This document summarizes an attempt to re-derive a useful relationship between the expected value of the exponential of a stochastic integral and the integral of the function, using the isometry property of the ITO integral. The derivation proceeds over multiple steps, expanding the stochastic integral as a limit of sums and applying properties of Brownian motion such as E[Wt - Ws] = 0. Mistakes are purposefully included for the reader to identify. The next section will highlight and explain the mistakes.
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 โsimpleโ 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
Letโs play a game.
Letโs 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
6. Luc_Faucheux_2021
Useful relationship through isometry
ยจ ๐ผ exp โซ
!"#
!"$
๐ ๐ . ๐๐ ๐ = exp[โซ
!"#
!"$ %
&
๐ ๐ &. ๐๐ ]
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) = โ'"#
'"( %
'!
. ๐'
ยจ ๐ผ exp โซ
!"#
!"$
๐ ๐ . ๐๐ ๐ = ๐ผ{โ'"#
'"( %
'!
. โซ
!"#
!"$
๐ ๐ . ๐๐ ๐
'
}
ยจ ๐ผ exp โซ
!"#
!"$
๐ ๐ . ๐๐ ๐ = โ'"#
'"( %
'!
. ๐ผ{ โซ
!"#
!"$
๐ ๐ . ๐๐ 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 ๐ผ{ โซ
!"#
!"$
๐ ๐ . ๐๐ ๐
&*+%
} = 0
ยจ ๐ผ{ โซ
!"#
!"$
๐ ๐ . ๐๐ ๐
&*
} = โซ
!"#
!"$
๐& ๐ . ๐๐
*
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 โซ
!"#
!"$
๐ ๐ . ๐๐ ๐ = โซ
!"#
!"$
๐ ๐ . ([). ๐๐ ๐
ยจ โซ
!"#
!"$
๐ ๐ . ๐๐ ๐ = lim
,โ(
โ."#
.",
๐ ๐ . . {๐ ๐ .+% โ ๐(๐ .)}
ยจ ๐ผ{โซ
!"#
!"$
๐ ๐ . ๐๐ ๐ } = ๐ผ{lim
,โ(
โ."#
.",
๐ ๐ . . {๐ ๐ .+% โ ๐(๐ .)} }
ยจ ๐ผ{โซ
!"#
!"$
๐ ๐ . ๐๐ ๐ } = lim
,โ(
โ."#
.",
๐ ๐ . . ๐ผ{๐ ๐ .+% โ ๐(๐ .)}
ยจ ๐ผ{โซ
!"#
!"$
๐ ๐ . ๐๐ ๐ } = lim
,โ(
โ."#
.",
๐ ๐ . . 0 = 0
7
10. Luc_Faucheux_2021
Useful relationship through isometry - V
ยจ ๐ผ{ โซ
!"#
!"$
๐ ๐ . ๐๐ ๐
&
} = โซ
!"#
!"$
๐ ๐ &. ๐๐
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 ๐ผ{ โซ
!"#
!"$
๐ ๐ . ([). ๐๐ ๐
&
} = โซ
!"#
!"$
๐ ๐ &. ๐๐
ยจ We can try to generalize to the higher orders: ๐ผ{ โซ
!"#
!"$
๐ ๐ . ๐๐ ๐
'
}
ยจ In particular can we write the following ?
ยจ ๐ผ{ โซ
!"#
!"$
๐ ๐ . ๐๐ ๐
&*
} = โซ
!"#
!"$
๐& ๐ . ๐๐
*
10
11. Luc_Faucheux_2021
Useful relationship through isometry - VI
ยจ ๐ผ โซ
!"#
!"$
๐ ๐ . ๐๐ ๐
'
= ๐ผ{ lim
,โ(
โ."#
.",
๐ ๐ . . {๐ ๐ .+% โ ๐(๐ .)}
'
}
ยจ In the case where ๐ is odd, we have ๐ = 2๐ + 1
ยจ ๐ผ โซ
!"#
!"$
๐ ๐ . ๐๐ ๐
&*+%
= ๐ผ{ lim
,โ(
โ."#
.",
๐ ๐ . . {๐ ๐ .+% โ ๐(๐ .)}
&*+%
}
ยจ That leaves us with a product of (2๐ + 1) limits of sums that looks something like that:
ยจ ๐ผ โซ
!"#
!"$
๐ ๐ . ๐๐ ๐
&*+%
= ๐ผ{โ'"%
'"&*+%
lim
,/โ(
โ./"#
./",/
๐ ๐ ./
. {๐ ๐ ./+% โ ๐(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 ๐ผ โซ
!"#
!"$
๐ ๐ . ๐๐ ๐
&*+%
= ๐ผ{โ'"%
'"&*+%
lim
,/โ(
โ./"#
./",/
๐ ๐ ./
. {๐ ๐ ./+% โ ๐(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 ๐ผ โซ
!"#
!"$
๐ ๐ . ๐๐ ๐
&*+%
= ๐ผ{lim
,โ(
โ."#
.", โ'0"%
'0"&*+%
๐ ๐ '0
. {๐ ๐ '012
โ ๐(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 ๐ผ โซ
!"#
!"$
๐ ๐ . ๐๐ ๐
&*+%
= lim
,โ(
โ."#
.",
๐ผ{โ'0"%
'0"&*+%
๐ ๐ '0
. {๐ ๐ '012
โ ๐(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 โisometryโ
12
13. Luc_Faucheux_2021
Useful relationship through isometry - VIII
ยจ ๐ผ โซ
!"#
!"$
๐ ๐ . ๐๐ ๐
&*+%
= lim
,โ(
โ."#
.",
๐ผ{โ'0"%
'0"&*+%
๐ ๐ '0
. {๐ ๐ '012
โ ๐(๐ '0
)}}
ยจ And we know that:
ยจ ๐ผ ๐ ๐ .+% โ ๐ ๐ . . [๐ ๐ /+% โ ๐(๐ /)] = ๐ฟ.,/. [๐ /+% โ ๐ /]
ยจ So we are just left with:
ยจ ๐ผ โซ
!"#
!"$
๐ ๐ . ๐๐ ๐
%&'(
= lim
)โ+
โ,"#
,")
๐ผ{โ-!"(
-!"%&'(
๐ฟ!"#
,!"!
๐ ๐ -!
. ๐ ๐ -#
. ๐ -!$%
โ ๐ -!
&
. {๐ ๐ -!$%
โ ๐(๐ -!
)}}
ยจ Since:
ยจ ๐ผ ๐ ๐ .+% โ ๐ ๐ . = 0
ยจ For the odd term we are left with:
ยจ ๐ผ โซ
!"#
!"$
๐ ๐ . ๐๐ 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 ๐ผ โซ
!"#
!"$
๐ ๐ . ๐๐ ๐
&*
= ๐ผ{ lim
,โ(
โ."#
.",
๐ ๐ . . {๐ ๐ .+% โ ๐(๐ .)}
&*
}
ยจ That leaves us with a product of (2๐ + 1) limits of sums that looks something like that:
ยจ ๐ผ โซ
!"#
!"$
๐ ๐ . ๐๐ ๐
&*
= ๐ผ{โ'"%
'"&*
lim
,/โ(
โ./"#
./",/
๐ ๐ ./
. {๐ ๐ ./+% โ ๐(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 ๐ผ โซ
!"#
!"$
๐ ๐ . ๐๐ ๐
&*
= ๐ผ{โ'"%
'"&*
lim
,/โ(
โ./"#
./",/
๐ ๐ ./
. {๐ ๐ ./+% โ ๐(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 ๐ผ โซ
!"#
!"$
๐ ๐ . ๐๐ ๐
&*
= ๐ผ{lim
,โ(
โ."#
.", โ'0"%
'0"&*
๐ ๐ '0
. {๐ ๐ '012
โ ๐(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 ๐ผ โซ
!"#
!"$
๐ ๐ . ๐๐ ๐
&*
= lim
,โ(
โ."#
.",
๐ผ{โ'0"%
'0"&*
๐ ๐ '0
. {๐ ๐ '012
โ ๐(๐ '0
)}}
15
16. Luc_Faucheux_2021
Useful relationship through isometry - XI
ยจ ๐ผ โซ
!"#
!"$
๐ ๐ . ๐๐ ๐
&*
= lim
,โ(
โ."#
.",
๐ผ{โ'0"%
'0"&*
๐ ๐ '0
. {๐ ๐ '012
โ ๐(๐ '0
)}}
ยจ And we know that:
ยจ ๐ผ ๐ ๐ .+% โ ๐ ๐ . . [๐ ๐ /+% โ ๐(๐ /)] = ๐ฟ.,/. [๐ /+% โ ๐ /]
ยจ So we are just left with:
ยจ ๐ผ โซ
!"#
!"$
๐ ๐ . ๐๐ ๐
%3
= lim
&โ(
โ)"#
)"&
โ4!"+
4!"%3
๐ฟ!"#
,!"!
๐ ๐ 4!
. ๐ ๐ 4#
. ๐ 4!$%
โ ๐ 4!
ยจ Which similar to the isometry derivation, reduces itself to:
ยจ ๐ผ โซ
!"#
!"$
๐ ๐ . ๐๐ ๐
&*
= lim
,โ(
โ."#
.", โ'0"%
'0"*
๐ ๐ '0
&
. ๐ '012
โ ๐ '0
ยจ We switch the sum and the product back and we get:
ยจ ๐ผ โซ
!"#
!"$
๐ ๐ . ๐๐ ๐
&*
= {โซ
!"#
!"$
๐ ๐ &. ๐๐ }*
16
18. Luc_Faucheux_2021
Useful relationship through isometry - XI
ยจ ๐ผ exp โซ
!"#
!"$
๐ ๐ . ๐๐ ๐ = โ*"#
*"( %
(&*)!
. {โซ
!"#
!"$
๐ ๐ &. ๐๐ }*
ยจ ๐ผ exp โซ
!"#
!"$
๐ ๐ . ๐๐ ๐ = โ*"#
*"( %
(*)!
. {
%
&
โซ
!"#
!"$
๐ ๐ &. ๐๐ }*
ยจ ๐ผ exp โซ
!"#
!"$
๐ ๐ . ๐๐ ๐ = exp(
%
&
โซ
!"#
!"$
๐ ๐ &. ๐๐ )
ยจ ๐ผ exp โซ
!"#
!"$
๐ ๐ . ๐๐ ๐ = exp(โซ
!"#
!"$ %
&
๐ ๐ &. ๐๐ )
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
20. Luc_Faucheux_2021
Quick side note
ยจ
Dโ
Dโ
= exp[โ โซ
!"#
!"$
๐ ๐ . ๐๐ ๐ โ
%
&
โซ
!"#
!"$
๐ ๐ &. ๐๐ ]
ยจ ๐ผ$
โ Dโ
Dโ
|๐ 0 = 1
ยจ We also have if we define :
ยจ ๐ ๐ก = ๐ ๐ก = 0 . exp(โซ
!"#
!"$
๐ ๐ . ๐๐ ๐ โ
%
&
โซ
!"#
!"$
๐ ๐ &. ๐๐ )
ยจ ๐ผ$
โ
๐ ๐ก |๐ 0 = ๐(0)
ยจ So the process
ยจ ๐ ๐ก = ๐ 0 . exp(โซ
!"#
!"$
๐ ๐ . ๐๐ ๐ โ
%
&
โซ
!"#
!"$
๐ ๐ &. ๐๐ )
ยจ Is a martingale under the โ-measure associated with the Brownian motion ๐
20
21. Luc_Faucheux_2021
Quick side note - II
ยจ ๐ ๐ก = ๐ 0 . exp(โซ
!"#
!"$
๐ ๐ . ๐๐ ๐ โ
%
&
โซ
!"#
!"$
๐ ๐ &. ๐๐ )
ยจ Is a martingale under the โ-measure associated with the Brownian motion ๐
ยจ So ๐ ๐ก is driftless and can be written (maybe) as the solution of an SDE that could look like:
ยจ ๐๐ ๐ก = 0. ๐๐ก + ๐ ๐ ๐ก , ๐ก . ([). ๐๐(๐ก)
ยจ Letโs use ITO lemma on:
ยจ ๐ ๐ก = ๐ 0 . exp(โซ
!"#
!"$
๐ ๐ . ๐๐ ๐ โ
%
&
โซ
!"#
!"$
๐ ๐ &. ๐๐ )
ยจ ๐ ๐ก = โซ
!"#
!"$
๐ ๐ . ๐๐ ๐ โ
%
&
โซ
!"#
!"$
๐ ๐ &. ๐๐
ยจ We apply ITO lemma to ๐ ๐ก = ๐ 0 . exp(๐ ๐ก )
21
22. Luc_Faucheux_2021
Quick side note - III
ยจ Applying Ito lemma:
ยจ ๐ ๐ ๐กG โ ๐ ๐ ๐กH = โซ
$"$H
$"$G IJ
IK
. ([). ๐๐(๐ก) + โซ
$"$H
$"$G %
&
.
