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# presentation-vol-arb

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### presentation-vol-arb

1. 1. Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via Particle Filtering presented by Mahsiul Khan, PhD Quantum Filtering Algorithm LLC, NY NYC Quant Research Group, Morgan Stanley Building June 15, 2015 presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 1 / 24
2. 2. Outline 1 Modeling Volatility 2 Dynamic Asset Price Model 3 Realized volatility 4 Implied volatility 5 Leverage SV Models 6 Particle Filtering (PF)-A Sequential Bayesian Filtering 7 Volatility Trading 8 Variance and Volatility Swaps 9 PF to predict Volatility on SP500 10 Backtesting with VIX 11 Conclusions presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 2 / 24
3. 3. Modeling Volatility Volatility refers to uncertainty. It is the standard deviation (σt) of asset price which is heteroscedastic & non-stationary. rt = µ + σtvt (1) yt = σt vt (2) yt ∼ N(0, σ2 t ) (3) σ2 t = α0 + α1y2 t−1 + β1σ2 t−1, GARCH(1, 1) (4) where St = price, rt = ln(St /St−1) log-return, µ= expected return, yt = rt − µ is excess return, vt is standard Gaussian noise process. presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 3 / 24
4. 4. Dynamic Asset Pricing Models Continuous Time Asset Pricing Model with SDE: dSt St = µdt + σt dWt , GBM(Geometric Brownian Motion)(5) d(ln St ) = (µ − 1 2 σ2 t )dt + σt dWt (with Ito′ s Lemma) (6) V(0,T) = E[ 1 T T 0 σ2 t dt] (7) ln( st s0 ) ∼ N[(µ − 1 2 σ2 t )t, σ2 t t] (8) where St = asset price with log-normal distribution, µ= expected return, σt =volatility, and Wt is Wiener process (Brownian motion). The V(0,T) is the expected future variance over [0, T]. presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 4 / 24
5. 5. Realized (Historical) Volatility The annualized Realized Volatility is calculated with daily return σrealized = 252 N − 1 N t=1 r2 t , assume E[rt ] = ¯rt = 0 (9) Where rt is log-return with log-normal distribution. It is also called historical (observed) volatility SV and GARCH family models are predictive models Mean zero assumption has little effect on volatility estimation Converting 1-Day vol to h-Day vol by scaling √ h causes overestimation on long-horizon vol (Diebold, 97) presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 5 / 24
6. 6. Implied Volatility Implied volatility parameter appear in Black-Scholes (BS) model. It is the expected future volatility of underlying. dSt St = µdt + σimplied dWt (10) C(St , t) = f(St, K, T − t, r, σimplied ), (BS call option model)(11) C(St , t) = e−r(T−t) E[payoff(ST , K)] (12) ˆσ¯C = f−1 ( ¯C, · · · ) (13) If ¯C(·) is the market-price of an option, then vol ˆσ¯C is implied by the market price ¯C(·), and is called impled vol. Objective: model theoretical vol with market-price of option In Idealized world there is only one volatility: σrealized = σimplied = σ Mispricing due to Model uncertainty and the estimation error creates arbitrage presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 6 / 24
7. 7. Leverage SV Models with Correlated Noise The standard SV model xt = β1 + β2xt−1 + σuut (state) (14) yt = ext /2 vt (obs) (ut , vt indep) (15) The Leverage SV model ut = ρvt−1 + 1 − ρ2u′ t (16) xt = β1 + β2xt−1 + β3yt−1e− xt−1 2 + ζu′ t (state) (17) yt = ext /2 vt (obs) (18) yt ∼ N(0, ext ) (19) p(xt|y1:t) = sequential ﬁltering (objective) p(xt, θ|y1:t) = sequential ﬁltering (Liu, & West) (20) where xt = log(σ2 t ) is log-volatility, corr(ut , vt−1) = ρ, and ut , vt ,& u′ t are standard Gaussian noise. θ is unknown static parameters. presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 7 / 24
8. 8. Particle Filtering(PF)-Sequential Bayesian Filtering 1 Kalman Filter-optimal when system is Linear & Gaussian 2 Extended Kalman Filter-when system is nonlinear & non-Gaussian. Non-linearity too high, approx. is poor. 3 PF is suboptimal when systems are highly nonlinear & non-Gaussian. It is a simulation based method under Bayes’ theorem for sequential realtime inferences. Let x0:t ≡ (x0, ..., xt) and y1:t ≡ (y1, ..., yt ) as state and observation sequences. Transition from prior to posterior as, p(x0:t|y1:t, Ψ) posterior = likelihood p(y1:t |x0:t, Ψ) prior p(x0:t |Ψ) p(y1:t|Ψ) evidence (21) {xi 0:t, wi t }N i=1 ≈ p(x0:t|y1:t) (22) presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 8 / 24
