measure because under this measure the Forward rate (simply compounded, not every forward rate!) spanning a period [๐‘ก., ๐‘ก/] is a martingale. ยจ Not super easy to convince yourself of, so worth looking at it again (it was a while since we did it, was in deck II and III) ยจ Also worth redoing it with the full notation that we have slowly developed as we went along ยจ This hopefully will be rigorous enough to stand the test of reading it back later on. ยจ Just to be on the safe side, we will write it: ยจ ๐”ผ$@ โ„ค($@) ยจ It is usually when the claims get PAID (the early one is when the claim gets SET or FIXED) 78
  • 79. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - II ยจ We have to do a little refresher on the notation (because remember unlike in Physics, what matters really in Finance is WHEN you get paid, not when you observe/fix/set the payment) ยจ ๐‘‰(๐‘ก) = ๐‘‰ ๐‘ก, $๐ป(๐‘ก), ๐‘ก., ๐‘ก/ 79 ๐‘ƒ๐‘Ž๐‘–๐‘‘ ๐‘Ž๐‘ก ๐‘ก๐‘–๐‘š๐‘’ ๐‘ก/ ๐น๐‘–๐‘ฅ๐‘’๐‘‘ ๐‘œ๐‘Ÿ ๐‘ ๐‘’๐‘ก ๐‘Ž๐‘ก ๐‘ก๐‘–๐‘š๐‘’ ๐‘ก. ๐บ๐‘’๐‘›๐‘’๐‘Ÿ๐‘Ž๐‘™ ๐‘ƒ๐‘Ž๐‘ฆ๐‘œ๐‘“๐‘“ ๐ป ๐‘ก ๐‘–๐‘› ๐‘๐‘ข๐‘Ÿ๐‘Ÿ๐‘’๐‘›๐‘๐‘ฆ $ ๐‘‰๐‘Ž๐‘™๐‘ข๐‘’ ๐‘œ๐‘“ ๐‘กโ„Ž๐‘’ ๐‘๐‘Ž๐‘ฆ๐‘œ๐‘“๐‘“ ๐‘œ๐‘๐‘ ๐‘’๐‘Ÿ๐‘ฃ๐‘’๐‘‘ ๐‘Ž๐‘ก ๐‘ก๐‘–๐‘š๐‘’ ๐‘ก
  • 80. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - III ยจ ๐‘‰(๐‘ก) = ๐‘‰ ๐‘ก, $๐ป(๐‘ก), ๐‘ก., ๐‘ก/ ยจ Of paramount importance is the payoff that ALWAYS pays $1 ยจ $๐ป ๐‘ก = $1 ยจ a $,$h($),$0,$@ P($,$,$@) is a martingale under ๐”ผ$@ โ„ค($@) ยจ a $,$h($),$0,$@ P($,$,$@) = ๐”ผ$@ โ„ค($@) a $@,$h $@ ,$0,$@ P $@,$@,$@ ๐”‰ ๐‘ก = ๐”ผ$@ โ„ค($@) ๐‘‰ ๐‘ก/, $๐ป ๐‘ก/ , ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก ยจ Because in a very convenient fashion: ยจ ๐‘ ๐‘ก/, ๐‘ก/, ๐‘ก/ = 1 ยจ For $๐ป ๐‘ก = $1 ยจ a $,$%,$0,$@ P($,$,$@) = ๐”ผ$@ โ„ค($@) a $@,$%,$0,$@ P $@,$@,$@ ๐”‰ ๐‘ก = ๐”ผ$@ โ„ค($@) ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = $1 80
  • 81. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - IV ยจ a $,$%,$0,$@ P($,$,$@) = ๐”ผ$@ โ„ค($@) a $@,$%,$0,$@ P $@,$@,$@ ๐”‰ ๐‘ก = ๐”ผ$@ โ„ค($@) ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = $1 ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก, ๐‘ก/) ยจ Note that this result is independent of the measure, and validates our intuition about the Zeros ยจ All right, letโ€™s nest those bad boys so that we recover the bootstrap relations 81
  • 82. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - V ยจ a $,$%,$0,$@ P($,$,$@) = ๐”ผ$@ โ„ค($@) a $@,$%,$0,$@ P $@,$@,$@ ๐”‰ ๐‘ก so: ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก, ๐‘ก/) ยจ a $,$%,$0,$0 P($,$,$0) = ๐”ผ$0 โ„ค($0) a $0,$%,$0,$0 P $0,$0,$0 ๐”‰ ๐‘ก so: ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก. = ๐‘(๐‘ก, ๐‘ก, ๐‘ก.) ยจ Note that the โ€œsettingโ€ or โ€œfixingโ€ time for a claim that always pays $1 is irrelevant, but letโ€™s keep it for now, we did not go through a couple of thousand slides of building a rigorous formalism to throw it all away now. ยจ a $,$%,$0,$@ P($,$,$0) = ๐”ผ$0 โ„ค($0) a $0,$%,$0,$@ P $0,$0,$0 ๐”‰ ๐‘ก ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐”ผ$0 โ„ค($0) a $0,$%,$0,$@ P $0,$0,$0 ๐”‰ ๐‘ก = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก 82
  • 83. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - VI ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก, ๐‘ก/) ยจ From the bootstrap relations we have by definition: ยจ ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐‘(๐‘ก, ๐‘ก., ๐‘ก/) ยจ So we have: ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐‘(๐‘ก, ๐‘ก., ๐‘ก/) ยจ Which leads us quite naturally to: ยจ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก 83
  • 84. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - VII ยจ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก ยจ That is where the nesting comes into the game: ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก, ๐‘ก/) ยจ And so by just changing the variables: ยจ ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ยจ Again when you read it aloud it makes sense: The value at time ๐‘ก. of a claim that will pay $1 no matter what at time ๐‘ก/ is equal to the value of the Zero Coupon Bond at time ๐‘ก. that pays $1 at time ๐‘ก/. ยจ That is almost a tautology, but we can nest that one into the first equation: ยจ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = ๐”ผ$0 โ„ค($0) ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก 84
  • 85. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - VIII ยจ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = ๐”ผ$0 โ„ค($0) ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก ยจ I know that we are mixing a little the Terminal/Forward measure ๐”ผ$@ โ„ค($@) and the early/discount measure ๐”ผ$0 โ„ค($0) but they are exactly the same just with a different end time for the expectation and the numeraire, but the above relationship is quite cool. ยจ In the early/discount measure ๐”ผ$0 โ„ค($0) , the Zeros are a martingale. And by Zeros we mean the ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ , so not every Zeros, careful about that. ยจ So the process of those guys will be driftless: ยจ ๐‘‘๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = 0. ๐‘‘๐‘ก + ๐‘ ๐‘œ๐‘š๐‘’๐‘กโ„Ž๐‘–๐‘›๐‘” . ๐‘‘๐‘Š $0 โ„ค $0 (๐‘ก) ยจ Where ๐‘Š $0 โ„ค $0 (๐‘ก) is the Brownian motion associated to the early measure ๐”ผ$0 โ„ค($0) 85
  • 86. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - IX ยจ OK, so we are like a fifth of the way there, so grab a coke and some popcorn ยจ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = ๐”ผ$0 โ„ค($0) ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก ยจ Now we did define the simply compounded Forward Rates as: ยจ ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ = % i $,$0,$@ . [ % PQ $,$0,$@ โˆ’ 1] ยจ Where ๐œ ๐‘ก, ๐‘ก., ๐‘ก/ is the daycount fraction that for sake of simplicity we will note ๐œ in this section. ยจ ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ = % i . [ % PQ $,$0,$@ โˆ’ 1] ยจ ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ = % %+i.j $,$0,$@ 86
  • 87. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - X ยจ Remember that we could have defined a number of other rates: ยจ Continuously compounded FORWARD : ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ = exp โˆ’๐‘… ๐‘ก, ๐‘ก., ๐‘ก/ . ๐œ ๐‘ก, ๐‘ก., ๐‘ก/ ยจ Simply compounded FORWARD: ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ = % %+i $,$0,$@ .j $,$0,$@ ยจ Annually compounded FORWARD : ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ = % (%+W $,$0,$@ ) I ?,?0,?@ ยจ ๐‘ž-times per year compounded FOWARD: ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ = % (%+ 2 J .WJ $,$0,$@ ) J.I ?,?0,?@ ยจ The function ๐œ ๐‘ก, ๐‘ก., ๐‘ก/ is the daycount fraction, will usually depends on what convention (ACT/ACT, ACT/360, 30/360,โ€ฆ) you will choose, and potentially adjustment for holidays and what holiday center 87
