Variance Reduction Techniques for Monte Carlo Simulations with Stochastic Volatility Models

Transcription

1 Variance Reduction Techniques for Monte Carlo Simulations with Stochastic Volatility Models Jean-Pierre Fouque North Carolina State University SAMSI November 3, 5 1

2 References: Variance Reduction for Monte Carlo Simulation in a Stochastic Volatility Environment with T. Tullie), Quantitative Finance. Variance Reduction for Monte Carlo Methods to Evaluate Option Prices under Multi-factor Stochastic Volatility Models with S. Han), Quantitative Finance 4. A Control Variate Method to Evaluate Option Prices under Multi-Factor Stochastic Volatility Models with S. Han), Submitted 4. Chapter in a forthcoming book. fouque

3 Essential in Monte Carlo Simulations Computing option prices IE { e rt HS T ) } 1 N over N independent realizations. N k=1 e rt HS k) T ) Computing measures of risk of portfolios, for instance IP {V T V < V ar} α 3

4 Stochastic Volatility Essential to account for: Distributions of Returns under physical measure IP) Smile/Skew in Implied Volatilities under risk neutral measure IP ) Volatility Time Scales at least two factors: one slow, one fast) 4

5 REQUIRED QUALITIES Parametrization of the Implied Volatility Surface It ; T, K) Universal Parsimonous Parameters: Model Independence Stability in Time: Predictive Power Easy Calibration: Practical Implementation Compatibility with Price Dynamics: Applicability to Pricing other Derivatives and Hedging 5

6 Model Under Risk Neutral ds t = rs t dt + fy t, Z t )S t dw ) t ) dy t = αm f Y t ) ν f α Λf Y t, Z t ) dt ) +ν f α ρ 1 dw ) t + 1 ρ 1) 1 dw t ) dz t = δm s Z t ) ν s δ Λs Y t, Z t ) +ν s δ ρ dw ) t + ρ 1 dw 1) t + dt 1 ρ ρ ) 1 dw t α large: Y t fast mean reverting on time scale 1/α << 1 δ small: Z t slow varying on time scale 1/δ >> 1 Λ f and Λ s : market prices of volatility risk ρ 1 < 1, ρ < 1, ρ 1 < 1 ρ : correlation coefficients ) 6

7 European Options: Options Pt, S t, Y t, Z t ) = IE { e rt t) HS T ) S t, Y t, Z t } Asian Call Options fixed or floating strike): P t = IE { e rt t) A T K 1 S T K) + F t } Arithmetic Average Asian Option AAO) A T = 1 T T S t dt Geometric Average Asian Option GAO) ) 1 T A T = exp lns t dt T More general sampling functions: dt dλt) 7

8 dv t = bt,v t )dt + at,v t )dη t v = x y z V t = S t Y t Z t η t = W ) t W 1) t W ) t bt, v) = rx αm f y) ν f α Λf y, z) δm s z) ν s δ Λs y, z) at, v) = fy, z)x ν f α ρ1 ν f α 1 ρ 1 ν s δ ρ ν s δ ρ1 ν s δ 1 ρ ρ 1 8

9 European Monte Carlo The price Pt, x, y, z) of an European option at time t is given by { } Pt, v) = IE e rt t) HS T ) V t = v A basic Monte Carlo approximation is based on calculating the sample mean Pt, x, y, z) 1 N N k=1 e rt t) HS k) T ) over N independent realizations of V starting at time t from v. 9

10 Importance Sampling Introduce the martingale t Q t = exp hs, V s ) dη s + 1 t and define a new probability IP equivalent to IP by d IP dip = Q T ) 1 ) hs, V s ) ds By Girsanov Theorem, under this new measure IP, the process η t = η t + is a standard Brownian motion. t hs, V s )ds 1

11 Monte Carlo under ĨP Pt, v) = IE { } e rt t) HS T )Q T V t = v where T Q T = exp hs, V s ) d η s 1 T hs, V s ) ds ) and the dynamics of our model becomes dv t = {bt, V t ) at, V t )ht, V t )}dt + at, V t )d η t where η t is a standard Brownian motion. Pt, x, y, z) 1 N N k=1 e rt t) HS k) T )Qk) T 11

