Multiscale Stochastic Volatility Models Heston 1.5

Transcription

1 Multiscale Stochastic Volatility Models Heston 1.5 Jean-Pierre Fouque Department of Statistics & Applied Probability University of California Santa Barbara Modeling and Managing Financial Risks Paris, January 10-13, 2011

2 References: Multiscale Stochastic Volatility for Equity, Interest-Rate and Credit Derivatives J.-P. Fouque, G. Papanicolaou, R. Sircar and K. Sølna Cambridge University Press (in press). A Fast Mean-Reverting Correction to Heston Stochastic Volatility Model J.-P. Fouque and M. Lorig SIAM Journal on Financial Mathematics (to appear).

3 Outline 1 Multiscale Models 2 Dynamics Why We like Heston Problems with Heston 3 Heston + Fast Factor Option Pricing 4

4 Outline 1 Multiscale Models 2 Dynamics Why We like Heston Problems with Heston 3 Heston + Fast Factor Option Pricing 4

5 Volatility Not Constant Multiscale Models 0.04 Daily Log Returns 0.02 S&P500 Simulated GBM

6 Skews of Implied Volatilities (SP500 on June 1, 2007) Implied Volatility Implied Volatilities on 1 June mo 2 mo 3 mo 6 mo 9 mo 12 mo 18 mo Moneyness=K/X

7 What s Wrong with Heston? Single factor of volatility running on single time scale not sufficient to describe dynamics of the volatilty process. Not just Heston...Any one-factor stochastic volatility model has trouble fitting implied volatility levels accross all strikes and maturities. Emperical evidence suggests existence of several stochastic volatility factors running on different time scales

8 Evidence Melino-Turnbull 1990, Dacorogna et al. 1997, Andersen-Bollerslev 1997, Fouque-Papanicolaou-Sircar 2000, Alizdeh-Brandt-Diebold 2001, Engle-Patton 2001, Lebaron 2001, Chernov-Gallant-Ghysels-Tauchen 2003, Fouque-Papanicolaou-Sircar-Solna 2003, Hillebrand 2003, Gatheral 2006, Christoffersen-Jacobs-Ornthanalai-Wang 2008.

9 Outline 1 Multiscale Models 2 Dynamics Why We like Heston Problems with Heston 3 Heston + Fast Factor Option Pricing 4

10 Multiscale SV Under Risk-Neutral dx t = rx t dt + f (Y t,z t )X t dw (0) t, ( 1 dy t = ε α(y t) 1 ) β(y t )Λ 1 (Y t,z t ) ε ( dz t = δc(z t ) ) δ g(z t )Λ 2 (Y t,z t ) Time scales: ε 1 1/δ Fast factor: Y Slow factor: Z Option pricing: dt + 1 ε β(y t )dw (1) t, dt + δ g(z t )dw (2) t. P ε,δ (t,x t,y t,z t ) = IE { e r(t t) h(x T ) X t,y t,z t }, P ε,δ t + L (X,Y,Z) P ε,δ rp ε,δ = 0.

11 Outline 1 Multiscale Models 2 Dynamics Why We like Heston Problems with Heston 3 Heston + Fast Factor Option Pricing 4

12 Expand Multiscale Models P ε,δ = i 0 ( ε) i ( δ) j P i,j, j 0 P 0 = P BS computed at volatility σ(z): σ 2 (z) = f 2 (,z) = f 2 (y,z)φ Y (dy). ε P1,0 δ P0,1 = V δ 0 = V ε 2 x 2 2 P BS x 2 P BS σ + V δ 1 + V3 ε x x x ( PBS ( x 2 2 P BS x 2 σ ). ), In terms of implied volatility: ( I b + τb δ) ( + a ε + τa δ) log(k/x), τ = T t. τ

13 Time Scales The previous asymptotics is a perturbation around Black-Scholes leading to Black-Scholes 2.0 where 2.0 = 1+ 2x(0.5) Why not keeping one volatility factor on a time scale of order one? This would be a perturbation around a one-factor stochastic volatility model, but which one? Heston of course!