  • 88. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - XI ยจ We are looking at the usual graph we had in section II and III 88 ๐‘ก ๐‘ก๐‘–๐‘š๐‘’ ๐‘™ ๐‘ก, ๐‘ก., ๐‘ก/ . ๐œ ๐‘ก, ๐‘ก., ๐‘ก/ ๐‘ก! = ๐‘ก ๐‘ก๐‘–๐‘š๐‘’ ๐‘ก! ๐‘ก" $1 $1 ๐‘ก"
  • 89. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - XII ยจ Letโ€™s construct a claim on a portfolio that pays $1 at time ๐‘ก. and pays back $1 at time ๐‘ก/ ยจ Note, by now that specific portfolio should not come as a surprise ยจ Letโ€™s note ฮ (๐‘ก) the value at time ๐‘ก of this portfolio ยจ ฮ  ๐‘ก = ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก. โˆ’ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ ยจ ฮ  ๐‘ก = ๐‘(๐‘ก, ๐‘ก, ๐‘ก.) โˆ’ ๐‘(๐‘ก, ๐‘ก, ๐‘ก/) ยจ ฮ  ๐‘ก = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ . { P $,$,$0 P($,$,$@) โˆ’ 1} ยจ ฮ  ๐‘ก = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ . { % P($,$0,$@) โˆ’ 1} ยจ ฮ  ๐‘ก = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ . 1 + ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ โˆ’ 1 = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ . ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ ยจ OK, keep that on the back of your minds for just a couple of slides 89
  • 90. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - XIII ยจ Because that claim is tradeable, the ratio of it to the numeraire is a martingale in the terminal measure: ยจ k $ P($,$,$@) = ๐”ผ$@ โ„ค($@) k $@ P $@,$@,$@ ๐”‰ ๐‘ก = ๐”ผ$@ โ„ค($@) ฮ  ๐‘ก/ ๐”‰ ๐‘ก ยจ ฮ  ๐‘ก = ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก. โˆ’ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ ยจ ฮ  ๐‘ก/ = ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. โˆ’ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก/ ยจ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก/ is the value at time ๐‘ก/ of a claim that pays $1 at time ๐‘ก/ ยจ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก/ = 1 ยจ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. is the value at time ๐‘ก/ of a claim that PAID $1 at time ๐‘ก. ยจ ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) is the value at time ๐‘ก. of a Zero coupon bond that pays $1 at time ๐‘ก/ 90
  • 91. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - XIV ยจ Letโ€™s make sure that we get this right: ยจ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. is the value at time ๐‘ก/ of a claim that PAID $1 at time ๐‘ก. ยจ ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) is the value at time ๐‘ก. of a Zero coupon bond that pays $1 at time ๐‘ก/ ยจ Letโ€™s say it another way ยจ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. is the value at time ๐‘ก/ of a claim that PAID $1 at time ๐‘ก. ยจ If you invested $1 at time ๐‘ก. you will receive ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. at time ๐‘ก/ ยจ ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) is the value at time ๐‘ก. of a Zero coupon bond that pays $1 at time ๐‘ก/ ยจ If you invested ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) at time ๐‘ก. you will receive $1 at time ๐‘ก/ 91
  • 92. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - XV ยจ If you invested $1 at time ๐‘ก. you will receive ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. at time ๐‘ก/ ยจ If you invested ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) at time ๐‘ก. you will receive $1 at time ๐‘ก/ ยจ And so: ยจ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. = % P($0,$0,$@) ยจ Really make sure that you are 100% convinced on that one. ยจ Just to be sure, letโ€™s break it down in the next slide: 92
  • 93. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - XVI ยจ If you invested $1 at time ๐‘ก. you will receive ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. at time ๐‘ก/ ยจ If you invested ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) at time ๐‘ก. you will receive $1 at time ๐‘ก/ ยจ If you invested ๐‘ ๐‘ก., ๐‘ก., ๐‘ก/ . { % P $0,$0,$@ } at time ๐‘ก. you will receive $1. { % P $0,$0,$@ } at time ๐‘ก/ ยจ If you invested { P $0,$0,$@ P $0,$0,$@ } at time ๐‘ก. you will receive $1. { % P $0,$0,$@ } at time ๐‘ก/ ยจ If you invested {$1} at time ๐‘ก. you will receive $1. { % P $0,$0,$@ } at time ๐‘ก/ ยจ But we also have the first relation: ยจ If you invested $1 at time ๐‘ก. you will receive ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. at time ๐‘ก/ ยจ And so: ยจ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. = % P $0,$0,$@ 93
  • 94. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - XVII ยจ Ok, we are 3 fifths of the way there ยจ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. = % P $0,$0,$@ ยจ But remember ! This is really saying: ยจ If you invested {$1} at time ๐‘ก. you will receive $1. { % P $0,$0,$@ } at time ๐’•๐’‹ ยจ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. = ๐‘‰(๐‘ก/, % P $0,$0,$@ , ๐‘ก., ๐‘ก/) ยจ The value at time ๐‘ก/ of receiving $1 that was PAID at time ๐‘ก., is equal to the value at time ๐‘ก/ of receiving $ % P $0,$0,$@ that was set at time ๐‘ก. and paid at time ๐‘ก/ 94
  • 95. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - XVIII ยจ Which we now plug back into: ยจ k $ P($,$,$@) = ๐”ผ$@ โ„ค($@) k $@ P $@,$@,$@ ๐”‰ ๐‘ก = ๐”ผ$@ โ„ค($@) ฮ  ๐‘ก/ ๐”‰ ๐‘ก ยจ ฮ  ๐‘ก = ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก. โˆ’ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ ยจ ฮ  ๐‘ก/ = ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. โˆ’ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก/ ยจ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก/ = $1 ยจ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. = ๐‘‰(๐‘ก/, % P $0,$0,$@ , ๐‘ก., ๐‘ก/) 95
  • 96. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - XIX ยจ We now go back to the bootstrap definition ยจ % P $0,$0,$@ = 1 + ๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ ยจ ฮ  ๐‘ก/ = ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. โˆ’ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก/ ยจ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก/ = $1 ยจ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. = ๐‘‰(๐‘ก/, $ % P $0,$0,$@ , ๐‘ก., ๐‘ก/) ยจ ฮ  ๐‘ก/ = ๐‘‰ ๐‘ก/, $ % P $0,$0,$@ , ๐‘ก., ๐‘ก/ โˆ’ $1 ยจ ฮ  ๐‘ก/ = ๐‘‰(๐‘ก/, $ % P $0,$0,$@ โˆ’ $1, ๐‘ก., ๐‘ก/) 96