I5J
IK5 . ([). (๐ฟ๐)&
ยจ In the โlimitโ of small me increments, this can be wri]en formally as the Ito lemma:
ยจ ๐ฟ๐ =
IJ
IK
. ([). ๐ฟ๐ +
%
&
.
I5J
IK5 . (๐ฟ๐)&
ยจ For a function of the Brownian motion ๐(๐ก):
ยจ ๐ ๐ ๐กG โ ๐ ๐ ๐กH = โซ
$"$H
$"$G IJ
IL
. ([). ๐๐(๐ก) + โซ
$"$H
$"$G %
&
.
I5J
IL5 . ([). ๐๐ก
22
28. Luc_Faucheux_2021
Quick side note - IX
ยจ The interesting thing is that:
ยจ ๐๐ ๐ก = ๐ ๐ก . ๐ ๐ก . [ . ๐๐ ๐ก
ยจ Is driftless, and the solution of it is:
ยจ ๐ ๐ก = ๐ 0 . exp(โซ
!"#
!"$
๐ ๐ . [ . ๐๐ ๐ โ
%
&
โซ
!"#
!"$
๐ ๐ &. ๐๐ )
ยจ Such that it is a martingale:
ยจ ๐ผ$
โ ๐ ๐ก |๐ 0 = ๐(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 ๐ผ$
โ exp(
%
&
โซ
!"#
!"$
๐ ๐ &. ๐๐ )|๐ 0 < โ
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
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
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
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 ๐๐ ๐ก, ๐ก, ๐ก = ๐ ๐ก . ๐๐ก โ ๐(๐ก). ([). ๐๐(๐ก)
ยจ The corresponding SIE is:
ยจ ๐ ๐ก, ๐ก, ๐ก โ ๐ ๐ , ๐ , ๐ = โซ
M"!
M"$
๐ ๐ข . ๐๐ข โ โซ
M"!
M"$
๐(๐ข). ([). ๐๐(๐ข)
ยจ In the risk-free measure โ(where we assumed that the Brownian motion ๐(๐ก) is the one
associated to this โ -measure)
ยจ ๐๐ถ ๐ก, ๐ก, ๐ก. = ๐ผ$0
โ N($)
N($0)
|๐(๐ก) = ๐ผ$0
โ
exp(โ โซ
!"$
!"$0
๐ ๐ , ๐ , ๐ . ๐๐ )|๐(๐ก)
36
38. Luc_Faucheux_2021
From short rate to Affine model - III
ยจ ๐๐ถ ๐ก, ๐ก, ๐ก) = exp โ๐ ๐ก, ๐ก, ๐ก . (๐ก) โ ๐ก) . ๐ผ$!
โ
exp(โ โซ
!"$
!"$!
{โซ
7"$
7"!
๐ ๐ข . ๐๐ข โ โซ
7"$
7"!
๐(๐ข). ([). ๐๐(๐ข)}. ๐๐ )|๐(๐ก)
ยจ ๐๐ถ ๐ก, ๐ก, ๐ก. = exp(๐ด ๐ก, ๐ก. โ ๐ต ๐ก, ๐ก. . ๐ ๐ก, ๐ก, ๐ก )
ยจ ๐๐ถ ๐ก, ๐ก, ๐ก. = exp โ๐ ๐ก, ๐ก, ๐ก . ๐ต ๐ก, ๐ก. . exp(๐ด ๐ก, ๐ก. )
ยจ In that formulation we see that naturally:
ยจ ๐ต ๐ก, ๐ก. = (๐ก. โ ๐ก)
ยจ exp ๐ด ๐ก, ๐ก. = ๐ผ$0
โ
exp(โ โซ
!"$
!"$0
{โซ
M"$
M"!
๐ ๐ข . ๐๐ข โ โซ
M"$
M"!
๐(๐ข). ([). ๐๐(๐ข)}. ๐๐ )|๐(๐ก)
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 ๐ต ๐ก, ๐ก. = (๐ก. โ ๐ก)
ยจ exp ๐ด ๐ก, ๐ก. = ๐ผ$0
โ
exp(โ โซ
!"$
!"$0
{โซ
M"$
M"!
๐ ๐ข . ๐๐ข โ โซ
M"$
M"!
๐(๐ข). ([). ๐๐(๐ข)}. ๐๐ )|๐(๐ก)
ยจ ๐ ๐ข = ๐
ยจ ๐ ๐ข =
I
IM
๐ 0, ๐ข, ๐ข + ๐&. ๐ข
ยจ Which led to:
ยจ ๐ด ๐ก, ๐ก. = โ โซ
!"$
!"$0
{[
IO #,M,M
IM
+ ๐&. ๐ข]. ๐ก. โ ๐ข โ
%
&
. (๐ก. โ ๐ข)&. ๐&}. ๐๐ข
ยจ ๐ด ๐ก, ๐ก. = ๐ 0, ๐ก, ๐ก . ๐ก. โ ๐ก + ln(
PQ #,#,$0
PQ #,#,$
) โ
R5
&
๐ก(๐ก. โ ๐ก)&
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 ๐ผ exp โซ
!"#
!"$
๐ ๐ . ๐๐ ๐ = exp[โซ
!"#
!"$ %
&
๐ ๐ &. ๐๐ ]
ยจ And apply it to:
ยจ exp ๐ด ๐ก, ๐ก. = ๐ผ$0
โ
exp(โ โซ
!"$
!"$0
{โซ
M"$
M"!
๐ ๐ข . ๐๐ข โ โซ
M"$
M"!
๐(๐ข). ([). ๐๐(๐ข)}. ๐๐ )|๐(๐ก)
ยจ Piterbarg (p.409) illustrates that approach on the even simpler model:
ยจ ๐๐ ๐ก, ๐ก, ๐ก = ๐. ([). ๐๐(๐ก)
ยจ Just for kicks, letโs redo it here also:
40
41. Luc_Faucheux_2021
From short rate to Affine model โ VI
ยจ ๐๐ ๐ก, ๐ก, ๐ก = ๐. ([). ๐๐(๐ก)
ยจ ๐ ๐ , ๐ , ๐ โ ๐ ๐ก, ๐ก, ๐ก = โ โซ
M"$
M"!
๐. ([). ๐๐(๐ข)
ยจ ๐๐ถ ๐ก, ๐ก, ๐ก. = ๐ผ$0
โ N($)
N($0)
|๐(๐ก) = ๐ผ$0
โ
exp(โ โซ
!"$
!"$0
๐ ๐ , ๐ , ๐ . ๐๐ )|๐(๐ก)
ยจ ๐๐ถ ๐ก, ๐ก, ๐ก. = ๐ผ$0
โ
exp(โ โซ
!"$
!"$0
{๐ (๐ก, ๐ก, ๐ก) โ โซ
M"$
M"!
๐. ([). ๐๐(๐ข)}. ๐๐ )|๐(๐ก)
ยจ ๐๐ถ ๐ก, ๐ก, ๐ก. = exp โ๐ ๐ก, ๐ก, ๐ก . (๐ก. โ ๐ก) . ๐ผ$0
โ
exp(โ โซ
!"$
!"$0
{โ โซ
M"$
M"!
๐. ([). ๐๐(๐ข)}. ๐๐ )|๐(๐ก)
ยจ We recognize here once again our good old friend Guido Fubini so that we can change the
order of integration:
ยจ ๐ = โซ
!"$
!"$0
โซ
M"$
M"!
๐. ([). ๐๐(๐ข) . ๐๐
41
42. Luc_Faucheux_2021
From short rate to Affine model โ VII
ยจ ๐ = โซ
!"$
!"$0
{โซ
M"$
M"!
๐. ([). ๐๐(๐ข)}. ๐๐
42
s
๐ = ๐ก!
u
s
u
๐ = ๐ก ๐ = ๐ก!
๐ = ๐ก
๐ = X
!"$
!"$0
{ X
M"$
M"!
๐. ([). ๐๐(๐ข)}. ๐๐ ๐ = X
M"$
M"$0
{ X
!"M
!"$0
๐. ๐๐ }. ([). ๐๐(๐ข)
48. Luc_Faucheux_2021
From short rate to Affine model โ XI
ยจ ๐ = โ๐ ๐ก, ๐ก, ๐ก . ๐ก. โ ๐ก + โซ
M"$
M"$0
๐. (๐ก. โ ๐ข). ([). ๐๐(๐ข)
ยจ Since the ITO integral is a martingale:
ยจ ๐ ๐ ๐ก = โ๐ ๐ก, ๐ก, ๐ก . ๐ก. โ ๐ก
ยจ So we then compute the variance:
ยจ ๐ ๐ ๐ก = ๐ผ (๐(๐ก)โ < ๐ >$)%
= ๐ผ (๐(๐ก) โ ๐ ๐ ๐ก )%
= ๐ผ (โซ
7"$
7"$!
๐. (๐ก) โ ๐ข). ([). ๐๐(๐ข))%
ยจ There, we can use once again the good old isometry property of the ITO integral:
ยจ ๐ผ (โซ
M"$
M"$0
๐. (๐ก. โ ๐ข). ([). ๐๐(๐ข))& = โซ
M"$
M"$0
{๐. (๐ก. โ ๐ข)}&. ๐๐ข =
R5
T
. ๐ก. โ ๐ก T
48
49. Luc_Faucheux_2021
From short rate to Affine model โ XII
ยจ ๐ = โ๐ ๐ก, ๐ก, ๐ก . ๐ก. โ ๐ก + โซ
M"$
M"$0
๐. (๐ก. โ ๐ข). ([). ๐๐(๐ข)
ยจ ๐ ๐ ๐ก = โ๐ ๐ก, ๐ก, ๐ก . ๐ก. โ ๐ก
ยจ ๐ ๐ ๐ก =
R5
T
. ๐ก. โ ๐ก T
ยจ ๐๐ถ ๐ก, ๐ก, ๐ก. = ๐ผ$0
โ
exp(๐(๐ก))|๐(๐ก) = exp ๐[๐(๐ก)] . exp
%
&
๐[๐(๐ก)]
ยจ ๐๐ถ ๐ก, ๐ก, ๐ก. = exp ๐[๐(๐ก)] . exp
%
&
๐[๐(๐ก)] = exp โ๐ ๐ก, ๐ก, ๐ก . ๐ก. โ ๐ก . exp
%
&
R5
T
. ๐ก. โ ๐ก T
ยจ ๐๐ถ ๐ก, ๐ก, ๐ก. = exp โ๐ ๐ก, ๐ก, ๐ก . ๐ก. โ ๐ก . exp
%
&
R5
T
. ๐ก. โ ๐ก T
ยจ ๐๐ถ ๐ก, ๐ก, ๐ก. = exp โ๐ ๐ก, ๐ก, ๐ก . ๐ก. โ ๐ก +
R5
U
. ๐ก. โ ๐ก T
ยจ And we do recover indeed the same result for that simple model
49
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 โdifferentialโ form it reads:
ยจ ๐ฟ๐ =
IJ
IK
. ๐ฟ๐ +
%
&
.
I5J
IK5 . (๐ฟ๐)&
ยจ It should really only be expressed as:
ยจ ๐ ๐ ๐กG โ ๐ ๐ ๐กH = โซ
$"$9
$"$: IJ
IV
. ([). ๐๐(๐ก) + โซ
$"$9
$"$: %
&
.