14 Dynamics Why We like Heston Problems with Heston Outline 1 Multiscale Models 2 Dynamics Why We like Heston Problems with Heston 3 Heston + Fast Factor Option Pricing 4

15 Dynamics Why We like Heston Problems with Heston Heston Under Risk-Neutral Measure dx t = rx t dt + Z t X t dw x t dz t = κ(θ Z t )dt + σ Z t dw z t d W x,w z t = ρdt One-factor stochastic volatility model Square of volatility, Z t, follows CIR process

16 Dynamics Why We like Heston Problems with Heston Outline 1 Multiscale Models 2 Dynamics Why We like Heston Problems with Heston 3 Heston + Fast Factor Option Pricing 4

17 Dynamics Why We like Heston Problems with Heston Formulas! Explicit formulas for european options: P H (t,x,z) = e rτ 1 2π q(t,x) = r(t t) + log x, ĥ(k) = e ikq h(e q )dq, Ĝ(τ,k,z) = e C(τ,k)+zD(τ,k) ikq e Ĝ(τ,k,z)ĥ(k)dk C(τ,k) and D(τ,k) solve ODEs in τ = T t.

18 Dynamics Why We like Heston Problems with Heston ODEs dd dτ (τ,k) = 1 2 σ2 D 2 (τ,k) (κ + ρσik)d(τ,k) + 1 ( k 2 + ik ), 2 D(0, k) = 0. dc (τ,k) = κθd(τ,k), dτ C(0,k) = 0,

19 Dynamics Why We like Heston Problems with Heston Pretty Pictures! Heston captures well-documented features of implied volatility surface: smile and skew T Σ K 150

20 Dynamics Why We like Heston Problems with Heston Outline 1 Multiscale Models 2 Dynamics Why We like Heston Problems with Heston 3 Heston + Fast Factor Option Pricing 4

21 Dynamics Why We like Heston Problems with Heston Captures Some...Not All Features of Smile Misprices far ITM and OTM European options [Fiorentini- Leon-Rubio 2002, Zhang-Shu ] Implied Volatility Days to Maturity = 583 Market Data Heston Fit log(k/x)

22 Dynamics Why We like Heston Problems with Heston Simultaneous Fit Across Expirations Is Poor Particular difficulty fitting short expirations [Gatheral 2006] Implied Volatility Days to Maturity = 65 Market Data Heston Fit log(k/x)

23 Heston + Fast Factor Option Pricing Outline 1 Multiscale Models 2 Dynamics Why We like Heston Problems with Heston 3 Heston + Fast Factor Option Pricing 4

24 Heston + Fast Factor Option Pricing Multiscale Under Risk-Neutral Measure dx t = rx t dt + Z t f (Y t )X t dw x dy t = Z t ε (m Y t)dt + ν 2 t Zt dz t = κ(θ Z t )dt + σ Z t dw z t d W i,w j t = ρ ijdt ε dw t y i,j {x,y,z} Volatility controled by product Z t f (Y t ) Y t modeled as OU process running on time-scale ε/z t Note: f (y) = 1 model reduces to Heston

25 Heston + Fast Factor Option Pricing Outline 1 Multiscale Models 2 Dynamics Why We like Heston Problems with Heston 3 Heston + Fast Factor Option Pricing 4

26 Heston + Fast Factor Option Pricing Option Pricing PDE Price of European Option Expressed as [ ] P t = E e r(t t) h(x T ) X t,y t,z t =: P ε (t,x t,y t,z t ) Using Feynman-Kac, derive following PDE for P ε L ε P ε (t,x,y,z) = 0, L ε = t + L (X,Y,Z) r, P ε (T,x,y,z) = h(x)

27 Heston + Fast Factor Option Pricing Some Book-Keeping of L ε L ε has convenient form (z factorizes) L ε = z ε L 0 + z ε L 1 + L 2, L 0 = ν (m y) y2 y L 1 = ρ yz σν 2 2 y z + ρ xyν 2 f (y)x 2 x y ) L 2 = t f 2 (y)zx 2 2 x 2 + r ( x x σ2 z 2 z 2 + κ(θ z) z + ρ xzσf (y)zx 2 x z.