  • 97. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - XX ยจ ฮ  ๐‘ก/ = ๐‘‰(๐‘ก/, $ % P $0,$0,$@ โˆ’ $1, ๐‘ก., ๐‘ก/) ยจ ฮ  ๐‘ก/ = ๐‘‰(๐‘ก/, $(1 + ๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ ) โˆ’ $1, ๐‘ก., ๐‘ก/) ยจ ฮ  ๐‘ก/ = ๐‘‰(๐‘ก/, $(1 + ๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ โˆ’ 1), ๐‘ก., ๐‘ก/) ยจ ฮ  ๐‘ก/ = ๐‘‰(๐‘ก/, $(๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ ), ๐‘ก., ๐‘ก/) ยจ ฮ  ๐‘ก/ = ๐‘‰(๐‘ก/, $๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/) ยจ ฮ  ๐‘ก/ = ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. โˆ’ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก/ ยจ That is essentially the intuition that we had built and illustrated in the graph 97
  • 98. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure โ€“ XXa ยจ ฮ  ๐‘ก = ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก. โˆ’ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ ยจ ฮ  ๐‘ก = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ . ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ ยจ ฮ  ๐‘ก/ = ๐‘‰(๐‘ก/, $๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/) ยจ ฮ  ๐‘ก/ = ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. โˆ’ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก/ ยจ ฮ  ๐‘ก = ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก. โˆ’ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ ยจ ฮ  ๐‘ก = ๐‘‰(๐‘ก, $๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/) 98
  • 99. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure โ€“ XXb ยจ ฮ  ๐‘ก = ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก. โˆ’ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ 99 ๐‘ก ๐‘ก๐‘–๐‘š๐‘’ ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ . ๐œ ๐‘ก, ๐‘ก., ๐‘ก/ ๐‘ก! = ๐‘ก ๐‘ก๐‘–๐‘š๐‘’ ๐‘ก! ๐‘ก" $1 $1 ๐‘ก"
  • 100. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure โ€“ XXc ยจ ฮ  ๐‘ก = ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก. โˆ’ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ ยจ ฮ  ๐‘ก = ๐‘‰(๐‘ก, $๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/) ยจ Receiving $1 at time ๐‘ก. and paying it back at time ๐‘ก/ is equivalent to: ยจ Receiving at time ๐‘ก/ the simply compounded forward rate ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ covering the period [๐‘ก., ๐‘ก/], multiplied by the appropriate daycount fraction. This forward rate sets at time ๐‘ก.. ยจ Simply compounded FORWARD: ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ = % %+i $,$0,$@ .j $,$0,$@ ยจ ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ = % %+i.j $,$0,$@ ยจ ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ = % PQ $,$0,$@ โˆ’ 1 100
  • 101. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure โ€“ XXd ยจ ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ = % PQ $,$0,$@ โˆ’ 1 before it sets for all time ๐‘ก < ๐‘ก. ยจ ๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ = % PQ $0,$0,$@ โˆ’ 1 when it sets at time ๐‘ก. ยจ ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ for all time after it did set ๐‘ก โ‰ฅ ๐‘ก. 101
  • 102. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - XXI ยจ ฮ  ๐‘ก/ = ๐‘‰(๐‘ก/, $๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/) ยจ ฮ  ๐‘ก/ = ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก. โˆ’ ๐‘‰ ๐‘ก/, $1, ๐‘ก., ๐‘ก/ ยจ The value at time ๐‘ก/ of a portfolio that consists of receiving $1 at time ๐‘ก. and paying it back at time ๐‘ก/ is the same value at time ๐‘ก/ of a portfolio paying at time ๐‘ก/ the simply compounded rate (times the daycount fraction), set at time ๐‘ก., and spanning the period [๐‘ก., ๐‘ก/] ยจ All right, past the halfway point: ยจ ฮ  ๐‘ก/ = ๐‘‰(๐‘ก/, $๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/) ยจ We also know that: ยจ k $ P($,$,$@) = ๐”ผ$@ โ„ค($@) k $@ P $@,$@,$@ ๐”‰ ๐‘ก = ๐”ผ$@ โ„ค($@) ฮ  ๐‘ก/ ๐”‰ ๐‘ก ยจ ฮ  ๐‘ก = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ . 1 + ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ โˆ’ 1 = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ . ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ 102
  • 103. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - XXII ยจ Letโ€™s put those together: ยจ k $ P($,$,$@) = ๐”ผ$@ โ„ค($@) k $@ P $@,$@,$@ ๐”‰ ๐‘ก = ๐”ผ$@ โ„ค($@) ฮ  ๐‘ก/ ๐”‰ ๐‘ก ยจ ฮ  ๐‘ก = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ . 1 + ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ โˆ’ 1 = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ . ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ ยจ k $ P($,$,$@) = P $,$,$@ .i.j $,$0,$@ P($,$,$@) = ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ ยจ ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$@ โ„ค($@) ฮ  ๐‘ก/ ๐”‰ ๐‘ก ยจ ฮ  ๐‘ก/ = ๐‘‰(๐‘ก/, $๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/) ยจ ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$@ โ„ค($@) ๐‘‰(๐‘ก/, $๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก 103
  • 104. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - XXIII ยจ ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$@ โ„ค($@) ๐‘‰(๐‘ก/, $๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก ยจ Letโ€™s drop the daycount fraction ยจ ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$@ โ„ค($@) ๐‘‰(๐‘ก/, $๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก ยจ We are almost tempted to say that the simply compounded forward rate is a martingale under the terminal measure ยจ Not quite ยจ What the above says is that the expectation under the ๐‘ก/-terminal measure of a claim that pays at time ๐‘ก/ the value of the simply compounded forward ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ set at time ๐‘ก. is the current value of the simply compounded forward spanning the period [๐‘ก., ๐‘ก/] ยจ We almost there, but before we perform the final step, letโ€™s take a small detour through swaps valuation 104