I5J
IK5 . ([). (๐ฟ๐)&
ยจ The ITO convention for the ITO integral is that we take the โLHSโ (Left Hand side) in the
partition as noted by: ([)
ยจ And the definition of the integral is:
ยจ โซ
$"$9
$"$:
๐ ๐(๐ก) . ๐๐ ๐ก = lim
,โ(
โ."#
.",
๐ ๐(๐ก.) . {๐ ๐ก.+% โ ๐(๐ก.)}
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 ๐ฟ๐ =
IJ
IK
. ๐ฟ๐ +
%
&
.
I5J
IK5 . (๐ฟ๐)&
ยจ And say โ 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 โTaylor
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 โdifferentialโ form, but by now we know better than
to trust is:
ยจ ๐ฟ๐ ๐, ๐ =
IJ
IV
. ๐ฟ๐ +
IJ
IW
. ๐ฟ๐ +
%
&
.
I5J
IV5 . ๐ฟ๐& +
%
&
.
I5J
IW5 . ๐ฟ๐& +
I5J
IVIW
. ๐ฟ๐. ๐ฟ๐
ยจ Note: should really be written as:
ยจ ๐ฟ๐ ๐, ๐ =
IJ
IK
. ๐ฟ๐ +
IJ
IX
. ๐ฟ๐ +
%
&
.
I5J
IK5 . ๐ฟ๐& +
%
&
.
I5J
IX5 . ๐ฟ๐& +
I5J
IKIX
. ๐ฟ๐. ๐ฟ๐
ยจ Lower case ๐ฅ is a regular variable
ยจ Upper case ๐ is a stochastic variable
ยจ ๐ ๐, ๐ is really noted ๐ ๐ฅ = ๐, ๐ฆ = ๐ 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 โซ
$"$H
$"$G
๐น ๐ ๐ก . ([). ๐๐(๐ก) = lim
Yโ(
{โ'"%
'"Y
๐น(๐(๐ก')). [๐(๐ก'+%) โ ๐(๐ก')]}
ยจ STRATO integral is defined as M (Middle)
ยจ โซ
$"$H
$"$G
๐น ๐ ๐ก . (โ). ๐๐(๐ก) = lim
Yโ(
{โ'"%
'"Y
๐น(
V($/12 +V($/)]
&
). [๐(๐ก'+%) โ ๐(๐ก')]}
ยจ For a simple Brownian motion
ยจ โซ
$"$H
$"$G
๐ ๐ ๐ก . (โ). ๐๐(๐ก) = โซ
$"$H
$"$G
๐ ๐ ๐ก . ([). ๐๐(๐ก) +
%
&
โซ
$"$H
$"$G IJ
IL
|L"[($). ๐๐ก
ยจ The integral in time โซ
$"$H
$"$G IJ
IL
|L"[($). ๐๐ก is the usual Riemann integral defined as
ยจ โซ
$"$H
$"$G
๐น ๐ ๐ก . ๐๐ก = lim
Yโ(
{โ'"%
'"Y
๐น(๐(๐[๐ก', ๐ก'+%])). [๐ก'+% โ ๐ก']}
55
56. Luc_Faucheux_2021
Useful tools โ ITO and STRATO correspondence - II
ยจ Where ๐[๐ก', ๐ก'+%] 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,โฆ)
ยจ For a more complicated stochastic process
ยจ ๐๐ ๐ก = ๐ ๐ก, ๐ ๐ก . ๐๐ก + ๐ ๐ก, ๐ ๐ก . ๐๐
ยจ We have:
ยจ โซ
$"$H
$"$G
๐ ๐ ๐ก . โ . ๐๐ ๐ก = โซ
$"$H
$"$G
๐ ๐ ๐ก . ([). ๐๐(๐ก) + โซ
$"$H
$"$G %
&
. ๐ ๐ก, ๐ ๐ก .
IJ
IK
|K"V($). ๐๐ก
56
62. Luc_Faucheux_2021
Useful tools - Fubini
ยจ ๐ = โซ
!"$
!"$0
{โซ
M"$
M"!
๐(๐ข). ([). ๐๐(๐ข)}. ๐๐
62
s
๐ = ๐ก!
u
s
u
๐ = ๐ก ๐ = ๐ก!
๐ = ๐ก
๐ = X
!"$
!"$0
{ X
M"$
M"!
๐(๐ข). ([). ๐๐(๐ข)}. ๐๐ ๐ = X
M"$
M"$0
{ X
!"M
!"$0
๐(๐ ). ๐๐ }. ([). ๐๐(๐ข)
63. Luc_Faucheux_2021
Useful tools โ how to always create a martingale
ยจ We use here the Tower property:
ยจ For any process ๐ ๐ก , we create:
ยจ ๐ ๐ก = ๐ผ
[
{๐(๐)|๐(๐ก)}
ยจ ๐ผ
[
๐ ๐ก ๐ ๐ = ๐ผ
[
๐ผ
[
๐ ๐ ๐ ๐ก ๐ ๐ = ๐ผ
[
๐ ๐ ๐ ๐ = ๐(๐ )
ยจ Because conditioning firstly on information back to time ๐ก then back to time ๐ is just the
same as conditioning back to time ๐ to start with.
ยจ ๐ผ
[
๐ ๐ก ๐ ๐ = ๐(๐ )
ยจ So ๐ ๐ก = ๐ผ
[
{๐(๐)|๐(๐ก)} is by construction a ๐-martingale
ยจ That is a neat little trick to always create a martingale process (Baxter p. 77)
63
67. Luc_Faucheux_2021
A few good measures
ยจ This section is a summary of some of the measures used in Finance, and their differences /
notation
67
68. Luc_Faucheux_2021
A few good measures โ The Physical measure
ยจ This is the โnaturalโ measure
ยจ It is usually noted โ (I guess the P stands for Physical)
ยจ Its characteristics (drift, variance) are usually calculated from historical data
68
69. Luc_Faucheux_2021
A few good measures โ The Risk-Neutral measure
ยจ This one usually comes right after the Physical measure
ยจ It is usually noted โ (I guess because in the alphabet Q comes right after P)
ยจ In Finance its Numeraire is the rolling Bank account (MMN-Money Market Numeraire)
ยจ As a first approximation (especially for equity derivative), the rates are assumed to be
deterministic (non-stochastic) and even further sometimes constant in time:
ยจ It is then usually noted as follows
ยจ ๐โณ ๐ก = ๐. โณ. ๐๐ก with: โณ ๐ก = ๐]$
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 = 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
ยจ If you had for a stock:
ยจ ๐๐ ๐ก = ๐โ. ๐. ๐๐ก + ๐โ. ๐. ๐๐โ
ยจ ๐โณ ๐ก = ๐. โณ. ๐๐ก
ยจ Then the ratio
^
โณ
will ALSO follows a geometric Brownian motion
ยจ ๐
^
โณ
= ๐โ โ ๐ .
^
โณ
. ๐๐ก + ๐โ.
^
โณ
. ๐๐โ
ยจ The ratio
^
โณ
is a martingale under the risk-free measure, it is thus driftless
ยจ ๐
^
โณ
= 0. ๐๐ก + {๐ ๐๐๐๐กโ๐๐๐}. ๐๐โ
70
71. Luc_Faucheux_2021
A few good measures โ The Risk-Neutral measure - IIa
ยจ ๐
^
โณ
= ๐โ โ ๐ .
^
โณ
. ๐๐ก + ๐โ.
^
โณ
. ๐๐โ
ยจ ๐
^
โณ
= 0. ๐๐ก + {๐ ๐๐๐๐กโ๐๐๐}. ๐๐โ
ยจ In this case, we see that we can define the โ-Brownian motion as:
ยจ ๐๐โ = ๐๐โ โ
`โS]
Rโ
. ๐๐ก
ยจ We then get:
ยจ ๐
^
โณ
= 0. ๐๐ก + ๐โ.
^
โณ
. ๐๐โ
ยจ We also see our old friend the โmarket price of riskโ, the excess return over the risk free
rate, normalized by the volatility of the asset.
ยจ ๐ =
`โS]
Rโ
71
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 ๐ต ๐ก = exp(โซ
!"#
!"$
๐ ๐ , ๐ , ๐ . ๐๐ )
ยจ In the extended Zeros framework of Mercurio and Lyashenko:
ยจ ๐๐ถ ๐ก, ๐ก, 0 = exp โซ
!"#
!"$
๐ ๐ , ๐ , ๐ . ๐๐ = ๐ต ๐ก
ยจ The rolling Bank Account has the useful property that:
ยจ ๐ต 0 = exp(โซ
!"#
!"#
๐ ๐ , ๐ , ๐ . ๐๐ ) = 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(#)
= ๐ผ$
โ a $,$%,$,$
N($)
|๐(0) = ๐ผ$
โ a $,$%,$,$
cde(โซ
<=>
<=?
O !,!,! .D!)
|๐(0)
ยจ
a #,$%,$,$
N(#)
=
a #,$%,$,$
%
= ๐๐ถ 0,0, ๐ก = ๐ผ$
โ a $,$%,$,$
cde(โซ
<=>
<=?
O !,!,! .D!)
|๐(0)
ยจ ๐๐ถ 0,0, ๐ก = ๐ผ$
โ a $,$%,$,$
cde(โซ
<=>
<=?
O !,!,! .D!)
|๐(0) = ๐ผ$
โ %
cde(โซ
<=>
<=?
O !,!,! .D!)
|๐(0)
ยจ ๐๐ถ 0,0, ๐ก = ๐ผ$
โ
exp(โ โซ
!"#
!"$
๐ ๐ , ๐ , ๐ . ๐๐ )|๐(0)
ยจ This is usually used when calibrating a model to the current time ๐ก = 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 ๐๐ถ 0,0, ๐ก =
PQ #,#,$
N(#)
= o
๐ 0,0, ๐ก = ๐ผ$
โ
exp(โ โซ
!"#
!"$
๐ ๐ , ๐ , ๐ . ๐๐ )|๐(0)
ยจ
PQ #,#,$@
N(#)
= o
๐ 0,0, ๐ก/ = ๐ผ$
โ PQ $,$,$@
N($)
|๐(0) = ๐ผ$
โ o
๐ ๐ก, ๐ก, ๐ก/ |๐(0)
ยจ
PQ $,$,$@
N($)
= o
๐ ๐ก, ๐ก, ๐ก/ = ๐ผ$0
โ PQ $0,$0,$@
N($0)
|๐(๐ก) = ๐ผ$0
โ o
๐ ๐ก., ๐ก., ๐ก/ |๐(๐ก) for: ๐ก < ๐ก.< ๐ก/
ยจ In particular for: ๐ก. = ๐ก/
ยจ
PQ $,$,$@
N($)
= o
๐ ๐ก, ๐ก, ๐ก/ = ๐ผ$@
โ PQ $@,$@,$@
N($@)
|๐(๐ก) = ๐ผ$@
โ o
๐ ๐ก/, ๐ก/, ๐ก/ |๐(๐ก) = ๐ผ$@
โ %
N($@)
|๐(๐ก)
ยจ ๐๐ถ ๐ก, ๐ก, ๐ก/ = ๐ต ๐ก . ๐ผ$@
โ %
N($@)
|๐(๐ก) = ๐ผ$@
โ
๐ต ๐ก .
%
N($@)
|๐(๐ก) = ๐ผ$@
โ N $
N($@)
|๐(๐ก)
75
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 ๐ก, ๐ก, ๐ก . ๐๐ก + ๐ ๐ก, ๐ก., ๐ก. . ([). ๐๐โ ๐ก
ยจ
D f
P $,$,$0
f
P $,$,$0
= ๐ ๐ก, ๐ก., ๐ก. . ([). ๐๐โ ๐ก
ยจ This is a driftless process. In particular:
ยจ ๐ o
๐ ๐ก, ๐ก, ๐ก. = o
๐ ๐ก, ๐ก, ๐ก. . ๐ ๐ก, ๐ก., ๐ก. . ([). ๐๐โ ๐ก
ยจ This is also a driftless process, hence the deflated Zeros are martingale under the risk-
neutral โ measure
ยจ ๐๐ต ๐ก = ๐ ๐ก, ๐ก, ๐ก . ๐ต ๐ก . ๐๐ก or ๐ต ๐ก = exp(โซ
!"#
!"$
๐ ๐ , ๐ , ๐ . ๐๐ )
ยจ p
๐๐ถ ๐ก, ๐ก, ๐ก. = o
๐ ๐ก, ๐ก, ๐ก. =
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 ๐๐ถ ๐ก, ๐ก, ๐ก/
ยจ Also referred to in textbooks as the ๐ก/-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 [๐ก., ๐ก/] 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 ๐ผ$@
โค($@)
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(๐ก) = ๐ ๐ก, $๐ป(๐ก), ๐ก., ๐ก/
79
๐๐๐๐ ๐๐ก ๐ก๐๐๐ ๐ก/
๐น๐๐ฅ๐๐ ๐๐ ๐ ๐๐ก ๐๐ก ๐ก๐๐๐ ๐ก.