28 Heston + Fast Factor Option Pricing Expand Perform singular perturbation with respect to ε P ε = P 0 + εp 1 + εp Look for P 0 and P 1 functions of t, x, and z only Find P 0 (t,x,z) = P H (t,x,z) with effective square volatility f 2 Z t = Z t, and effective correlation ρ ρ xz f

29 Heston + Fast Factor Option Pricing More Formulas! P 1 (t,x,z) = e rτ 2π f0 (τ,k) = f1 (τ,k) = e ikq( ) κθ f 0 (τ,k) + z f 1 (τ,k) Ĝ(τ,k,z)ĥ(k)dk, τ 0 τ 0 f1 (t,k)dt, b(s,k)e A(τ,k,s) ds. b(τ,k) = ( V 1 D(τ,k) ( k 2 + ik ) + V 2 D 2 (τ,k)( ik) +V 3 ( ik 3 + k 2) + V 4 D(τ,k) ( k 2)). A(τ,k,s) solves ODE in τ

30 Heston + Fast Factor Option Pricing Goup Parameters: the V s P 1 is linear function of four constants V 1 = ρ yz σν 2 φ, V 2 = ρ xz ρ yz σ 2 ν 2 ψ, V 3 = ρ xy ν 2 f φ, V 4 = ρ xy ρ xz σν 2 f ψ. ψ(y) and φ(y) solve Poisson equations L 0 φ = 1 ( f 2 f 2 ), 2 L 0 ψ = f f. Each V i has unique effect on implied volatility

31 Outline 1 Multiscale Models 2 Dynamics Why We like Heston Problems with Heston 3 Heston + Fast Factor Option Pricing 4

32 Effect of V 1 and V 2 on Implied Volatility V 1 Effect V 2 Effect σ imp σ imp log(k/x) log(k/x) V i = 0 corresponds to Heston

33 Effect of V 3 and V 4 on Implied Volatility V 3 Effect V 4 Effect σ imp 0.18 σ imp log(k/x) log(k/x) V i = 0 corresponds to Heston

34 Outline 1 Multiscale Models 2 Dynamics Why We like Heston Problems with Heston 3 Heston + Fast Factor Option Pricing 4

35 Captures More Features of Smile Better fit for far ITM and OTM European options (SPX, T=121 days, on May 17, 2006) Implied Volatility Market Data Heston Fit Multiscale Fit log(k/x)

36 Simultaneous Fit Across Expirations Is Improved Vast improvement for short expirations 65 days to maturity Implied Volatility Market Data Heston Fit Multiscale Fit log(k/x)

37 Accuracy of Approximation: Smooth Payoffs Usual technique for smooth payoffs (Schwartz space of rapidly decaying functions): Write a PDE for the residual (model price minus next-order approximated price) Use probabilistic representation for parabolic PDEs with source (small of order ε) and terminal condition (small of order ε) Deduce residual is small of order ε Well, not exactly because moments of Y t need to be controlled uniformly in ε

38 Moments of Y dy t = Z t ε (m Y t)dt + ν 2 Zt ε dw t y Y t = m + (y m)e 1 ε R t 0 Zsds + ν 2 e 1 R t t ε 0 Zudu e 1 R s ε 0 Zudu ν Z s dw ε s y 0 We show that for any given α (1/2,1) there is a constant C such that. and therefore an accuracy of ε 1 E Y t C ε α 1,

39 Numerical Illustration for Call Options Monte Carlo simulation for a European call option: standard Euler scheme, time step of 10 5 years, 10 6 sample paths with ε = 10 3 so that εv 3 = is of the same order as V3 ε, the largest of the V ε i s obtained in the data calibration example. The parameters used in the simulation are: x = 100,z = 0.24,r = 0.05,κ = 1,θ = 1,σ = 0.39,ρ xz = 0.35, y = 0.06,m = 0.06,ν = 1,ρ xy = 0.35,ρ yz = 0.35, τ = 1,K = 100, and f (y) = e y m ν2 so that f 2 = 1. ε εp1 P 0 + ε P 1 PMC σ MC P 0 + ε P 1 P MC

40 Summary Heston model provides easy way to calculate option prices in stochastic volatility setting, but fails to capture some features of implied volatility surface Multiscale model offers improved fit to implied volatility surface while maintaining convenience of option pricing forumulas Present work with Matt Lorig to appear in the SIAM Journal on Financial Mathematics Work in preparation for Risk: formulas for fast and slow time scales around Heston (Heston 2.0).