  • 105. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - XXIV ยจ ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$@ โ„ค($@) ๐‘‰(๐‘ก/, $๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก ยจ ๐”ผ$@ โ„ค($@) ๐‘‰(๐‘ก/, $๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก = ๐”ผ$@ โ„ค($@) a($@,$i.j $0,$0,$@ ,$0,$@) P($@,$@,$@) ๐”‰ ๐‘ก ยจ ๐”ผ$@ โ„ค($@) a($@,$i.j $0,$0,$@ ,$0,$@) P($@,$@,$@) ๐”‰ ๐‘ก = a($,$i.j $0,$0,$@ ,$0,$@) P($,$,$@) = ๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ ยจ ๐‘‰ ๐‘ก, $๐œ. ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ . {๐œ. ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ } ยจ This is why we can value swaps without a volatility curve just using the yield curve and nothing else (remember for REGULAR swaps) ยจ A swaplet pays at time ๐‘ก/ (end of the period) a Libor rate set at time ๐‘ก. (beginning of the period) times the appropriate daycount fraction? Boom, the current value of that swaplet is the current value of that simply compounded forward rate ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ times the daycount fraction time the discount factor observed in the current discount curve between now (time ๐‘ก) and the payment date ๐‘ก/, that discount factor is ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ 105
  • 106. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - XXV ยจ So this is really what underpins the valuation of all swaps (REGULAR, no funny business about paying and setting at different dates that the ones wee just talked about!), fixed cash flows of course and all that. ยจ Fairly cool right ? ยจ Almost there about why this is called the forward measure. 106
  • 107. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - XXVI ยจ ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$@ โ„ค($@) ๐‘‰(๐‘ก/, $๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก ยจ We know take the limit ๐‘ก/ โ†’ ๐‘ก. in order to recover the usual instantaneous forward ยจ Just for sake of completeness letโ€™s refresh our knowledge from deck V-a 107
  • 108. Luc_Faucheux_2021 Another summary - XVII ยจ From the variable ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ , we are absolutely free to define a bunch of other variables, and we certainly did not deprive ourselves of doing so: ยจ Continuously compounded FORWARD : ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ = exp โˆ’๐‘… ๐‘ก, ๐‘ก., ๐‘ก/ . ๐œ ๐‘ก, ๐‘ก., ๐‘ก/ ยจ Simply compounded FORWARD: ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ = % %+i $,$0,$@ .j $,$0,$@ ยจ Annually compounded FORWARD : ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ = % (%+W $,$0,$@ ) I ?,?0,?@ ยจ ๐‘ž-times per year compounded FOWARD: ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ = % (%+ 2 J .WJ $,$0,$@ ) J.I ?,?0,?@ ยจ The function ๐œ ๐‘ก, ๐‘ก., ๐‘ก/ is the daycount fraction, will usually depends on what convention (ACT/ACT, ACT/360, 30/360,โ€ฆ) you will choose, and potentially adjustment for holidays and what holiday center 108
  • 109. Luc_Faucheux_2021 Another summary - XVIII ยจ In the small ๐œ ๐‘ก, ๐‘ก., ๐‘ก/ โ†’ 0 limit (also if the rates themselves are such that they are <<1) ยจ In bootstrap form which is the intuitive way: ยจ Continuously compounded spot: ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ = 1 โˆ’ ๐‘… ๐‘ก, ๐‘ก., ๐‘ก/ . ๐œ + ๐’ช(๐œ&. ๐‘…&) ยจ Simply compounded spot: ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ = 1 โˆ’ ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ . ๐œ + ๐’ช(๐œ&. ๐‘™&) ยจ Annually compounded spot: ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ = 1 โˆ’ ๐‘Œ ๐‘ก, ๐‘ก., ๐‘ก/ . ๐œ + ๐’ช(๐œ&. ๐‘ฆ&) ยจ ๐‘ž-times per year compounded spot ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ = 1 โˆ’ ๐‘Œm ๐‘ก, ๐‘ก., ๐‘ก/ . ๐œ + ๐’ช(๐œ&. ๐‘ฆm &) ยจ So in the limit of small ๐œ ๐‘ก, ๐‘ก., ๐‘ก/ , (and also small rates), in particular when: ๐‘ก/ โ†’ ๐‘ก., all rates converge to the same limit we call ยจ ๐ฟ๐‘–๐‘š ๐‘ก/ โ†’ ๐‘ก. = lim $@โ†’$0 ( %SPQ $,$0,$@ i $,$0,$@ ) that we will note Instantaneous Forward Rate 109
  • 110. Luc_Faucheux_2021 Another summary - XIX ยจ ๐ฟ๐‘–๐‘š ๐‘ก/ โ†’ ๐‘ก. = lim $@โ†’$0 ( %SPQ $,$0,$@ i $,$0,$@ ) that we will note Instantaneous Forward Rate ยจ ๐ผ๐น๐‘ค๐‘… ๐‘ก, ๐‘ก., ๐‘ก/ = ๐ผ๐น๐‘ค๐‘… ๐‘ก, ๐‘ก., ๐‘ก.+ = ๐ผ๐น๐‘ค๐‘… ๐‘ก, ๐‘ก. = lim $@โ†’$0 ( %SPQ $,$0,$@ i $,$0,$@ ) ยจ In the small ๐œ ๐‘ก, ๐‘ก., ๐‘ก/ limit, (and also small rates) since really what matters is how small the product of the defined rate by the daycount fraction, ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ is close to 1. ยจ ๐ฟ๐‘–๐‘š ๐‘ก/ โ†’ ๐‘ก. = lim $@โ†’$0 ( %SPQ $,$0,$@ i $,$0,$@ ) = lim $@โ†’$0 ( Sno(PQ $,$0,$@ ) i $,$0,$@ ) ยจ Usually most textbooks will assume without explicitly telling you that in that limit we will also have: ยจ lim $@โ†’$0 (๐œ ๐‘ก, ๐‘ก., ๐‘ก/ ) = (๐‘ก/ โˆ’ ๐‘ก.), so that ๐ฟ๐‘–๐‘š ๐‘ก/ โ†’ ๐‘ก. = lim $@โ†’$0 S no PQ $,$0,$@ i $,$0,$@ 110
  • 111. Luc_Faucheux_2021 Another summary - XX ยจ ๐ผ๐น๐‘ค๐‘… ๐‘ก, ๐‘ก., ๐‘ก/ = ๐ผ๐น๐‘ค๐‘… ๐‘ก, ๐‘ก., ๐‘ก.+ = ๐ผ๐น๐‘ค๐‘… ๐‘ก, ๐‘ก. = lim $@โ†’$0 ( %SPQ $,$0,$@ i $,$0,$@ ) ยจ ๐ผ๐น๐‘ค๐‘… ๐‘ก, ๐‘ก. = lim $@โ†’$0 ( %SPQ $,$0,$@ i $,$0,$@ ) = lim $@โ†’$0 (๐‘… ๐‘ก, ๐‘ก., ๐‘ก/ ) = ๐‘… ๐‘ก, ๐‘ก., ๐‘ก. ยจ ๐ผ๐น๐‘ค๐‘… ๐‘ก, ๐‘ก. = lim $@โ†’$0 ( %SPQ $,$0,$@ i $,$0,$@ ) = lim $@โ†’$0 (๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ ) = ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก. ยจ ๐ผ๐น๐‘ค๐‘… ๐‘ก, ๐‘ก. = lim $@โ†’$0 ( %SPQ $,$0,$@ i $,$0,$@ ) = lim $@โ†’$0 (๐‘Œ ๐‘ก, ๐‘ก., ๐‘ก/ ) = ๐‘Œ ๐‘ก, ๐‘ก., ๐‘ก. ยจ ๐ผ๐น๐‘ค๐‘… ๐‘ก, ๐‘ก. = lim $@โ†’$0 ( %SPQ $,$0,$@ i $,$0,$@ ) = lim $@โ†’$0 (๐‘Œm ๐‘ก, ๐‘ก., ๐‘ก/ ) = ๐‘Œm ๐‘ก, ๐‘ก., ๐‘ก. 111
  • 112. Luc_Faucheux_2021 Another summary - XXI ยจ ๐‘… ๐‘ก, ๐‘ก., ๐‘ก. = ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก. = ๐‘Œ ๐‘ก, ๐‘ก., ๐‘ก. = ๐‘Œm ๐‘ก, ๐‘ก., ๐‘ก. = ๐‘“(๐‘ก, ๐‘ก.) as per the notation in most textbooks ยจ lim $@โ†’$0 ( %SPQ $,$0,$@ i $,$0,$@ ) = lim $@โ†’$0 ( Sno(PQ $,$0,$@ ) i $,$0,$@ ) ยจ From bootstrap: ยจ ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก/ /๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. ยจ ln(๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ = ln(๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก/ โˆ’ ln(๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. ยจ lim $@โ†’$0 ( Sno(PQ $,$0,$@ ) i $,$0,$@ ) = โˆ’ lim $@โ†’$0 no(PQ $,$,$@ Sno(PQ $,$,$0 i $,$0,$@ = โˆ’ lim $@โ†’$0 ( no(PQ $,$,$@ Sno(PQ $,$,$0 $@ S $0 ) ยจ lim $@โ†’$0 ( Sno(PQ $,$0,$@ ) i $,$0,$@ ) = โˆ’ Ino(PQ $,$,$0 I$0 112