๐บ๐๐๐๐๐๐ ๐๐๐ฆ๐๐๐ ๐ป ๐ก ๐๐ ๐๐ข๐๐๐๐๐๐ฆ $
๐๐๐๐ข๐ ๐๐ ๐กโ๐ ๐๐๐ฆ๐๐๐ ๐๐๐ ๐๐๐ฃ๐๐ ๐๐ก ๐ก๐๐๐ ๐ก
80. Luc_Faucheux_2021
A few good measures โ The Terminal/Forward measure - III
ยจ ๐(๐ก) = ๐ ๐ก, $๐ป(๐ก), ๐ก., ๐ก/
ยจ Of paramount importance is the payoff that ALWAYS pays $1
ยจ $๐ป ๐ก = $1
ยจ
a $,$h($),$0,$@
P($,$,$@)
is a martingale under ๐ผ$@
โค($@)
ยจ
a $,$h($),$0,$@
P($,$,$@)
= ๐ผ$@
โค($@) a $@,$h $@ ,$0,$@
P $@,$@,$@
๐ ๐ก = ๐ผ$@
โค($@)
๐ ๐ก/, $๐ป ๐ก/ , ๐ก., ๐ก/ ๐ ๐ก
ยจ Because in a very convenient fashion:
ยจ ๐ ๐ก/, ๐ก/, ๐ก/ = 1
ยจ For $๐ป ๐ก = $1
ยจ
a $,$%,$0,$@
P($,$,$@)
= ๐ผ$@
โค($@) a $@,$%,$0,$@
P $@,$@,$@
๐ ๐ก = ๐ผ$@
โค($@)
๐ ๐ก/, $1, ๐ก., ๐ก/ ๐ ๐ก = $1
80
81. Luc_Faucheux_2021
A few good measures โ The Terminal/Forward measure - IV
ยจ
a $,$%,$0,$@
P($,$,$@)
= ๐ผ$@
โค($@) a $@,$%,$0,$@
P $@,$@,$@
๐ ๐ก = ๐ผ$@
โค($@)
๐ ๐ก/, $1, ๐ก., ๐ก/ ๐ ๐ก = $1
ยจ ๐ ๐ก, $1, ๐ก., ๐ก/ = ๐(๐ก, ๐ก, ๐ก/)
ยจ Note that this result is independent of the measure, and validates our intuition about the
Zeros
ยจ All right, letโs 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($,$,$@)
= ๐ผ$@
โค($@) a $@,$%,$0,$@
P $@,$@,$@
๐ ๐ก so: ๐ ๐ก, $1, ๐ก., ๐ก/ = ๐(๐ก, ๐ก, ๐ก/)
ยจ
a $,$%,$0,$0
P($,$,$0)
= ๐ผ$0
โค($0) a $0,$%,$0,$0
P $0,$0,$0
๐ ๐ก so: ๐ ๐ก, $1, ๐ก., ๐ก. = ๐(๐ก, ๐ก, ๐ก.)
ยจ Note that the โsettingโ or โfixingโ time for a claim that always pays $1 is irrelevant, but letโs
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)
= ๐ผ$0
โค($0) a $0,$%,$0,$@
P $0,$0,$0
๐ ๐ก
ยจ ๐ ๐ก, $1, ๐ก., ๐ก/ = ๐ ๐ก, ๐ก, ๐ก. . ๐ผ$0
โค($0) a $0,$%,$0,$@
P $0,$0,$0
๐ ๐ก = ๐ ๐ก, ๐ก, ๐ก. . ๐ผ$0
โค($0)
๐ ๐ก., $1, ๐ก., ๐ก/ ๐ ๐ก
82
83. Luc_Faucheux_2021
A few good measures โ The Terminal/Forward measure - VI
ยจ ๐ ๐ก, $1, ๐ก., ๐ก/ = ๐ ๐ก, ๐ก, ๐ก. . ๐ผ$0
โค($0)
๐ ๐ก., $1, ๐ก., ๐ก/ ๐ ๐ก
ยจ ๐ ๐ก, $1, ๐ก., ๐ก/ = ๐(๐ก, ๐ก, ๐ก/)
ยจ From the bootstrap relations we have by definition:
ยจ ๐ ๐ก, ๐ก, ๐ก/ = ๐ ๐ก, ๐ก, ๐ก. . ๐(๐ก, ๐ก., ๐ก/)
ยจ So we have:
ยจ ๐ ๐ก, $1, ๐ก., ๐ก/ = ๐ ๐ก, ๐ก, ๐ก. . ๐ผ$0
โค($0)
๐ ๐ก., $1, ๐ก., ๐ก/ ๐ ๐ก
ยจ ๐ ๐ก, $1, ๐ก., ๐ก/ = ๐ ๐ก, ๐ก, ๐ก. . ๐(๐ก, ๐ก., ๐ก/)
ยจ Which leads us quite naturally to:
ยจ ๐ ๐ก, ๐ก., ๐ก/ = ๐ผ$0
โค($0)
๐ ๐ก., $1, ๐ก., ๐ก/ ๐ ๐ก
83
84. Luc_Faucheux_2021
A few good measures โ The Terminal/Forward measure - VII
ยจ ๐ ๐ก, ๐ก., ๐ก/ = ๐ผ$0
โค($0)
๐ ๐ก., $1, ๐ก., ๐ก/ ๐ ๐ก
ยจ That is where the nesting comes into the game:
ยจ ๐ ๐ก, $1, ๐ก., ๐ก/ = ๐(๐ก, ๐ก, ๐ก/)
ยจ And so by just changing the variables:
ยจ ๐ ๐ก., $1, ๐ก., ๐ก/ = ๐(๐ก., ๐ก., ๐ก/)
ยจ Again when you read it aloud it makes sense: The value at time ๐ก. of a claim that will pay $1
no matter what at time ๐ก/ is equal to the value of the Zero Coupon Bond at time ๐ก. that pays
$1 at time ๐ก/.
ยจ That is almost a tautology, but we can nest that one into the first equation:
ยจ ๐ ๐ก, ๐ก., ๐ก/ = ๐ผ$0
โค($0)
๐ ๐ก., $1, ๐ก., ๐ก/ ๐ ๐ก = ๐ผ$0
โค($0)
๐(๐ก., ๐ก., ๐ก/) ๐ ๐ก
84
85. Luc_Faucheux_2021
A few good measures โ The Terminal/Forward measure - VIII
ยจ ๐ ๐ก, ๐ก., ๐ก/ = ๐ผ$0
โค($0)
๐ ๐ก., $1, ๐ก., ๐ก/ ๐ ๐ก = ๐ผ$0
โค($0)
๐(๐ก., ๐ก., ๐ก/) ๐ ๐ก
ยจ I know that we are mixing a little the Terminal/Forward measure ๐ผ$@
โค($@)
and the
early/discount measure ๐ผ$0
โค($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 ๐ผ$0
โค($0)
, the Zeros are a martingale. And by Zeros we mean the
๐ ๐ก, ๐ก., ๐ก/ , so not every Zeros, careful about that.
ยจ So the process of those guys will be driftless:
ยจ ๐๐ ๐ก, ๐ก., ๐ก/ = 0. ๐๐ก + ๐ ๐๐๐๐กโ๐๐๐ . ๐๐
$0
โค $0
(๐ก)
ยจ Where ๐
$0
โค $0
(๐ก) is the Brownian motion associated to the early measure ๐ผ$0
โค($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 ๐ก, ๐ก., ๐ก/ = ๐ผ$0
โค($0)
๐ ๐ก., $1, ๐ก., ๐ก/ ๐ ๐ก = ๐ผ$0
โค($0)
๐(๐ก., ๐ก., ๐ก/) ๐ ๐ก
ยจ Now we did define the simply compounded Forward Rates as:
ยจ ๐ฟ ๐ก, ๐ก., ๐ก/ =
%
i $,$0,$@
. [
%
PQ $,$0,$@
โ 1]
ยจ Where ๐ ๐ก, ๐ก., ๐ก/ is the daycount fraction that for sake of simplicity we will note ๐ in this
section.
ยจ ๐ฟ ๐ก, ๐ก., ๐ก/ =
%
i
. [
%
PQ $,$0,$@
โ 1]
ยจ ๐๐ถ ๐ก, ๐ก., ๐ก/ =
%
%+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 : ๐๐ถ ๐ก, ๐ก., ๐ก/ = exp โ๐ ๐ก, ๐ก., ๐ก/ . ๐ ๐ก, ๐ก., ๐ก/
ยจ Simply compounded FORWARD: ๐๐ถ ๐ก, ๐ก., ๐ก/ =
%
%+i $,$0,$@ .j $,$0,$@
ยจ Annually compounded FORWARD : ๐๐ถ ๐ก, ๐ก., ๐ก/ =
%
(%+W $,$0,$@ )
I ?,?0,?@
ยจ ๐-times per year compounded FOWARD: ๐๐ถ ๐ก, ๐ก., ๐ก/ =
%
(%+
2
J
.WJ $,$0,$@ )
J.I ?,?0,?@
ยจ The function ๐ ๐ก, ๐ก., ๐ก/ is the daycount fraction, will usually depends on what convention
(ACT/ACT, ACT/360, 30/360,โฆ) 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
๐ก
๐ก๐๐๐
๐ ๐ก, ๐ก., ๐ก/ . ๐ ๐ก, ๐ก., ๐ก/
๐ก!
=
๐ก
๐ก๐๐๐
๐ก!
๐ก"
$1
$1
๐ก"
89. Luc_Faucheux_2021
A few good measures โ The Terminal/Forward measure - XII
ยจ Letโs construct a claim on a portfolio that pays $1 at time ๐ก. and pays back $1 at time ๐ก/
ยจ Note, by now that specific portfolio should not come as a surprise
ยจ Letโs note ฮ (๐ก) the value at time ๐ก of this portfolio
ยจ ฮ ๐ก = ๐ ๐ก, $1, ๐ก., ๐ก. โ ๐ ๐ก, $1, ๐ก., ๐ก/
ยจ ฮ ๐ก = ๐(๐ก, ๐ก, ๐ก.) โ ๐(๐ก, ๐ก, ๐ก/)
ยจ ฮ ๐ก = ๐ ๐ก, ๐ก, ๐ก/ . {
P $,$,$0
P($,$,$@)
โ 1}
ยจ ฮ ๐ก = ๐ ๐ก, ๐ก, ๐ก/ . {
%
P($,$0,$@)
โ 1}
ยจ ฮ ๐ก = ๐ ๐ก, ๐ก, ๐ก/ . 1 + ๐. ๐ฟ ๐ก, ๐ก., ๐ก/ โ 1 = ๐ ๐ก, ๐ก, ๐ก/ . ๐. ๐ฟ ๐ก, ๐ก., ๐ก/
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($,$,$@)
= ๐ผ$@
โค($@) k $@
P $@,$@,$@
๐ ๐ก = ๐ผ$@
โค($@)
ฮ ๐ก/ ๐ ๐ก
ยจ ฮ ๐ก = ๐ ๐ก, $1, ๐ก., ๐ก. โ ๐ ๐ก, $1, ๐ก., ๐ก/
ยจ ฮ ๐ก/ = ๐ ๐ก/, $1, ๐ก., ๐ก. โ ๐ ๐ก/, $1, ๐ก., ๐ก/
ยจ ๐ ๐ก/, $1, ๐ก., ๐ก/ is the value at time ๐ก/ of a claim that pays $1 at time ๐ก/
ยจ ๐ ๐ก/, $1, ๐ก., ๐ก/ = 1
ยจ ๐ ๐ก/, $1, ๐ก., ๐ก. is the value at time ๐ก/ of a claim that PAID $1 at time ๐ก.