9. 9. Particle Filtering-continue Posterior : p(x0:t|y1:t) = p(x0:t−1|y1:t−1) p(yt |xt )p(xt |xt−1) p(yt |y1:t−1) Filtering : p(xt |y1:t) = p(yt |xt )p(xt |y1:t−1) p(yt |y1:t−1) Prediction : p(xt |y1:t−1) = p(xt |xt−1)p(xt−1|y1:t−1)dxt−1 Update : p(xt |y1:t) ∝ p(yt |xt )p(xt |y1:t−1) wi t ∝ wi t−1 p(yt |xi t )p(xi t |xi t−1) q(xi t|xi t−1, yt ) {xi 0:t, wi t }N i=1 a.s −→ p(x0:t|y1:t) as N → ∞ {xi 0:t}N i=1 are streams of particles (realizations) with associated weights {wi t }N i=1 to approximate posterior PDF p(x0:t|y1:t). presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 9 / 24
10. 10. PF:Pictorial Representation (from Van Der Merwe) presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 10 / 24
11. 11. Volatility Trading Most volatility arbitrage strategies take advantage of the difference between the implied and realized vol of an asset. The strategy often implemented through delta neutral method consisting options and its underlying assets. Long on vol ⇒ buy (call) option + sell underlying Short on vol ⇒ sell option + buy underlying A long vol position have positive gain if realized vol is higher than implied vol at the time of the trade Delta neutral is not neutral over time, as price changes Dynamic re-hedging constitute transaction costs Delta-hedging is not pure exposure to volatility, also depends on underlying price presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 11 / 24
12. 12. Variance and Volatility Swap-Pure Vol Exposure Variance (volatility) swap is considered an asset class with pure exposures to volatility Variance swaps is a forward contracts on annualized variance (vol) when trader A pays strike variance (vol) Kvar , and receives realized variance (vol) σ2 R at Maturity T. A long (short) variance swap position will proﬁt if the realized variance(vol) of the underlying is greater (smaller) than the strike Kvar at Maturity. Variance swap trade on spread of Realized vs Implied vol Variance swaps are actively traded on major equity indices: S&P500, Nasdaq, Nikkei, EuroStoxx50, ... Drawbacks: It is OTC products, lack of liquidity presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 12 / 24
13. 13. Variance and Volatility Swap What is the difference bet variance and volatility swap? Kvar = strike variance=Expected future variance on [0,T] Kvol = strike volatility=Expected future volatility on [0,T] Variance Swap Payoff: Nvar x(σ2 R − Kvar ), Nvar = vega notional 2xKvar Volatility Swap Payoff: Nvolx(σR − Kvol) ≈ Nvol 2Kvol (σ2 R − K2 vol) Kvar = E[1 T T 0 σ2 t dt], Kvol = E[1 T T 0 σtdt] Kvol ≈ √ Kvar , Why approximation ? Because, E[ √ variance] <= E[variance] (Jensen’s Ineq). Most trade variance swap, pricing volatility swap is not linear. presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 13 / 24
14. 14. Variance Swap:Mark-to-Market At inception, swap value is zero: E[e−r(T−t0)(σ2 R − Kvar )] = 0 Mark-to-Market(MTM) value computed dynamically on [t0, T] Decompose variance V(t0, T) at time t V(t0, T)(T − t0) = V(t0, t)(t − t0) realized + V(t, T)(T − t) future V(t0, T) = λV(t0, t) + (1 − λ)V(t, T), λ = t−t0 T−t0 MTM(t) = Ne−r(T−t)[λ(V(t0, t) − Kvar ) + (1 − λ)(Kt − Kvar )] λ= proportion of time elapsed by t Kt=new strike var available at t MTM(t) is used to adjust the position presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 14 / 24
15. 15. Pricing Variance/Volatility Swaps Estimating Kvar is the most important and difﬁcult, closed form, numerical and MC methods are used. Javaheri, Wilmott, Haug (2004) proposed closed form solution with mean-reverting OU process Demeterﬁ et al (1999b) also proposed closed form solution replicating portfolio of options under BS model Heston’s (1993) SV model also used for estimating Kvar Demeterﬁ rule of thumb, Kvar = σATMF 1 + 3Txskew2, σATMF =at-the-money-forward vol, T=maturity in years CBOE vol index VIX is fair value strike (Kvar ) for variance swap on S&P500 index presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 15 / 24