  • 113. Luc_Faucheux_2021 Another summary - XXII ยจ ๐‘… ๐‘ก, ๐‘ก., ๐‘ก. = ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก. = ๐‘Œ ๐‘ก, ๐‘ก., ๐‘ก. = ๐‘Œm ๐‘ก, ๐‘ก., ๐‘ก. = ๐‘“ ๐‘ก, ๐‘ก. = โˆ’ Ino(PQ $,$,$0 I$0 ยจ A lot of models loooove to use the Instantaneous Forward Rate (HJM) ยจ We can also take another limit, the Instantaneous Short Rate defined as: ยจ ๐ผ๐‘†โ„Ž๐‘… ๐‘ก, ๐‘ก., ๐‘ก/ = ๐ผ๐‘†โ„Ž๐‘… ๐‘ก, ๐‘ก+, ๐‘ก + = ๐ผ๐‘†โ„Ž๐‘… ๐‘ก = lim $@โ†’$0,$@โ†’$ ( %SPQ $,$0,$@ i $,$0,$@ ) ยจ ๐ผ๐‘†โ„Ž๐‘… ๐‘ก = lim $0โ†’$ ๐ผ๐น๐‘ค๐‘… ๐‘ก, ๐‘ก. = ๐‘… ๐‘ก, ๐‘ก, ๐‘ก = ๐ฟ ๐‘ก, ๐‘ก, ๐‘ก = ๐‘Œ ๐‘ก, ๐‘ก, ๐‘ก = ๐‘Œm ๐‘ก, ๐‘ก, ๐‘ก = ๐‘“ ๐‘ก, ๐‘ก = ๐‘Ÿ(๐‘ก) ยจ Most of the early models were built on the short rate, and then a lot of models were โ€œaffine modelsโ€ meaning that there were assumptions of linearity for a lot of the functions. 113
  • 114. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - XXVII ยจ ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$@ โ„ค($@) ๐‘‰(๐‘ก/, $๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก ยจ lim $@โ†’$0 ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐‘… ๐‘ก, ๐‘ก., ๐‘ก. sometimes noted (Mercurio) ๐‘… ๐‘ก, ๐‘ก., ๐‘ก. + ยจ lim $@โ†’$0 [๐”ผ$@ โ„ค($@) ๐‘‰(๐‘ก/, $๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก ] = ๐”ผ$0 โ„ค($0) ๐‘‰(๐‘ก., $๐ฟ ๐‘ก., ๐‘ก., ๐‘ก. , ๐‘ก., ๐‘ก.) ๐”‰ ๐‘ก ยจ This is also true, as you can visualize essentially squeezing the period [๐‘ก., ๐‘ก/] to [๐‘ก., ๐‘ก.] ยจ Payment date ๐‘ก/ goes to ๐‘ก. ยจ Because ๐‘‰(๐‘ก., $๐ฟ ๐‘ก., ๐‘ก., ๐‘ก. , ๐‘ก., ๐‘ก.) is the value at time ๐‘ก. of a claim that pays , $๐ฟ ๐‘ก., ๐‘ก., ๐‘ก. , set at time ๐‘ก. and paid at time ๐‘ก., all the dates are the same and thus it is legitimate to write: ยจ ๐‘‰ ๐‘ก., $๐ฟ ๐‘ก., ๐‘ก., ๐‘ก. , ๐‘ก., ๐‘ก. = ๐ฟ ๐‘ก., ๐‘ก., ๐‘ก. = ๐‘… ๐‘ก., ๐‘ก., ๐‘ก. ยจ This is reminding us of Physics where variables are observed, defined and โ€œpaidโ€ at the same time so we do not have to go through that notation. 114
  • 115. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - XXVIII ยจ ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$@ โ„ค($@) ๐‘‰(๐‘ก/, $๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก ยจ lim $@โ†’$0 ๐ฟ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐‘… ๐‘ก, ๐‘ก., ๐‘ก. ยจ lim $@โ†’$0 [๐”ผ$@ โ„ค($@) ๐‘‰(๐‘ก/, $๐ฟ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก ] = ๐”ผ$0 โ„ค($0) ๐‘‰(๐‘ก., $๐ฟ ๐‘ก., ๐‘ก., ๐‘ก. , ๐‘ก., ๐‘ก.) ๐”‰ ๐‘ก ยจ ๐‘… ๐‘ก, ๐‘ก., ๐‘ก. = ๐”ผ$0 โ„ค($0) ๐‘‰(๐‘ก., $๐ฟ ๐‘ก., ๐‘ก., ๐‘ก. , ๐‘ก., ๐‘ก.) ๐”‰ ๐‘ก = ๐”ผ$0 โ„ค($0) ๐‘…(๐‘ก., ๐‘ก., ๐‘ก.) ๐”‰ ๐‘ก ยจ This is why is it called the forward measure ยจ Essentially in textbooks you will see it as (for example Mercurio p. 34) โ€œthe expected value of any future instantaneous spot interest rate, under the corresponding measure, is equal to the related instantaneous forward rateโ€ ยจ We almost there where we recognize our good old friend the instantaneous forward: ยจ ๐‘… ๐‘ก, ๐‘ก., ๐‘ก. = โˆ’ Ino(PQ $,$,$0 I$0 115
  • 116. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - XXIX ยจ ๐‘… ๐‘ก, ๐‘ก., ๐‘ก. = โˆ’ Ino(PQ $,$,$0 I$0 ยจ Just to make it simpler ยจ ๐‘… ๐‘ก, ๐‘ข, ๐‘ข = โˆ’ Ino(PQ $,$,M IM ยจ โˆซ M"$ M"$0 ๐‘… ๐‘ก, ๐‘ข, ๐‘ข . ๐‘‘๐‘ข = โˆซ M"$ M"$0 โˆ’ Ino(PQ $,$,M IM . ๐‘‘๐‘ข = [โˆ’ln(๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ข )]M"$ M"$0 ยจ โˆซ M"$ M"$0 ๐‘… ๐‘ก, ๐‘ข, ๐‘ข . ๐‘‘๐‘ข = [โˆ’ln(๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. ) + ln ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก ) = โˆ’ln(๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. ) ยจ โˆ’ ln ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = โˆซ M"$ M"$0 ๐‘… ๐‘ก, ๐‘ข, ๐‘ข . ๐‘‘๐‘ข ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. = exp(โˆ’ โˆซ M"$ M"$0 ๐‘… ๐‘ก, ๐‘ข, ๐‘ข . ๐‘‘๐‘ข) 116
  • 117. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - XXX ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. = exp(โˆ’ โˆซ M"$ M"$0 ๐‘… ๐‘ก, ๐‘ข, ๐‘ข . ๐‘‘๐‘ข) ยจ In particular for ๐‘ก = 0 ยจ ๐‘๐ถ 0,0, ๐‘ก. = ๐‘ 0,0, ๐‘ก. = exp(โˆ’ โˆซ M"# M"$0 ๐‘… 0, ๐‘ข, ๐‘ข . ๐‘‘๐‘ข) ยจ Almost there ยจ ๐‘… ๐‘ก, ๐‘ก., ๐‘ก. = ๐”ผ$0 โ„ค($0) ๐‘‰(๐‘ก., $๐ฟ ๐‘ก., ๐‘ก., ๐‘ก. , ๐‘ก., ๐‘ก.) ๐”‰ ๐‘ก = ๐”ผ$0 โ„ค($0) ๐‘…(๐‘ก., ๐‘ก., ๐‘ก.) ๐”‰ ๐‘ก ยจ ๐‘… ๐‘ก, ๐‘ข, ๐‘ข = ๐”ผM โ„ค(M) ๐‘…(๐‘ข, ๐‘ข, ๐‘ข) ๐”‰ ๐‘ก so ๐‘… 0, ๐‘ข, ๐‘ข = ๐”ผM โ„ค(M) ๐‘…(๐‘ข, ๐‘ข, ๐‘ข) ๐”‰ 0 ยจ ๐‘ 0,0, ๐‘ก. = exp(โˆ’ โˆซ M"# M"$0 ๐”ผM โ„ค(M) ๐‘…(๐‘ข, ๐‘ข, ๐‘ข) ๐”‰ 0 . ๐‘‘๐‘ข) ยจ And remember we had in the Risk free measure ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก/ = ๐”ผ$@ โ„š ๐‘’S โˆซ <=? <=?@ O !,!,! .D! |๐”‰(๐‘ก) 117
  • 118. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - XXXI ยจ ๐‘ 0,0, ๐‘ก. = exp(โˆ’ โˆซ M"# M"$0 ๐”ผM โ„ค(M) ๐‘…(๐‘ข, ๐‘ข, ๐‘ข) ๐”‰ 0 . ๐‘‘๐‘ข) ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก/ = ๐”ผ$@ โ„š exp[โˆ’ โˆซ !"$ !"$@ ๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ]|๐”‰(๐‘ก) ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = ๐”ผ$0 โ„š exp[โˆ’ โˆซ !"$ !"$0 ๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ]|๐”‰(๐‘ก) ยจ ๐‘ 0,0, ๐‘ก. = exp(โˆ’ โˆซ M"# M"$0 ๐”ผM โ„ค(M) ๐‘…(๐‘ข, ๐‘ข, ๐‘ข) ๐”‰ 0 . ๐‘‘๐‘ข) ยจ ๐‘ 0,0, ๐‘ก = exp(โˆ’ โˆซ M"# M"$ ๐”ผM โ„ค(M) ๐‘…(๐‘ข, ๐‘ข, ๐‘ข) ๐”‰ 0 . ๐‘‘๐‘ข) ยจ ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. = ๐”ผ$0 โ„š exp[โˆ’ โˆซ !"$ !"$0 ๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ]|๐”‰(๐‘ก) ยจ ๐‘๐ถ 0,0, ๐‘ก = ๐”ผ$ โ„š exp[โˆ’ โˆซ !"# !"$ ๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ]|๐”‰(0) 118
  • 119. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - XXXII ยจ Those are the famous relations that look almost the same but are quite different: ยจ ๐‘ 0,0, ๐‘ก = exp(โˆ’ โˆซ !"# !"$ ๐”ผ! โ„ค(!) ๐‘…(๐‘ , ๐‘ , ๐‘ ) ๐”‰ 0 . ๐‘‘๐‘ ) ยจ ๐‘๐ถ 0,0, ๐‘ก = ๐”ผ$ โ„š exp[โˆ’ โˆซ !"# !"$ ๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ]|๐”‰(0) ยจ To quote Rebonato (p. 33): โ€œconsiderable grief has come to erstwhile happy individuals and their families by confusing the two equations aboveโ€ 119
  • 120. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - XXXIII ยจ ๐‘ 0,0, ๐‘ก = exp(โˆ’ โˆซ !"# !"$ ๐”ผ! โ„ค(!) ๐‘…(๐‘ , ๐‘ , ๐‘ ) ๐”‰ 0 . ๐‘‘๐‘ ) ยจ ๐‘๐ถ 0,0, ๐‘ก = ๐”ผ$ โ„š exp[โˆ’ โˆซ !"# !"$ ๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ]|๐”‰(0) ยจ So you are in good company if by now you are confused about the equations above as it looks like we switched the Expectation, the integral and the exponential in just a different order. ยจ Remember that those are NOT the same measures ยจ In fact in the first equation, we integrate over an infinite number of different measures ยจ Also just to make it clear remember to write the first one as: ยจ ๐‘ 0,0, ๐‘ก = exp(โˆ’ โˆซ !"# !"$ ๐”ผ! โ„ค(!) ๐‘‰(๐‘ , $๐‘… ๐‘ , ๐‘ , ๐‘  , ๐‘ , ๐‘ ) ๐”‰ 0 . ๐‘‘๐‘ ) 120