ยจ ๐(๐ก., ๐ก., ๐ก/) is the value at time ๐ก. of a Zero coupon bond that pays $1 at time ๐ก/
90
91. Luc_Faucheux_2021
A few good measures โ The Terminal/Forward measure - XIV
ยจ Letโs make sure that we get this right:
ยจ ๐ ๐ก/, $1, ๐ก., ๐ก. is the value at time ๐ก/ of a claim that PAID $1 at time ๐ก.
ยจ ๐(๐ก., ๐ก., ๐ก/) is the value at time ๐ก. of a Zero coupon bond that pays $1 at time ๐ก/
ยจ Letโs say it another way
ยจ ๐ ๐ก/, $1, ๐ก., ๐ก. is the value at time ๐ก/ of a claim that PAID $1 at time ๐ก.
ยจ If you invested $1 at time ๐ก. you will receive ๐ ๐ก/, $1, ๐ก., ๐ก. at time ๐ก/
ยจ ๐(๐ก., ๐ก., ๐ก/) is the value at time ๐ก. of a Zero coupon bond that pays $1 at time ๐ก/
ยจ If you invested ๐(๐ก., ๐ก., ๐ก/) at time ๐ก. you will receive $1 at time ๐ก/
91
92. Luc_Faucheux_2021
A few good measures โ The Terminal/Forward measure - XV
ยจ If you invested $1 at time ๐ก. you will receive ๐ ๐ก/, $1, ๐ก., ๐ก. at time ๐ก/
ยจ If you invested ๐(๐ก., ๐ก., ๐ก/) at time ๐ก. you will receive $1 at time ๐ก/
ยจ And so:
ยจ ๐ ๐ก/, $1, ๐ก., ๐ก. =
%
P($0,$0,$@)
ยจ Really make sure that you are 100% convinced on that one.
ยจ Just to be sure, letโs 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 ๐ก. you will receive ๐ ๐ก/, $1, ๐ก., ๐ก. at time ๐ก/
ยจ If you invested ๐(๐ก., ๐ก., ๐ก/) at time ๐ก. you will receive $1 at time ๐ก/
ยจ If you invested ๐ ๐ก., ๐ก., ๐ก/ . {
%
P $0,$0,$@
} at time ๐ก. you will receive $1. {
%
P $0,$0,$@
} at time ๐ก/
ยจ If you invested {
P $0,$0,$@
P $0,$0,$@
} at time ๐ก. you will receive $1. {
%
P $0,$0,$@
} at time ๐ก/
ยจ If you invested {$1} at time ๐ก. you will receive $1. {
%
P $0,$0,$@
} at time ๐ก/
ยจ But we also have the first relation:
ยจ If you invested $1 at time ๐ก. you will receive ๐ ๐ก/, $1, ๐ก., ๐ก. at time ๐ก/
ยจ And so:
ยจ ๐ ๐ก/, $1, ๐ก., ๐ก. =
%
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 ๐ก/, $1, ๐ก., ๐ก. =
%
P $0,$0,$@
ยจ But remember ! This is really saying:
ยจ If you invested {$1} at time ๐ก. you will receive $1. {
%
P $0,$0,$@
} at time ๐๐
ยจ ๐ ๐ก/, $1, ๐ก., ๐ก. = ๐(๐ก/,
%
P $0,$0,$@
, ๐ก., ๐ก/)
ยจ The value at time ๐ก/ of receiving $1 that was PAID at time ๐ก., is equal to the value at time ๐ก/
of receiving $
%
P $0,$0,$@
that was set at time ๐ก. and paid at time ๐ก/
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($,$,$@)
= ๐ผ$@
โค($@) k $@
P $@,$@,$@
๐ ๐ก = ๐ผ$@
โค($@)
ฮ ๐ก/ ๐ ๐ก
ยจ ฮ ๐ก = ๐ ๐ก, $1, ๐ก., ๐ก. โ ๐ ๐ก, $1, ๐ก., ๐ก/
ยจ ฮ ๐ก/ = ๐ ๐ก/, $1, ๐ก., ๐ก. โ ๐ ๐ก/, $1, ๐ก., ๐ก/
ยจ ๐ ๐ก/, $1, ๐ก., ๐ก/ = $1
ยจ ๐ ๐ก/, $1, ๐ก., ๐ก. = ๐(๐ก/,
%
P $0,$0,$@
, ๐ก., ๐ก/)
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. ๐ฟ ๐ก., ๐ก., ๐ก/
ยจ ฮ ๐ก/ = ๐ ๐ก/, $1, ๐ก., ๐ก. โ ๐ ๐ก/, $1, ๐ก., ๐ก/
ยจ ๐ ๐ก/, $1, ๐ก., ๐ก/ = $1
ยจ ๐ ๐ก/, $1, ๐ก., ๐ก. = ๐(๐ก/, $
%
P $0,$0,$@
, ๐ก., ๐ก/)
ยจ ฮ ๐ก/ = ๐ ๐ก/, $
%
P $0,$0,$@
, ๐ก., ๐ก/ โ $1
ยจ ฮ ๐ก/ = ๐(๐ก/, $
%
P $0,$0,$@
โ $1, ๐ก., ๐ก/)
96
97. Luc_Faucheux_2021
A few good measures โ The Terminal/Forward measure - XX
ยจ ฮ ๐ก/ = ๐(๐ก/, $
%
P $0,$0,$@
โ $1, ๐ก., ๐ก/)
ยจ ฮ ๐ก/ = ๐(๐ก/, $(1 + ๐. ๐ฟ ๐ก., ๐ก., ๐ก/ ) โ $1, ๐ก., ๐ก/)
ยจ ฮ ๐ก/ = ๐(๐ก/, $(1 + ๐. ๐ฟ ๐ก., ๐ก., ๐ก/ โ 1), ๐ก., ๐ก/)
ยจ ฮ ๐ก/ = ๐(๐ก/, $(๐. ๐ฟ ๐ก., ๐ก., ๐ก/ ), ๐ก., ๐ก/)
ยจ ฮ ๐ก/ = ๐(๐ก/, $๐. ๐ฟ ๐ก., ๐ก., ๐ก/ , ๐ก., ๐ก/)
ยจ ฮ ๐ก/ = ๐ ๐ก/, $1, ๐ก., ๐ก. โ ๐ ๐ก/, $1, ๐ก., ๐ก/
ยจ That is essentially the intuition that we had built and illustrated in the graph
97
100. Luc_Faucheux_2021
A few good measures โ The Terminal/Forward measure โ XXc
ยจ ฮ ๐ก = ๐ ๐ก, $1, ๐ก., ๐ก. โ ๐ ๐ก, $1, ๐ก., ๐ก/
ยจ ฮ ๐ก = ๐(๐ก, $๐. ๐ฟ ๐ก, ๐ก., ๐ก/ , ๐ก., ๐ก/)
ยจ Receiving $1 at time ๐ก. and paying it back at time ๐ก/ is equivalent to:
ยจ Receiving at time ๐ก/ the simply compounded forward rate ๐ฟ ๐ก, ๐ก., ๐ก/ covering the period
[๐ก., ๐ก/], multiplied by the appropriate daycount fraction. This forward rate sets at time ๐ก..
ยจ Simply compounded FORWARD: ๐๐ถ ๐ก, ๐ก., ๐ก/ =
%
%+i $,$0,$@ .j $,$0,$@
ยจ ๐๐ถ ๐ก, ๐ก., ๐ก/ =
%
%+i.j $,$0,$@
ยจ ๐. ๐ฟ ๐ก, ๐ก., ๐ก/ =
%
PQ $,$0,$@
โ 1
100
101. Luc_Faucheux_2021
A few good measures โ The Terminal/Forward measure โ XXd
ยจ ๐. ๐ฟ ๐ก, ๐ก., ๐ก/ =
%
PQ $,$0,$@
โ 1 before it sets for all time ๐ก < ๐ก.
ยจ ๐. ๐ฟ ๐ก., ๐ก., ๐ก/ =
%
PQ $0,$0,$@
โ 1 when it sets at time ๐ก.
ยจ ๐. ๐ฟ ๐ก, ๐ก., ๐ก/ = ๐. ๐ฟ ๐ก., ๐ก., ๐ก/ for all time after it did set ๐ก โฅ ๐ก.
101
102. Luc_Faucheux_2021
A few good measures โ The Terminal/Forward measure - XXI
ยจ ฮ ๐ก/ = ๐(๐ก/, $๐. ๐ฟ ๐ก., ๐ก., ๐ก/ , ๐ก., ๐ก/)
ยจ ฮ ๐ก/ = ๐ ๐ก/, $1, ๐ก., ๐ก. โ ๐ ๐ก/, $1, ๐ก., ๐ก/
ยจ The value at time ๐ก/ of a portfolio that consists of receiving $1 at time ๐ก. and paying it back at
time ๐ก/ is the same value at time ๐ก/ of a portfolio paying at time ๐ก/ the simply compounded rate
(times the daycount fraction), set at time ๐ก., and spanning the period [๐ก., ๐ก/]
ยจ All right, past the halfway point:
ยจ ฮ ๐ก/ = ๐(๐ก/, $๐. ๐ฟ ๐ก., ๐ก., ๐ก/ , ๐ก., ๐ก/)
ยจ We also know that:
ยจ
k $
P($,$,$@)
= ๐ผ$@
โค($@) k $@
P $@,$@,$@
๐ ๐ก = ๐ผ$@
โค($@)
ฮ ๐ก/ ๐ ๐ก
ยจ ฮ ๐ก = ๐ ๐ก, ๐ก, ๐ก/ . 1 + ๐. ๐ฟ ๐ก, ๐ก., ๐ก/ โ 1 = ๐ ๐ก, ๐ก, ๐ก/ . ๐. ๐ฟ ๐ก, ๐ก., ๐ก/
102
104. Luc_Faucheux_2021
A few good measures โ The Terminal/Forward measure - XXIII
ยจ ๐. ๐ฟ ๐ก, ๐ก., ๐ก/ = ๐ผ$@
โค($@)
๐(๐ก/, $๐. ๐ฟ ๐ก., ๐ก., ๐ก/ , ๐ก., ๐ก/) ๐ ๐ก
ยจ Letโs drop the daycount fraction
ยจ ๐ฟ ๐ก, ๐ก., ๐ก/ = ๐ผ$@
โค($@)
๐(๐ก/, $๐ฟ ๐ก., ๐ก., ๐ก/ , ๐ก., ๐ก/) ๐ ๐ก
ยจ We are almost tempted to say that the simply compounded forward rate is a martingale
under the terminal measure
ยจ Not quite
ยจ What the above says is that the expectation under the ๐ก/-terminal measure of a claim that
pays at time ๐ก/ the value of the simply compounded forward ๐ฟ ๐ก., ๐ก., ๐ก/ set at time ๐ก. is the
current value of the simply compounded forward spanning the period [๐ก., ๐ก/]
ยจ We almost there, but before we perform the final step, letโs take a small detour through
swaps valuation
104
105. Luc_Faucheux_2021
A few good measures โ The Terminal/Forward measure - XXIV
ยจ ๐. ๐ฟ ๐ก, ๐ก., ๐ก/ = ๐ผ$@
โค($@)
๐(๐ก/, $๐. ๐ฟ ๐ก., ๐ก., ๐ก/ , ๐ก., ๐ก/) ๐ ๐ก
ยจ ๐ผ$@
โค($@)
๐(๐ก/, $๐. ๐ฟ ๐ก., ๐ก., ๐ก/ , ๐ก., ๐ก/) ๐ ๐ก = ๐ผ$@
โค($@) a($@,$i.j $0,$0,$@ ,$0,$@)
P($@,$@,$@)
๐ ๐ก
ยจ ๐ผ$@
โค($@) a($@,$i.j $0,$0,$@ ,$0,$@)
P($@,$@,$@)
๐ ๐ก =
a($,$i.j $0,$0,$@ ,$0,$@)
P($,$,$@)
= ๐. ๐ฟ ๐ก, ๐ก., ๐ก/
ยจ ๐ ๐ก, $๐. ๐ฟ ๐ก., ๐ก., ๐ก/ , ๐ก., ๐ก/ = ๐ ๐ก, ๐ก, ๐ก/ . {๐. ๐ฟ ๐ก, ๐ก., ๐ก/ }
ยจ This is why we can value swaps without a volatility curve just using the yield curve and
nothing else (remember for REGULAR swaps)
ยจ A swaplet pays at time ๐ก/ (end of the period) a Libor rate set at time ๐ก. (beginning of the
period) times the appropriate daycount fraction? Boom, the current value of that swaplet is
the current value of that simply compounded forward rate ๐ฟ ๐ก, ๐ก., ๐ก/ times the daycount
fraction time the discount factor observed in the current discount curve between now (time
๐ก) and the payment date ๐ก/, that discount factor is ๐ ๐ก, ๐ก, ๐ก/
105
106. Luc_Faucheux_2021
A few good measures โ The Terminal/Forward measure - XXV
ยจ So this is really what underpins the valuation of all swaps (REGULAR, no funny business
about paying and setting at different dates that the ones wee just talked about!), fixed cash
flows of course and all that.