12 Choice of h Applying Ito s formula to Pt, V t )Q t = HV T )Q T = Pt, v) + T t Q s a v P + P h)s, V s )d η s Variance of the payoff HV T )Q T { T VarĨP {HV T )Q T } = IE t Q s a v P + P h ds }. Indeed, if the quantity P to be computed was known, one could obtain a zero variance by choosing h = 1 P a v P) Our strategy is to use approximations to the exact value P. 1

13 Pricing Equation δ << 1 and ε = 1/α << 1 { } P ε,δ t, x, y, z) = IE e rt t) HS T ) S t = x, Y t = y, Z t = z 1 ε L + 1 ε L 1 + L + δm 1 + δm + P ε,δ T, x, y, z) = Hx) L = m f y) y + ν f L 1 = ν f ρ 1 fx L = t + 1 f x x + r x y Λ f x x ) δ ε M 3 P ε,δ = y M 1 = ν s ) M = m s z) y z + ν s ) ) Λ s z + ρ fx x z z M 3 = ν f ν s ρ 1 ρ + ρ 1 1 ρ 1 ) 13

14 European Options Approximations Combination of singular and regular perturbations = P ε,δ t, x, y, z) P BS t,x;t, σ) + T t) V δ P BS σ + V 1 δ x P BS x σ +T t) V ε x P BS x + V ε 3 x x Leading order Black-Scholes price P BS t,x; σz)): L BS σz))p BS = P BS T, x; σz)) = Hx) at the z-dependent effective volatility σz): σ z) = f, z) x P BS x where the brackets denote the average with respect to the invariant distribution Nm f, ν f ). ) )) 14

15 The small parameters V δ, V1 δ, V ε, V3 ε ) are given by V δ = ν s δ Λ s σ = ν f ε φ Λ f y V ε V δ 1 = ρ ν s δ f σ V3 ε ν f ε = ρ 1 f φ y σ = d σ/dz, and φy, z) is a solution of the Poisson equation L φy, z) = f y, z) σ z). Accuracy: Smooth payoffs: error = Oε + δ) Calls kinks): error = Oε log ε + δ) Digitals jumps): error = Oε /3 log ε + δ) 15

16 P ε 1t, x, z) = T t) solves L BS σ)p ε 1 + Corrections Equations V ε x P BS x V ε x P BS x + V ε 3 x x + V ε 3 x x )) x P BS x )) x P BS x =, P ε 1T, x, z) = ) P1 δ t, x, z) = T t) V δ P BS σ + V 1 δ x P BS x σ solves L BS σ)p δ 1 + V δ P BS σ ) + V 1 δ x P BS x σ =, P δ 1 T, x) = for European options: P BS σ = T t)σx P BS x ) 16

17 Computation of h using P BS only = 1 P BS ht, x, y, z) = fy, z)x 1 P BS t, x; σz)) a P BS t, x; σz)) ρ 1 ν f ε ν f ε 1 ρ 1 ρ ν s δ ρ 1 ν s δ ν s δ 1 ρ ρ 1 = fy, z)x P BS x P BS ν σ z) P BS s δ σ P BS ρ ρ 1 P BS x P BS y P BS z 1 ρ ρ 1 where we have used that P BS does not depend on y. The Vega is given by P BS σ = x T t N d 1 x, z)). 17

18 Computation of h using P = P BS + P ε 1 + P δ 1 = 1 Pt, x, z) P x Pt, x, z) ht, x, y, z) = fy, z)x 1 Pt, x, z) a Pt, x, z) ρ 1 ν f ε ν f ε 1 ρ 1 ρ ν s δ ρ 1 ν s δ ν s δ 1 ρ ρ 1 fy, z)x ν σ z) P BS s δ σ P BS t, x; σz)) ρ ρ 1 P x P y P z 1 ρ ρ 1 where we have used again that P does not depend on y, and neglected terms of higher order. 18

19 Numerical Results European Calls) r m f m s ν f ν s ρ 1 ρ ρ 1 Λ f Λ s fy, z) 1% expy + z) $S Y Z $K T years α δ P MC P BS P P IS P BS ) P IS P) ) ) ) ) ) ) ) ) ) ) ) ) 5 realizations Euler scheme with time step t =.5). 19

20 Two-Step Variance Reduction for Asian Options ) } + Pt, S t, Y t, Z t, I t ) = IE {e rt t) IT T K 1S T K S t, Y t, Z t, I t Basic monte Carlo for AAO s P P MC = e rt t) N where I t = t S udu or di t = S t dt). N k=1 I k) T T K 1S k) T K Unlike the case of European options, the approximated prices of AAOs do not have closed-form solutions F-Han 3). ) + Importance sampling using approximated AAO prices requires numerical PDE solutions = not efficient. = two-step variance reduction strategy by combining control variates and importance sampling.