  • 121. Luc_Faucheux_2021 A few good measures โ€“ The Terminal/Forward measure - XXXIV ยจ So juts one more time (hopefully the last time) ยจ ๐‘ 0,0, ๐‘ก = exp(โˆ’ โˆซ !"# !"$ ๐”ผ! โ„ค(!) ๐‘…(๐‘ , ๐‘ , ๐‘ ) ๐”‰ 0 . ๐‘‘๐‘ ) ยจ ๐‘๐ถ 0,0, ๐‘ก = ๐”ผ$ โ„š exp[โˆ’ โˆซ !"# !"$ ๐‘… ๐‘ , ๐‘ , ๐‘  . ๐‘‘๐‘ ]|๐”‰(0) 121
  • 122. Luc_Faucheux_2021 A few good measures โ€“ The early/discount measure ยจ This is the measure where the numeraire is the Zero ๐‘๐ถ ๐‘ก, ๐‘ก, ๐‘ก. ยจ Also referred to in textbooks as the ๐‘ก.-Terminal or sometimes ๐‘‡.-terminal ยจ ๐”ผ$0 โ„ค($0) ยจ It is usually when the claims get SET at the beginning of the period (the forward one is when the claim gets PAID at the end of the period) ยจ We sort of did it as we went over the terminal measure but worth noting that number of relations ยจ So letโ€™s redo a couple of the early slides with the early (or ๐‘ก.-Terminal) measure in mind 122
  • 123. Luc_Faucheux_2021 A few good measures โ€“ The early/discount measure - II ยจ a $,$%,$0,$@ P($,$,$@) = ๐”ผ$@ โ„ค($@) a $@,$%,$0,$@ P $@,$@,$@ ๐”‰ ๐‘ก so: ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก, ๐‘ก/) ยจ a $,$%,$0,$0 P($,$,$0) = ๐”ผ$0 โ„ค($0) a $0,$%,$0,$0 P $0,$0,$0 ๐”‰ ๐‘ก so: ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก. = ๐‘(๐‘ก, ๐‘ก, ๐‘ก.) ยจ Note that the โ€œsettingโ€ or โ€œfixingโ€ time for a claim that always pays $1 is irrelevant, but letโ€™s keep it for now, we did not go through a couple of thousand slides of building a rigorous formalism to throw it all away now. ยจ a $,$%,$0,$@ P($,$,$0) = ๐”ผ$0 โ„ค($0) a $0,$%,$0,$@ P $0,$0,$0 ๐”‰ ๐‘ก ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐”ผ$0 โ„ค($0) a $0,$%,$0,$@ P $0,$0,$0 ๐”‰ ๐‘ก = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก ยจ To be super specific, we should really if we want write: ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘‰ ๐‘ก, $๐‘(๐‘ก, ๐‘ก, ๐‘ก/), ๐‘ก, ๐‘ก = $๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก, ๐‘ก/) 123
  • 124. Luc_Faucheux_2021 A few good measures โ€“ The early/discount measure - III ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก, ๐‘ก/) ยจ From the bootstrap relations we have by definition: ยจ ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐‘(๐‘ก, ๐‘ก., ๐‘ก/) ยจ So we have: ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐‘(๐‘ก, ๐‘ก., ๐‘ก/) ยจ Which leads us quite naturally to: ยจ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก 124
  • 125. Luc_Faucheux_2021 A few good measures โ€“ The early/discount measure - IV ยจ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก ยจ That is where the nesting comes into the game: ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก, ๐‘ก/) ยจ And so by just changing the variables: ยจ ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ยจ Again when you read it aloud it makes sense: The value at time ๐‘ก. of a claim that will pay $1 no matter what at time ๐‘ก/ is equal to the value of the Zero Coupon Bond at time ๐‘ก. that pays $1 at time ๐‘ก/. ยจ That is almost a tautology, but we can nest that one into the first equation: ยจ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = ๐”ผ$0 โ„ค($0) ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก 125
  • 126. Luc_Faucheux_2021 A few good measures โ€“ The early/discount measure - V ยจ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = ๐”ผ$0 โ„ค($0) ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก ยจ I know that we are mixing a little the Terminal/Forward measure ๐”ผ$@ โ„ค($@) and the early/discount measure ๐”ผ$0 โ„ค($0) but they are exactly the same just with a different end time for the expectation and the numeraire, but the above relationship is quite cool. ยจ In the early/discount measure ๐”ผ$0 โ„ค($0) , the Zeros are a martingale. And by Zeros we mean the ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ , so not every Zeros, careful about that. ยจ So the process of those guys will be driftless: ยจ ๐‘‘๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = 0. ๐‘‘๐‘ก + ๐‘ ๐‘œ๐‘š๐‘’๐‘กโ„Ž๐‘–๐‘›๐‘” . ๐‘‘๐‘Š $0 โ„ค $0 (๐‘ก) ยจ Where ๐‘Š $0 โ„ค $0 (๐‘ก) is the Brownian motion associated to the early measure ๐”ผ$0 โ„ค($0) 126
  • 127. Luc_Faucheux_2021 A few good measures โ€“ The early/discount measure - VI ยจ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = ๐”ผ$0 โ„ค($0) ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก ยจ Simply compounded FORWARD at time ๐‘ก: ๐‘๐ถ ๐‘ก, ๐‘ก., ๐‘ก/ = % %+i $,$0,$@ .j $,$0,$@ ยจ Simply compounded FORWARD at time ๐‘ก. : ๐‘๐ถ ๐‘ก., ๐‘ก., ๐‘ก/ = % %+i $0,$0,$@ .j $0,$0,$@ ยจ With for sake of simplicity: ๐œ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐œ ยจ ๐”ผ$0 โ„ค($0) % %+i.j $0,$0,$@ ๐”‰ ๐‘ก = % %+i.j $,$0,$@ ยจ ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $ % %+i.j $,$0,$@ , ๐‘ก., ๐‘ก. ๐”‰ ๐‘ก = % %+i.j $,$0,$@ 127
  • 128. Luc_Faucheux_2021 A few good measures โ€“ The early/discount measure - VII ยจ ๐”ผ$0 โ„ค($0) % %+i.j $0,$0,$@ ๐”‰ ๐‘ก = % %+i.j $,$0,$@ ยจ We use the magic equation: ๐‘‹ โˆ’ ๐‘‹ = 0 which is true whatever the value of ๐‘‹ ยจ % %+i.j $0,$0,$@ = %+i.j $0,$0,$@ Si.j $0,$0,$@ %+i.j $0,$0,$@ = 1 โˆ’ i.j $0,$0,$@ %+i.j $0,$0,$@ ยจ Same on the right hand side at time ๐‘ก ยจ % %+i.j $,$0,$@ = %+i.j $,$0,$@ Si.j $,$0,$@ %+i.j $,$0,$@ = 1 โˆ’ i.j $,$0,$@ %+i.j $,$0,$@ ยจ ๐”ผ$0 โ„ค($0) 1 โˆ’ i.j $0,$0,$@ %+i.j $0,$0,$@ ๐”‰ ๐‘ก = 1 โˆ’ ๐”ผ$0 โ„ค $0 i.j $0,$0,$@ %+i.j $0,$0,$@ ๐”‰ ๐‘ก = 1 โˆ’ i.j $,$0,$@ %+i.j $,$0,$@ ยจ ๐”ผ$0 โ„ค $0 i.j $0,$0,$@ %+i.j $0,$0,$@ ๐”‰ ๐‘ก = i.j $,$0,$@ %+i.j $,$0,$@ 128
  • 129. Luc_Faucheux_2021 A few good measures โ€“ The early/discount measure - VIII ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ = ๐‘‰ ๐‘ก, $๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ , ๐‘ก, ๐‘ก ยจ The value of a claim at time ๐‘ก that pays $1 at time ๐‘ก/ is ๐‘(๐‘ก, ๐‘ก, ๐‘ก/) ยจ If you pay $๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ at time ๐‘ก, you will receive $1 at time ๐‘ก/ ยจ If you pay $1 = $๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ . { % P $,$,$@ } at time ๐‘ก, you will receive $1. % P $,$,$@ at time ๐‘ก/ 129 ๐‘ก" ๐‘ก! ๐‘ก ๐‘ก๐‘–๐‘š๐‘’ $1 { 1 ๐‘(๐‘ก, ๐‘ก, ๐‘ก/) } { 1 ๐‘(๐‘ก, ๐‘ก, ๐‘ก.) } {? }
  • 130. Luc_Faucheux_2021 A few good measures โ€“ The early/discount measure - IX ยจ If you pay $1 = $๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ . { % P $,$,$@ } at time ๐‘ก, you will receive $1. % P $,$,$@ at time ๐‘ก/ ยจ If you pay $1 at time ๐‘ก, you will receive $ % P $,$,$@ at time ๐‘ก/ ยจ If you pay $1 at time ๐‘ก, you will receive $ % P $,$,$0 at time ๐‘ก. 130 ๐‘ก๐‘–๐‘š๐‘’ $1 { 1 ๐‘(๐‘ก, ๐‘ก, ๐‘ก/) } { 1 ๐‘(๐‘ก, ๐‘ก, ๐‘ก.) } {? } ๐‘ก" ๐‘ก! ๐‘ก