ยจ Fairly cool right ?
ยจ Almost there about why this is called the forward measure.
106
107. Luc_Faucheux_2021
A few good measures โ The Terminal/Forward measure - XXVI
ยจ ๐ฟ ๐ก, ๐ก., ๐ก/ = ๐ผ$@
โค($@)
๐(๐ก/, $๐ฟ ๐ก., ๐ก., ๐ก/ , ๐ก., ๐ก/) ๐ ๐ก
ยจ We know take the limit ๐ก/ โ ๐ก. in order to recover the usual instantaneous forward
ยจ Just for sake of completeness letโs refresh our knowledge from deck V-a
107
108. Luc_Faucheux_2021
Another summary - XVII
ยจ From the variable ๐๐ถ ๐ก, ๐ก., ๐ก/ , 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 : ๐๐ถ ๐ก, ๐ก., ๐ก/ = exp โ๐ ๐ก, ๐ก., ๐ก/ . ๐ ๐ก, ๐ก., ๐ก/
ยจ Simply compounded FORWARD: ๐๐ถ ๐ก, ๐ก., ๐ก/ =
%
%+i $,$0,$@ .j $,$0,$@
ยจ Annually compounded FORWARD : ๐๐ถ ๐ก, ๐ก., ๐ก/ =
%
(%+W $,$0,$@ )
I ?,?0,?@
ยจ ๐-times per year compounded FOWARD: ๐๐ถ ๐ก, ๐ก., ๐ก/ =
%
(%+
2
J
.WJ $,$0,$@ )
J.I ?,?0,?@
ยจ The function ๐ ๐ก, ๐ก., ๐ก/ is the daycount fraction, will usually depends on what convention
(ACT/ACT, ACT/360, 30/360,โฆ) 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 ๐ก, ๐ก., ๐ก/ 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: ๐๐ถ ๐ก, ๐ก., ๐ก/ = 1 โ ๐ ๐ก, ๐ก., ๐ก/ . ๐ + ๐ช(๐&. ๐ &)
ยจ Simply compounded spot: ๐๐ถ ๐ก, ๐ก., ๐ก/ = 1 โ ๐ฟ ๐ก, ๐ก., ๐ก/ . ๐ + ๐ช(๐&. ๐&)
ยจ Annually compounded spot: ๐๐ถ ๐ก, ๐ก., ๐ก/ = 1 โ ๐ ๐ก, ๐ก., ๐ก/ . ๐ + ๐ช(๐&. ๐ฆ&)
ยจ ๐-times per year compounded spot ๐๐ถ ๐ก, ๐ก., ๐ก/ = 1 โ ๐m ๐ก, ๐ก., ๐ก/ . ๐ + ๐ช(๐&. ๐ฆm
&)
ยจ So in the limit of small ๐ ๐ก, ๐ก., ๐ก/ , (and also small rates), in particular when: ๐ก/ โ ๐ก., all rates
converge to the same limit we call
ยจ ๐ฟ๐๐ ๐ก/ โ ๐ก. = lim
$@โ$0
(
%SPQ $,$0,$@
i $,$0,$@
) that we will note Instantaneous Forward Rate
109
110. Luc_Faucheux_2021
Another summary - XIX
ยจ ๐ฟ๐๐ ๐ก/ โ ๐ก. = lim
$@โ$0
(
%SPQ $,$0,$@
i $,$0,$@
) that we will note Instantaneous Forward Rate
ยจ ๐ผ๐น๐ค๐ ๐ก, ๐ก., ๐ก/ = ๐ผ๐น๐ค๐ ๐ก, ๐ก., ๐ก.+ = ๐ผ๐น๐ค๐ ๐ก, ๐ก. = lim
$@โ$0
(
%SPQ $,$0,$@
i $,$0,$@
)
ยจ In the small ๐ ๐ก, ๐ก., ๐ก/ limit, (and also small rates) since really what matters is how small the
product of the defined rate by the daycount fraction, ๐๐ถ ๐ก, ๐ก., ๐ก/ is close to 1.
ยจ ๐ฟ๐๐ ๐ก/ โ ๐ก. = 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 ๐ก, ๐ก., ๐ก/ ) = (๐ก/ โ ๐ก.), so that ๐ฟ๐๐ ๐ก/ โ ๐ก. = lim
$@โ$0
S no PQ $,$0,$@
i $,$0,$@
110
112. Luc_Faucheux_2021
Another summary - XXI
ยจ ๐ ๐ก, ๐ก., ๐ก. = ๐ฟ ๐ก, ๐ก., ๐ก. = ๐ ๐ก, ๐ก., ๐ก. = ๐m ๐ก, ๐ก., ๐ก. = ๐(๐ก, ๐ก.) 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 ๐๐ถ ๐ก, ๐ก., ๐ก/ = ๐๐ถ ๐ก, ๐ก, ๐ก/ /๐๐ถ ๐ก, ๐ก, ๐ก.
ยจ ln(๐๐ถ ๐ก, ๐ก., ๐ก/ = ln(๐๐ถ ๐ก, ๐ก, ๐ก/ โ ln(๐๐ถ ๐ก, ๐ก, ๐ก.
ยจ lim
$@โ$0
(
Sno(PQ $,$0,$@ )
i $,$0,$@
) = โ lim
$@โ$0
no(PQ $,$,$@ Sno(PQ $,$,$0
i $,$0,$@
= โ lim
$@โ$0
(
no(PQ $,$,$@ Sno(PQ $,$,$0
$@ S $0
)
ยจ lim
$@โ$0
(
Sno(PQ $,$0,$@ )
i $,$0,$@
) = โ
Ino(PQ $,$,$0
I$0
112
113. Luc_Faucheux_2021
Another summary - XXII
ยจ ๐ ๐ก, ๐ก., ๐ก. = ๐ฟ ๐ก, ๐ก., ๐ก. = ๐ ๐ก, ๐ก., ๐ก. = ๐m ๐ก, ๐ก., ๐ก. = ๐ ๐ก, ๐ก. = โ
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 ๐ผ๐โ๐ ๐ก, ๐ก., ๐ก/ = ๐ผ๐โ๐ ๐ก, ๐ก+, ๐ก + = ๐ผ๐โ๐ ๐ก = lim
$@โ$0,$@โ$
(
%SPQ $,$0,$@
i $,$0,$@
)
ยจ ๐ผ๐โ๐ ๐ก = lim
$0โ$
๐ผ๐น๐ค๐ ๐ก, ๐ก. = ๐ ๐ก, ๐ก, ๐ก = ๐ฟ ๐ก, ๐ก, ๐ก = ๐ ๐ก, ๐ก, ๐ก = ๐m ๐ก, ๐ก, ๐ก = ๐ ๐ก, ๐ก = ๐(๐ก)
ยจ Most of the early models were built on the short rate, and then a lot of models were โaffine
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 ๐ฟ ๐ก, ๐ก., ๐ก/ = ๐ผ$@
โค($@)
๐(๐ก/, $๐ฟ ๐ก., ๐ก., ๐ก/ , ๐ก., ๐ก/) ๐ ๐ก
ยจ lim
$@โ$0
๐ฟ ๐ก, ๐ก., ๐ก/ = ๐ ๐ก, ๐ก., ๐ก. sometimes noted (Mercurio) ๐ ๐ก, ๐ก., ๐ก. +
ยจ lim
$@โ$0
[๐ผ$@
โค($@)
๐(๐ก/, $๐ฟ ๐ก., ๐ก., ๐ก/ , ๐ก., ๐ก/) ๐ ๐ก ] = ๐ผ$0
โค($0)
๐(๐ก., $๐ฟ ๐ก., ๐ก., ๐ก. , ๐ก., ๐ก.) ๐ ๐ก
ยจ This is also true, as you can visualize essentially squeezing the period [๐ก., ๐ก/] to [๐ก., ๐ก.]
ยจ Payment date ๐ก/ goes to ๐ก.
ยจ Because ๐(๐ก., $๐ฟ ๐ก., ๐ก., ๐ก. , ๐ก., ๐ก.) is the value at time ๐ก. of a claim that pays , $๐ฟ ๐ก., ๐ก., ๐ก. , set
at time ๐ก. and paid at time ๐ก., all the dates are the same and thus it is legitimate to write:
ยจ ๐ ๐ก., $๐ฟ ๐ก., ๐ก., ๐ก. , ๐ก., ๐ก. = ๐ฟ ๐ก., ๐ก., ๐ก. = ๐ ๐ก., ๐ก., ๐ก.
ยจ This is reminding us of Physics where variables are observed, defined and โpaidโ at the same
time so we do not have to go through that notation.
114
115. Luc_Faucheux_2021
A few good measures โ The Terminal/Forward measure - XXVIII
ยจ ๐ฟ ๐ก, ๐ก., ๐ก/ = ๐ผ$@
โค($@)
๐(๐ก/, $๐ฟ ๐ก., ๐ก., ๐ก/ , ๐ก., ๐ก/) ๐ ๐ก
ยจ lim
$@โ$0
๐ฟ ๐ก, ๐ก., ๐ก/ = ๐ ๐ก, ๐ก., ๐ก.
ยจ lim
$@โ$0
[๐ผ$@
โค($@)
๐(๐ก/, $๐ฟ ๐ก., ๐ก., ๐ก/ , ๐ก., ๐ก/) ๐ ๐ก ] = ๐ผ$0
โค($0)
๐(๐ก., $๐ฟ ๐ก., ๐ก., ๐ก. , ๐ก., ๐ก.) ๐ ๐ก
ยจ ๐ ๐ก, ๐ก., ๐ก. = ๐ผ$0
โค($0)
๐(๐ก., $๐ฟ ๐ก., ๐ก., ๐ก. , ๐ก., ๐ก.) ๐ ๐ก = ๐ผ$0
โค($0)
๐ (๐ก., ๐ก., ๐ก.) ๐ ๐ก
ยจ This is why is it called the forward measure
ยจ Essentially in textbooks you will see it as (for example Mercurio p. 34) โthe 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 ๐ก, ๐ก., ๐ก. = โ
Ino(PQ $,$,$0
I$0
115
119. Luc_Faucheux_2021
A few good measures โ The Terminal/Forward measure - XXXII
ยจ Those are the famous relations that look almost the same but are quite different:
ยจ ๐ 0,0, ๐ก = exp(โ โซ
!"#
!"$
๐ผ!
โค(!)
๐ (๐ , ๐ , ๐ ) ๐ 0 . ๐๐ )
ยจ ๐๐ถ 0,0, ๐ก = ๐ผ$
โ
exp[โ โซ
!"#
!"$
๐ ๐ , ๐ , ๐ . ๐๐ ]|๐(0)
ยจ To quote Rebonato (p. 33): โconsiderable 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, ๐ก = exp(โ โซ
!"#
!"$
๐ผ!
โค(!)
๐ (๐ , ๐ , ๐ ) ๐ 0 . ๐๐ )
ยจ ๐๐ถ 0,0, ๐ก = ๐ผ$
โ
exp[โ โซ
!"#
!"$
๐ ๐ , ๐ , ๐ . ๐๐ ]|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, ๐ก = exp(โ โซ
!"#
!"$
๐ผ!