  • 131. Luc_Faucheux_2021 A few good measures โ€“ The early/discount measure - X ยจ If you pay $1 at time ๐‘ก, you will receive $ % P $,$,$@ at time ๐‘ก/ ยจ If you pay $1 at time ๐‘ก, you will receive $ % P $,$,$0 at time ๐‘ก. ยจ From the bootstrap definition: ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐‘(๐‘ก, ๐‘ก., ๐‘ก/) 131 ๐‘ก๐‘–๐‘š๐‘’ $1 { 1 ๐‘(๐‘ก, ๐‘ก, ๐‘ก/) } { 1 ๐‘(๐‘ก, ๐‘ก, ๐‘ก.) } {? } ๐‘ก" ๐‘ก! ๐‘ก
  • 132. Luc_Faucheux_2021 A few good measures โ€“ The early/discount measure - XI ยจ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = ๐”ผ$0 โ„ค($0) ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก ยจ If you pay $1 at time ๐‘ก, you will receive $ % P $,$,$0 at time ๐‘ก. ยจ If you then re-invest that amount at time ๐‘ก. until time ๐‘ก/, what would you expect to receive then? 132 ๐‘ก๐‘–๐‘š๐‘’ $1 { 1 ๐‘(๐‘ก, ๐‘ก, ๐‘ก/) } { 1 ๐‘(๐‘ก, ๐‘ก, ๐‘ก.) } {? } ๐‘ก" ๐‘ก! ๐‘ก
  • 133. Luc_Faucheux_2021 ยจ a $,$%,$0,$@ P($,$,$@) = ๐”ผ$@ โ„ค($@) a $@,$%,$0,$@ P $@,$@,$@ ๐”‰ ๐‘ก = ๐”ผ$@ โ„ค($@) % % ๐”‰ ๐‘ก = ๐”ผ$@ โ„ค($@) 1 ๐”‰ ๐‘ก = 1 ยจ so: ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก, ๐‘ก/) ยจ a $,$%,$0,$@ P($,$,$@) = ๐”ผ$@ โ„ค($@) a $@,$%,$0,$@ P $@,$@,$@ ๐”‰ ๐‘ก so: ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก, ๐‘ก/) 133
  • 134. Luc_Faucheux_2021 A few good measures โ€“ The early/discount measure - XI 134 ๐‘ก๐‘–๐‘š๐‘’ $1 { 1 ๐‘(๐‘ก, ๐‘ก, ๐‘ก/) } { 1 ๐‘(๐‘ก, ๐‘ก, ๐‘ก.) } {? } ๐‘ก" ๐‘ก! ๐‘ก
  • 135. Luc_Faucheux_2021 ยจ a $,$%,$0,$0 P($,$,$0) = ๐”ผ$0 โ„ค($0) a $0,$%,$0,$0 P $0,$0,$0 ๐”‰ ๐‘ก = ๐”ผ$0 โ„ค($0) % % ๐”‰ ๐‘ก = ๐”ผ$0 โ„ค($0) 1 ๐”‰ ๐‘ก = 1 ยจ so: ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก. = ๐‘(๐‘ก, ๐‘ก, ๐‘ก.) ยจ So: ๐‘‰ ๐‘ก, $ % P($,$,$0) , ๐‘ก., ๐‘ก. = % P $,$,$0 . ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก. = % P $,$,$0 . ๐‘(๐‘ก, ๐‘ก, ๐‘ก.) = $1 A few good measures โ€“ The early/discount measure - XI 135 ๐‘ก๐‘–๐‘š๐‘’ $1 { 1 ๐‘(๐‘ก, ๐‘ก, ๐‘ก.) } ๐‘ก" ๐‘ก! ๐‘ก
  • 136. Luc_Faucheux_2021 ยจ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = ๐”ผ$0 โ„ค($0) ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก A few good measures โ€“ The early/discount measure - XI 136 ๐‘ก๐‘–๐‘š๐‘’ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ ๐”ผ$0 โ„ค($0) ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก ๐‘ก" ๐‘ก! ๐‘ก
  • 137. Luc_Faucheux_2021 A few good measures โ€“ The early/discount measure - XII ยจ First of all, it is always useful from time to time to go back to the โ€œZero volatility worldโ€ or deterministic, or โ€œyield curve worldโ€ where nothing is assumed to be stochastic, and all functions are deterministic ยจ This is nice because then we do not have to worry about ITO and STRATO and all the strangeness and alienness of stochastic calculus ยจ It is also nice because it is a nice check of our understanding and intuition ยจ The really cool thing about 0 volatility is that there is no convexity adjustment ยจ ๐”ผ % V = % ๐”ผ p ยจ So you cannot really mess up anything thereโ€ฆ. 137
  • 138. Luc_Faucheux_2021 ยจ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = ๐”ผ$0 โ„ค($0) ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก ยจ At Zero volatility, ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = ๐‘‰# ๐‘ก., $1, ๐‘ก., ๐‘ก/ = ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ A few good measures โ€“ The early/discount measure - XIII 138 ๐‘ก๐‘–๐‘š๐‘’ $1 { 1 ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. } ๐‘ก" ๐‘ก! ๐‘ก
  • 139. Luc_Faucheux_2021 ยจ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = ๐”ผ$0 โ„ค($0) ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก ยจ At Zero volatility,๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = ๐‘# ๐‘ก., ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก., ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก., ๐‘ก/) A few good measures โ€“ The early/discount measure - XIV 139 ๐‘ก๐‘–๐‘š๐‘’ ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ ๐”ผ$0 โ„ค($0) ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก = ๐‘ ๐‘ก., ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก., ๐‘ก/) ๐‘ก" ๐‘ก! ๐‘ก
  • 140. Luc_Faucheux_2021 ยจ At Zero volatility ? = % P $,$0,$@ = % P $0,$0,$@ ยจ At Zero volatility, ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = ๐‘# ๐‘ก., ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก., ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก., ๐‘ก/) ยจ At Zero volatility, ๐”ผ$0 โ„ค($0) % a $0,$%,$0,$@ ๐”‰ ๐‘ก = % P> $0,$0,$@ = % P $0,$0,$@ = % P $,$0,$@ A few good measures โ€“ The early/discount measure - XV 140 ๐‘ก๐‘–๐‘š๐‘’ $1 { 1 ๐‘(๐‘ก, ๐‘ก, ๐‘ก/) } { 1 ๐‘(๐‘ก, ๐‘ก, ๐‘ก.) } ๐‘ก" ๐‘ก! ๐‘ก { 1 ๐‘(๐‘ก, ๐‘ก., ๐‘ก/) }
  • 141. Luc_Faucheux_2021 A few good measures โ€“ The early/discount measure - XVI ยจ For a non-zero volatility (outside the pure yield curve world), we cannot have a unique value for {?}. There is a distribution to the possible middle points at time ๐‘ก. as we explained in section III ยจ So the graph below is misleading, we really have a distribution at time ๐‘ก. and also at time ๐‘ก/ 141 ๐‘ก๐‘–๐‘š๐‘’ $1 { 1 ๐‘(๐‘ก, ๐‘ก, ๐‘ก/) } { 1 ๐‘(๐‘ก, ๐‘ก, ๐‘ก.) } {? } ๐‘ก" ๐‘ก! ๐‘ก
  • 142. Luc_Faucheux_2021 A few good measures โ€“ The early/discount measure - Deferred premium ยจ Suppose that we have a general claim , $๐ป ๐‘ก that only depends on thins that happen before the time it finally โ€œsetsโ€ at time ๐‘ก. ยจ I wish that there was a way to express the sentence above. ยจ Oh wait there is actually, the filtration ๐”‰(๐‘ก.) ยจ So we can write that claim as $๐ป ๐‘ก = $๐ป ๐”‰(๐‘ก.) ยจ Suppose now that this claims is paid at a time ๐‘ก/ > ๐‘ก. ยจ The deferred premium formula (part III) can be expressed as: ยจ ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $๐ป ๐”‰(๐‘ก.) , ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก) = ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $๐ป ๐”‰(๐‘ก.) . ๐‘๐ถ(๐‘ก., ๐‘ก., ๐‘ก/), ๐‘ก., ๐‘ก. |๐”‰(๐‘ก) ยจ ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $๐ป ๐”‰(๐‘ก.) , ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก) = ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $๐ป ๐”‰(๐‘ก.) . ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก. |๐”‰(๐‘ก) ยจ Note the nesting in the function ๐‘‰ 142