โค(!)
๐(๐ , $๐ ๐ , ๐ , ๐ , ๐ , ๐ ) ๐ 0 . ๐๐ )
120
121. Luc_Faucheux_2021
A few good measures โ The Terminal/Forward measure - XXXIV
ยจ So juts one more time (hopefully the last time)
ยจ ๐ 0,0, ๐ก = exp(โ โซ
!"#
!"$
๐ผ!
โค(!)
๐ (๐ , ๐ , ๐ ) ๐ 0 . ๐๐ )
ยจ ๐๐ถ 0,0, ๐ก = ๐ผ$
โ
exp[โ โซ
!"#
!"$
๐ ๐ , ๐ , ๐ . ๐๐ ]|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 ๐๐ถ ๐ก, ๐ก, ๐ก.
ยจ Also referred to in textbooks as the ๐ก.-Terminal or sometimes ๐.-terminal
ยจ ๐ผ$0
โค($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 letโs redo a couple of the early slides with the early (or ๐ก.-Terminal) measure in mind
122
123. Luc_Faucheux_2021
A few good measures โ The early/discount measure - II
ยจ
a $,$%,$0,$@
P($,$,$@)
= ๐ผ$@
โค($@) a $@,$%,$0,$@
P $@,$@,$@
๐ ๐ก so: ๐ ๐ก, $1, ๐ก., ๐ก/ = ๐(๐ก, ๐ก, ๐ก/)
ยจ
a $,$%,$0,$0
P($,$,$0)
= ๐ผ$0
โค($0) a $0,$%,$0,$0
P $0,$0,$0
๐ ๐ก so: ๐ ๐ก, $1, ๐ก., ๐ก. = ๐(๐ก, ๐ก, ๐ก.)
ยจ Note that the โsettingโ or โfixingโ time for a claim that always pays $1 is irrelevant, but letโs
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)
= ๐ผ$0
โค($0) a $0,$%,$0,$@
P $0,$0,$0
๐ ๐ก
ยจ ๐ ๐ก, $1, ๐ก., ๐ก/ = ๐ ๐ก, ๐ก, ๐ก. . ๐ผ$0
โค($0) a $0,$%,$0,$@
P $0,$0,$0
๐ ๐ก = ๐ ๐ก, ๐ก, ๐ก. . ๐ผ$0
โค($0)
๐ ๐ก., $1, ๐ก., ๐ก/ ๐ ๐ก
ยจ To be super specific, we should really if we want write:
ยจ ๐ ๐ก, $1, ๐ก., ๐ก/ = ๐ ๐ก, $๐(๐ก, ๐ก, ๐ก/), ๐ก, ๐ก = $๐ ๐ก, ๐ก, ๐ก/ = ๐(๐ก, ๐ก, ๐ก/)
123
124. Luc_Faucheux_2021
A few good measures โ The early/discount measure - III
ยจ ๐ ๐ก, $1, ๐ก., ๐ก/ = ๐ ๐ก, ๐ก, ๐ก. . ๐ผ$0
โค($0)
๐ ๐ก., $1, ๐ก., ๐ก/ ๐ ๐ก
ยจ ๐ ๐ก, $1, ๐ก., ๐ก/ = ๐(๐ก, ๐ก, ๐ก/)
ยจ From the bootstrap relations we have by definition:
ยจ ๐ ๐ก, ๐ก, ๐ก/ = ๐ ๐ก, ๐ก, ๐ก. . ๐(๐ก, ๐ก., ๐ก/)
ยจ So we have:
ยจ ๐ ๐ก, $1, ๐ก., ๐ก/ = ๐ ๐ก, ๐ก, ๐ก. . ๐ผ$0
โค($0)
๐ ๐ก., $1, ๐ก., ๐ก/ ๐ ๐ก
ยจ ๐ ๐ก, $1, ๐ก., ๐ก/ = ๐ ๐ก, ๐ก, ๐ก. . ๐(๐ก, ๐ก., ๐ก/)
ยจ Which leads us quite naturally to:
ยจ ๐ ๐ก, ๐ก., ๐ก/ = ๐ผ$0
โค($0)
๐ ๐ก., $1, ๐ก., ๐ก/ ๐ ๐ก
124
125. Luc_Faucheux_2021
A few good measures โ The early/discount measure - IV
ยจ ๐ ๐ก, ๐ก., ๐ก/ = ๐ผ$0
โค($0)
๐ ๐ก., $1, ๐ก., ๐ก/ ๐ ๐ก
ยจ That is where the nesting comes into the game:
ยจ ๐ ๐ก, $1, ๐ก., ๐ก/ = ๐(๐ก, ๐ก, ๐ก/)
ยจ And so by just changing the variables:
ยจ ๐ ๐ก., $1, ๐ก., ๐ก/ = ๐(๐ก., ๐ก., ๐ก/)
ยจ Again when you read it aloud it makes sense: The value at time ๐ก. of a claim that will pay $1
no matter what at time ๐ก/ is equal to the value of the Zero Coupon Bond at time ๐ก. that pays
$1 at time ๐ก/.
ยจ That is almost a tautology, but we can nest that one into the first equation:
ยจ ๐ ๐ก, ๐ก., ๐ก/ = ๐ผ$0
โค($0)
๐ ๐ก., $1, ๐ก., ๐ก/ ๐ ๐ก = ๐ผ$0
โค($0)
๐(๐ก., ๐ก., ๐ก/) ๐ ๐ก
125
126. Luc_Faucheux_2021
A few good measures โ The early/discount measure - V
ยจ ๐ ๐ก, ๐ก., ๐ก/ = ๐ผ$0
โค($0)
๐ ๐ก., $1, ๐ก., ๐ก/ ๐ ๐ก = ๐ผ$0
โค($0)
๐(๐ก., ๐ก., ๐ก/) ๐ ๐ก
ยจ I know that we are mixing a little the Terminal/Forward measure ๐ผ$@
โค($@)
and the
early/discount measure ๐ผ$0
โค($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 ๐ผ$0
โค($0)
, the Zeros are a martingale. And by Zeros we mean the
๐ ๐ก, ๐ก., ๐ก/ , so not every Zeros, careful about that.
ยจ So the process of those guys will be driftless:
ยจ ๐๐ ๐ก, ๐ก., ๐ก/ = 0. ๐๐ก + ๐ ๐๐๐๐กโ๐๐๐ . ๐๐
$0
โค $0
(๐ก)
ยจ Where ๐
$0
โค $0
(๐ก) is the Brownian motion associated to the early measure ๐ผ$0
โค($0)
126
127. Luc_Faucheux_2021
A few good measures โ The early/discount measure - VI
ยจ ๐ ๐ก, ๐ก., ๐ก/ = ๐ผ$0
โค($0)
๐ ๐ก., $1, ๐ก., ๐ก/ ๐ ๐ก = ๐ผ$0
โค($0)
๐(๐ก., ๐ก., ๐ก/) ๐ ๐ก
ยจ Simply compounded FORWARD at time ๐ก: ๐๐ถ ๐ก, ๐ก., ๐ก/ =
%
%+i $,$0,$@ .j $,$0,$@
ยจ Simply compounded FORWARD at time ๐ก. : ๐๐ถ ๐ก., ๐ก., ๐ก/ =
%
%+i $0,$0,$@ .j $0,$0,$@
ยจ With for sake of simplicity: ๐ ๐ก, ๐ก., ๐ก/ = ๐
ยจ ๐ผ$0
โค($0) %
%+i.j $0,$0,$@
๐ ๐ก =
%
%+i.j $,$0,$@
ยจ ๐ผ$0
โค($0)
๐ ๐ก., $
%
%+i.j $,$0,$@
, ๐ก., ๐ก. ๐ ๐ก =
%
%+i.j $,$0,$@
127
128. Luc_Faucheux_2021
A few good measures โ The early/discount measure - VII
ยจ ๐ผ$0
โค($0) %
%+i.j $0,$0,$@
๐ ๐ก =
%
%+i.j $,$0,$@
ยจ We use the magic equation: ๐ โ 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 โ
i.j $0,$0,$@
%+i.j $0,$0,$@
ยจ Same on the right hand side at time ๐ก
ยจ
%
%+i.j $,$0,$@
=
%+i.j $,$0,$@ Si.j $,$0,$@
%+i.j $,$0,$@
= 1 โ
i.j $,$0,$@
%+i.j $,$0,$@
ยจ ๐ผ$0
โค($0)
1 โ
i.j $0,$0,$@
%+i.j $0,$0,$@
๐ ๐ก = 1 โ ๐ผ$0
โค $0 i.j $0,$0,$@
%+i.j $0,$0,$@
๐ ๐ก = 1 โ
i.j $,$0,$@
%+i.j $,$0,$@
ยจ ๐ผ$0
โค $0 i.j $0,$0,$@
%+i.j $0,$0,$@
๐ ๐ก =
i.j $,$0,$@
%+i.j $,$0,$@
128
129. Luc_Faucheux_2021
A few good measures โ The early/discount measure - VIII
ยจ ๐ ๐ก, $1, ๐ก., ๐ก/ = ๐ ๐ก, ๐ก, ๐ก/ = ๐ ๐ก, $๐ ๐ก, ๐ก, ๐ก/ , ๐ก, ๐ก
ยจ The value of a claim at time ๐ก that pays $1 at time ๐ก/ is ๐(๐ก, ๐ก, ๐ก/)
ยจ If you pay $๐ ๐ก, ๐ก, ๐ก/ at time ๐ก, you will receive $1 at time ๐ก/
ยจ If you pay $1 = $๐ ๐ก, ๐ก, ๐ก/ . {
%
P $,$,$@
} at time ๐ก, you will receive $1.
%
P $,$,$@
at time ๐ก/
129
๐ก"
๐ก!
๐ก
๐ก๐๐๐
$1
{
1
๐(๐ก, ๐ก, ๐ก/)
}
{
1
๐(๐ก, ๐ก, ๐ก.)
} {? }
130. Luc_Faucheux_2021
A few good measures โ The early/discount measure - IX
ยจ If you pay $1 = $๐ ๐ก, ๐ก, ๐ก/ . {
%
P $,$,$@
} at time ๐ก, you will receive $1.
%
P $,$,$@
at time ๐ก/
ยจ If you pay $1 at time ๐ก, you will receive $
%
P $,$,$@
at time ๐ก/
ยจ If you pay $1 at time ๐ก, you will receive $
%
P $,$,$0
at time ๐ก.
130
๐ก๐๐๐
$1
{
1
๐(๐ก, ๐ก, ๐ก/)
}
{
1
๐(๐ก, ๐ก, ๐ก.)
} {? }
๐ก"
๐ก!
๐ก
131. Luc_Faucheux_2021
A few good measures โ The early/discount measure - X
ยจ If you pay $1 at time ๐ก, you will receive $
%
P $,$,$@
at time ๐ก/
ยจ If you pay $1 at time ๐ก, you will receive $
%
P $,$,$0
at time ๐ก.
ยจ From the bootstrap definition: ๐ ๐ก, ๐ก, ๐ก/ = ๐ ๐ก, ๐ก, ๐ก. . ๐(๐ก, ๐ก., ๐ก/)
131
๐ก๐๐๐
$1
{
1
๐(๐ก, ๐ก, ๐ก/)
}
{
1
๐(๐ก, ๐ก, ๐ก.)
} {? }
๐ก"
๐ก!
๐ก
132. Luc_Faucheux_2021
A few good measures โ The early/discount measure - XI
ยจ ๐ ๐ก, ๐ก., ๐ก/ = ๐ผ$0
โค($0)
๐ ๐ก., $1, ๐ก., ๐ก/ ๐ ๐ก = ๐ผ$0
โค($0)
๐(๐ก., ๐ก., ๐ก/) ๐ ๐ก
ยจ If you pay $1 at time ๐ก, you will receive $
%
P $,$,$0
at time ๐ก.
ยจ If you then re-invest that amount at time ๐ก. until time ๐ก/, what would you expect to receive
then?