  • 143. Luc_Faucheux_2021 A few good measures โ€“ The early/discount measure - Deferred premium - II ยจ We have reduced the calculation of the claim to an expectation at time ๐‘ก. under the early (discount) measure. ยจ This is super useful when building trees and discounting back on the tree (in order to price callable for example), that way when discounting back you pick up the value of the claim at the time ๐‘ก., as opposed to picking it up at time ๐‘ก/ (where you would not necessiraly know how to spread it on the tree nodes because you would have to forward propagate it from ๐‘ก. to ๐‘ก/in the first place, which is the thing that you are trying to avoid) ยจ This is why in 99% of the tree valuation models out there, the measure being used is the โ€œearly/discountโ€, which maybe we should just rename the โ€œtree measureโ€ 143
  • 144. Luc_Faucheux_2021 A few good measures โ€“ The early/discount measure - Deferred premium - III 144 ๐‘ก" ๐‘ก! ๐‘ก๐‘–๐‘š๐‘’ ๐‘ก
  • 145. Luc_Faucheux_2021 A few good measures โ€“ The early/discount measure - Deferred premium - IV 145 ๐‘ก" ๐‘ก! ๐‘ก๐‘–๐‘š๐‘’ ๐ป(๐”‰ ๐‘ก) ) sets at time ๐‘ก) and depends only on ๐”‰ ๐‘ก) , what happens before ๐‘ก) ๐‘‰ ๐‘ก., $๐ป(๐”‰ ๐‘ก. ), ๐‘ก., ๐‘ก/
  • 146. Luc_Faucheux_2021 A few good measures โ€“ The early/discount measure - Deferred premium - V ยจ ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $๐ป ๐”‰(๐‘ก.) , ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก) = ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $๐ป ๐”‰(๐‘ก.) . ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก. |๐”‰(๐‘ก) ยจ In particular if we set : $๐ป ๐”‰(๐‘ก.) = $1 ยจ ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก) = ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1. ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก. |๐”‰(๐‘ก) ยจ ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก) = ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก. |๐”‰(๐‘ก) ยจ ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก) = ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $๐‘ ๐‘ก., ๐‘ก., ๐‘ก/ , ๐‘ก., ๐‘ก. |๐”‰(๐‘ก) ยจ ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก) = ๐”ผ$0 โ„ค($0) ๐‘ ๐‘ก., ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก) ยจ ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก) = ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ ยจ % P($,$,$0) . ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก) = % P($,$,$0) . ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = P $,$0,$@ P($,$,$0) 146
  • 147. Luc_Faucheux_2021 A few good measures โ€“ The early/discount measure - Deferred premium - VI ยจ % P($,$,$0) . ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก) = % P($,$,$0) . ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = P $,$0,$@ P($,$,$0) ยจ ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $ % P($,$,$0) , ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก) = % P($,$,$0) . ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = P $,$0,$@ P($,$,$0) ยจ Which is equivalent to setting: $๐ป ๐”‰(๐‘ก.) = $ % P($,$,$0) 147
  • 148. Luc_Faucheux_2021 A few good measures โ€“ The early/discount measure - Deferred premium - VII ยจ a $,$%,$0,$@ P($,$,$@) = ๐”ผ$@ โ„ค($@) a $@,$%,$0,$@ P $@,$@,$@ ๐”‰ ๐‘ก = ๐”ผ$@ โ„ค($@) % % ๐”‰ ๐‘ก = ๐”ผ$@ โ„ค($@) 1 ๐”‰ ๐‘ก = 1 ยจ so: ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก, ๐‘ก/) ยจ So: ๐‘‰ ๐‘ก, $ % P($,$,$@) , ๐‘ก., ๐‘ก/ = % P $,$,$@ . ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = % P $,$,$@ . ๐‘(๐‘ก, ๐‘ก, ๐‘ก/) = $1 ยจ If you invest $1 today until time ๐‘ก/, on average you expect to get back % P $,$,$@ ยจ a $,$%,$0,$@ P($,$,$0) = ๐”ผ$0 โ„ค($0) a $0,$%,$0,$@ P $0,$0,$0 ๐”‰ ๐‘ก = ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก) ยจ a $,$%,$0,$@ P($,$,$0) = ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1. ๐‘(๐‘ก., ๐‘ก., ๐‘ก/), ๐‘ก., ๐‘ก. |๐”‰(๐‘ก) = ๐”ผ$0 โ„ค($0) ๐‘(๐‘ก., ๐‘ก., ๐‘ก/)|๐”‰(๐‘ก) = ๐‘(๐‘ก, ๐‘ก., ๐‘ก/) ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก/ 148
  • 149. Luc_Faucheux_2021 A few good measures โ€“ The early/discount measure - Deferred premium - VIII ยจ a $,$%,$0,$@ P($,$,$0) = ๐”ผ$0 โ„ค($0) a $0,$%,$0,$@ P $0,$0,$0 ๐”‰ ๐‘ก = ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก) ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐”ผ$0 โ„ค($0) ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก ยจ ๐‘‰ ๐‘ก, $ % P $,$,$0 , ๐‘ก., ๐‘ก/ = ๐”ผ$0 โ„ค($0) ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก = ๐‘(๐‘ก, ๐‘ก., ๐‘ก/) ยจ ๐‘‰ ๐‘ก, $ % P $,$,$0 , ๐‘ก., ๐‘ก. = $1 ยจ If you invest $1 today until time ๐‘ก., on average you will expect % P $,$,$0 ยจ If you invest $1 today until time ๐‘ก/, on average you expect to get back % P $,$,$@ 149
  • 150. Luc_Faucheux_2021 A few good measures โ€“ The early/discount measure - Deferred premium - IX ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ |๐”‰(๐‘ก) ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $๐‘(๐‘ก., ๐‘ก., ๐‘ก/), ๐‘ก., ๐‘ก. ๐”‰ ๐‘ก ยจ ๐‘‰ ๐‘ก, $ % P $,$,$@ , ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $ % P $,$,$@ . ๐‘(๐‘ก., ๐‘ก., ๐‘ก/), ๐‘ก., ๐‘ก. ๐”‰ ๐‘ก ยจ ๐‘‰ ๐‘ก, $ % P $,$,$@ , ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . % P $,$,$@ . ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1. ๐‘(๐‘ก., ๐‘ก., ๐‘ก/), ๐‘ก., ๐‘ก. ๐”‰ ๐‘ก ยจ ๐‘‰ ๐‘ก, $ % P $,$,$@ , ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . % P $,$,$@ . ๐”ผ$0 โ„ค($0) ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก ยจ ๐‘‰ ๐‘ก, $ % P $,$,$@ , ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . % P $,$,$@ . ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . % P $,$,$0 โˆ—P $,$0,$@ . ๐‘ ๐‘ก, ๐‘ก., ๐‘ก/ = $1 ยจ If you invest $1 at time ๐‘ก, invest it until time ๐‘ก. THEN re-invest back util time ๐‘ก/, you will STILL expect to receive on average % P $,$,$@ 150
  • 151. Luc_Faucheux_2021 A few good measures โ€“ The early/discount measure - Deferred premium - X ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘(๐‘ก, ๐‘ก, ๐‘ก/) ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก. = ๐‘(๐‘ก, ๐‘ก, ๐‘ก.) ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $๐‘(๐‘ก., ๐‘ก., ๐‘ก/), ๐‘ก., ๐‘ก. ๐”‰ ๐‘ก ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘ ๐‘ก, ๐‘ก, ๐‘ก. . ๐”ผ$0 โ„ค($0) ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก. . ๐”ผ$0 โ„ค($0) ๐‘(๐‘ก., ๐‘ก., ๐‘ก/) ๐”‰ ๐‘ก ยจ ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก/ = ๐‘‰ ๐‘ก, $1, ๐‘ก., ๐‘ก. . ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก ยจ ๐”ผ$0 โ„ค($0) ๐‘‰ ๐‘ก., $1, ๐‘ก., ๐‘ก/ ๐”‰ ๐‘ก = a $,$%,$0,$@ a $,$%,$0,$0 151