132
๐ก๐๐๐
$1
{
1
๐(๐ก, ๐ก, ๐ก/)
}
{
1
๐(๐ก, ๐ก, ๐ก.)
} {? }
๐ก"
๐ก!
๐ก
133. Luc_Faucheux_2021
ยจ
a $,$%,$0,$@
P($,$,$@)
= ๐ผ$@
โค($@) a $@,$%,$0,$@
P $@,$@,$@
๐ ๐ก = ๐ผ$@
โค($@) %
%
๐ ๐ก = ๐ผ$@
โค($@)
1 ๐ ๐ก = 1
ยจ so: ๐ ๐ก, $1, ๐ก., ๐ก/ = ๐(๐ก, ๐ก, ๐ก/)
ยจ
a $,$%,$0,$@
P($,$,$@)
= ๐ผ$@
โค($@) a $@,$%,$0,$@
P $@,$@,$@
๐ ๐ก so: ๐ ๐ก, $1, ๐ก., ๐ก/ = ๐(๐ก, ๐ก, ๐ก/)
133
134. Luc_Faucheux_2021
A few good measures โ The early/discount measure - XI
134
๐ก๐๐๐
$1
{
1
๐(๐ก, ๐ก, ๐ก/)
}
{
1
๐(๐ก, ๐ก, ๐ก.)
}
{? }
๐ก"
๐ก!
๐ก
135. Luc_Faucheux_2021
ยจ
a $,$%,$0,$0
P($,$,$0)
= ๐ผ$0
โค($0) a $0,$%,$0,$0
P $0,$0,$0
๐ ๐ก = ๐ผ$0
โค($0) %
%
๐ ๐ก = ๐ผ$0
โค($0)
1 ๐ ๐ก = 1
ยจ so: ๐ ๐ก, $1, ๐ก., ๐ก. = ๐(๐ก, ๐ก, ๐ก.)
ยจ So: ๐ ๐ก, $
%
P($,$,$0)
, ๐ก., ๐ก. =
%
P $,$,$0
. ๐ ๐ก, $1, ๐ก., ๐ก. =
%
P $,$,$0
. ๐(๐ก, ๐ก, ๐ก.) = $1
A few good measures โ The early/discount measure - XI
135
๐ก๐๐๐
$1
{
1
๐(๐ก, ๐ก, ๐ก.)
}
๐ก"
๐ก!
๐ก
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 โZero volatility worldโ or
deterministic, or โyield 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 ๐ผ
%
V
=
%
๐ผ p
ยจ So you cannot really mess up anything thereโฆ.
137
140. Luc_Faucheux_2021
ยจ At Zero volatility ? =
%
P $,$0,$@
=
%
P $0,$0,$@
ยจ At Zero volatility, ๐ผ$0
โค($0)
๐ ๐ก., $1, ๐ก., ๐ก/ ๐ ๐ก = ๐# ๐ก., ๐ก., ๐ก/ = ๐ ๐ก., ๐ก., ๐ก/ = ๐(๐ก, ๐ก., ๐ก/)
ยจ At Zero volatility, ๐ผ$0
โค($0) %
a $0,$%,$0,$@
๐ ๐ก =
%
P> $0,$0,$@
=
%
P $0,$0,$@
=
%
P $,$0,$@
A few good measures โ The early/discount measure - XV
140
๐ก๐๐๐
$1
{
1
๐(๐ก, ๐ก, ๐ก/)
}
{
1
๐(๐ก, ๐ก, ๐ก.)
}
๐ก"
๐ก!
๐ก
{
1
๐(๐ก, ๐ก., ๐ก/)
}
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 ๐ก. as we explained in
section III
ยจ So the graph below is misleading, we really have a distribution at time ๐ก. and also at time ๐ก/
141
๐ก๐๐๐
$1
{
1
๐(๐ก, ๐ก, ๐ก/)
}
{
1
๐(๐ก, ๐ก, ๐ก.)
} {? }
๐ก"
๐ก!
๐ก
142. Luc_Faucheux_2021
A few good measures โ The early/discount measure -
Deferred premium
ยจ Suppose that we have a general claim , $๐ป ๐ก that only depends on thins that happen before
the time it finally โsetsโ at time ๐ก.
ยจ I wish that there was a way to express the sentence above.
ยจ Oh wait there is actually, the filtration ๐(๐ก.)
ยจ So we can write that claim as $๐ป ๐ก = $๐ป ๐(๐ก.)
ยจ Suppose now that this claims is paid at a time ๐ก/ > ๐ก.
ยจ The deferred premium formula (part III) can be expressed as:
ยจ ๐ผ$0
โค($0)
๐ ๐ก., $๐ป ๐(๐ก.) , ๐ก., ๐ก/ |๐(๐ก) = ๐ผ$0
โค($0)
๐ ๐ก., $๐ป ๐(๐ก.) . ๐๐ถ(๐ก., ๐ก., ๐ก/), ๐ก., ๐ก. |๐(๐ก)
ยจ ๐ผ$0
โค($0)
๐ ๐ก., $๐ป ๐(๐ก.) , ๐ก., ๐ก/ |๐(๐ก) = ๐ผ$0
โค($0)
๐ ๐ก., $๐ป ๐(๐ก.) . ๐ ๐ก., $1, ๐ก., ๐ก/ , ๐ก., ๐ก. |๐(๐ก)
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 ๐ก. 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 ๐ก., as opposed to picking it up at time ๐ก/ (where you would not necessiraly know
how to spread it on the tree nodes because you would have to forward propagate it from ๐ก.
to ๐ก/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
โearly/discountโ, which maybe we should just rename the โtree measureโ
143
144. Luc_Faucheux_2021
A few good measures โ The early/discount measure -
Deferred premium - III
144
๐ก"
๐ก!
๐ก๐๐๐
๐ก
145. Luc_Faucheux_2021
A few good measures โ The early/discount measure -
Deferred premium - IV
145
๐ก"
๐ก!
๐ก๐๐๐
๐ป(๐ ๐ก) ) sets at time ๐ก) and depends only on ๐ ๐ก) , what happens before ๐ก)
๐ ๐ก., $๐ป(๐ ๐ก. ), ๐ก., ๐ก/
147. Luc_Faucheux_2021
A few good measures โ The early/discount measure -
Deferred premium - VI
ยจ
%
P($,$,$0)
. ๐ผ$0
โค($0)
๐ ๐ก., $1, ๐ก., ๐ก/ |๐(๐ก) =
%
P($,$,$0)
. ๐ ๐ก, ๐ก., ๐ก/ =
P $,$0,$@
P($,$,$0)
ยจ ๐ผ$0
โค($0)
๐ ๐ก., $
%
P($,$,$0)
, ๐ก., ๐ก/ |๐(๐ก) =
%
P($,$,$0)
. ๐ ๐ก, ๐ก., ๐ก/ =
P $,$0,$@
P($,$,$0)
ยจ Which is equivalent to setting: $๐ป ๐(๐ก.) = $
%
P($,$,$0)
147
148. Luc_Faucheux_2021
A few good measures โ The early/discount measure -
Deferred premium - VII
ยจ
a $,$%,$0,$@
P($,$,$@)
= ๐ผ$@
โค($@) a $@,$%,$0,$@
P $@,$@,$@
๐ ๐ก = ๐ผ$@
โค($@) %
%
๐ ๐ก = ๐ผ$@
โค($@)
1 ๐ ๐ก = 1
ยจ so: ๐ ๐ก, $1, ๐ก., ๐ก/ = ๐(๐ก, ๐ก, ๐ก/)
ยจ So: ๐ ๐ก, $
%
P($,$,$@)
, ๐ก., ๐ก/ =
%
P $,$,$@
. ๐ ๐ก, $1, ๐ก., ๐ก/ =
%
P $,$,$@
. ๐(๐ก, ๐ก, ๐ก/) = $1
ยจ If you invest $1 today until time ๐ก/, on average you expect to get back
%
P $,$,$@
ยจ
a $,$%,$0,$@
P($,$,$0)
= ๐ผ$0
โค($0) a $0,$%,$0,$@
P $0,$0,$0
๐ ๐ก = ๐ผ$0
โค($0)
๐ ๐ก., $1, ๐ก., ๐ก/ |๐(๐ก)
ยจ
a $,$%,$0,$@
P($,$,$0)
= ๐ผ$0
โค($0)
๐ ๐ก., $1. ๐(๐ก., ๐ก., ๐ก/), ๐ก., ๐ก. |๐(๐ก) = ๐ผ$0
โค($0)
๐(๐ก., ๐ก., ๐ก/)|๐(๐ก) = ๐(๐ก, ๐ก., ๐ก/)
ยจ ๐ ๐ก, $1, ๐ก., ๐ก/ = ๐ ๐ก, ๐ก, ๐ก/ = ๐ ๐ก, ๐ก, ๐ก. . ๐ ๐ก, ๐ก., ๐ก/ = ๐ ๐ก, ๐ก, ๐ก/
148
149. Luc_Faucheux_2021
A few good measures โ The early/discount measure -
Deferred premium - VIII
ยจ
a $,$%,$0,$@
P($,$,$0)
= ๐ผ$0
โค($0) a $0,$%,$0,$@
P $0,$0,$0
๐ ๐ก = ๐ผ$0
โค($0)
๐ ๐ก., $1, ๐ก., ๐ก/ |๐(๐ก)
ยจ ๐ ๐ก, $1, ๐ก., ๐ก/ = ๐ ๐ก, ๐ก, ๐ก. . ๐ผ$0
โค($0)
๐(๐ก., ๐ก., ๐ก/) ๐ ๐ก
ยจ ๐ ๐ก, $
%
P $,$,$0
, ๐ก., ๐ก/ = ๐ผ$0
โค($0)
๐(๐ก., ๐ก., ๐ก/) ๐ ๐ก = ๐(๐ก, ๐ก., ๐ก/)
ยจ ๐ ๐ก, $
%
P $,$,$0
, ๐ก., ๐ก. = $1
ยจ If you invest $1 today until time ๐ก., on average you will expect
%
P $,$,$0
ยจ If you invest $1 today until time ๐ก/, 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 ๐ก, $1, ๐ก., ๐ก/ = ๐ ๐ก, ๐ก, ๐ก. . ๐ผ$0
โค($0)
๐ ๐ก., $1, ๐ก., ๐ก/ |๐(๐ก)
ยจ ๐ ๐ก, $1, ๐ก., ๐ก/ = ๐ ๐ก, ๐ก, ๐ก. . ๐ผ$0
โค($0)
๐ ๐ก., $๐(๐ก., ๐ก., ๐ก/), ๐ก., ๐ก. ๐ ๐ก
ยจ ๐ ๐ก, $
%
P $,$,$@
, ๐ก., ๐ก/ = ๐ ๐ก, ๐ก, ๐ก. . ๐ผ$0
โค($0)
๐ ๐ก., $
%
P $,$,$@
. ๐(๐ก., ๐ก., ๐ก/), ๐ก., ๐ก. ๐ ๐ก
ยจ ๐ ๐ก, $
%
P $,$,$@
, ๐ก., ๐ก/ = ๐ ๐ก, ๐ก, ๐ก. .
%
P $,$,$@
. ๐ผ$0
โค($0)
๐ ๐ก., $1. ๐(๐ก., ๐ก., ๐ก/), ๐ก., ๐ก. ๐ ๐ก
ยจ ๐ ๐ก, $
%
P $,$,$@
, ๐ก., ๐ก/ = ๐ ๐ก, ๐ก, ๐ก. .
%
P $,$,$@
. ๐ผ$0
โค($0)
๐(๐ก., ๐ก., ๐ก/) ๐ ๐ก
ยจ ๐ ๐ก, $
%
P $,$,$@
, ๐ก., ๐ก/ = ๐ ๐ก, ๐ก, ๐ก. .
%
P $,$,$@
. ๐ ๐ก, ๐ก., ๐ก/ = ๐ ๐ก, ๐ก, ๐ก. .
%
P $,$,$0 โP $,$0,$@
. ๐ ๐ก, ๐ก., ๐ก/ = $1
ยจ If you invest $1 at time ๐ก, invest it until time ๐ก. THEN re-invest back util time ๐ก/, you will STILL expect
to receive on average
%
P $,$,$@
150