21 Control Variates for Arithmetic Average Asian Options Constant volatility: Boyle-Broadie-Glasserman proposed a variance reduction method for AAOs based on using GAOs as control variates: P CV A = P MC A + λp MC G P G ) PG MC is the unbiased Monte Carlo estimator of the GAO price computed using the same run as for PA MC. The company price P G, i.e. the counterpart geometric average Asian option, has an analytic solution. The parameter λ is chosen to minimize the sample variance. For Asian options, λ is often chosen equal to -1. 1

22 Two-Step Strategy in the Stochastic Volatility Case Implement GAOs control variates. No closed-form solutions for GAOs. Evaluate GAOs by Monte Carlo using the importance sampling variance reduction technique presented and tested in the European options case. { P G t, V t, L t ) = IE e exp rt t) LT T ) ) } + K 1 S T K V t, L t where V t = S t, Y t, Z t ), and the running sum process L t ) is given by dl t = ln S t dt

23 Approximate GAOs Prices fixed strike case K 1 = ) P G t,x,y,z,l) P G t,x,z,l) where P G = P fix V δ T t) V ε T t) P fix x P fix σ + V ε V ε 3 ) T t) V 1 δ T t)x P fix x σ P fix x + V ε 3 T t) P fix x 3 The leading order term is the homogenized Black-Scholes price: ) P fix L t lnx t, x, L; σ) = exp + lnx + R Nd 1 ) Ke rt t) Nd ) T P fix σ = T t 3 σx P fix x T t 6 σx P fix x with formulas for Rt, T, z), d 1 x, z, L) and d x, z, L). 3

24 Approximate the perfect h G Computation of h G h G t, x, y, z, L) = 1 P G a v,l) P G by approximating P G t, x, y, z, L) by P G t, x, z, L) = h G t, x, z, L) = P G x P G t, x, z, L) fy, z)x ν s δ σz) P fix σ P fix t, x, z, L) ρ ρ 1 1 ρ ρ 1 4

25 Importance Sampling GAOs Monte Carlo r m f m s ν f ν s ρ 1 ρ ρ 1 Λ f Λ s fy, z) 1% expy + z) $S Y Z L $K T years α δ P MC G P IS G PG ) ) ) ) ) ) ) ) ) 5

26 AAOs Monte Carlo by C.V. GAOs and I.S Prices 5 basic Monte Carlo Control Variates with importance sampling Number of realizations ) P CV A = PMC A 1 P MC G PIS G P G ) with α = 75 and δ =.1 Variance reduced from )1 4 to 1.61)1 6 with respective sample means and t =.5, N = 5). 6

27 Control Variates Conventional control variate setup with m multiple controls: P CV = P MC + m λ i ˆP i C P i C) i=1 P MC : unbiased estimator of P obtained by the sample mean of outputs from N i.i.d. realizations. Each ˆP i C represents a sample mean obtained from the same realizations than those used in P MC. ˆP i C is an unbiased estimator of Pi C expression. which admits a closed-form The control variate P CV is an unbiased estimator of P. The control parameters λ i s need to be chosen to minimize the variance of P CV see Glasserman). 7

28 Building New Control Variates Using Ito s formula, the discounted option price satisfies d e rs Ps,S s,y s,z s ) ) = e rs a P)s,S s,y s,z s ) dη s Integrating between the initial time and the maturity time T, and using the terminal condition PT,S T,Y T,Z T ) = HS T ), we have P,S,Y,Z ) = e rt HS T ) T e rs a P)s,S s,y s,z s ) dη s This suggests that the martingale term is the perfect control variate. However it requires the unknown pricing function Ps,x,y,z) through its gradient in the stochastic integrals. 8

29 The approximation Using Approximations Pt,x,y,z) Pt,x,z) P = P BS + P ε 1 + P δ 1 suggests multiple control variates given by the centered martingales: M 1 P) = T M P) = ν δ T e rs P x s, S s, Z s )fy s, Z s )S s dw s ) rs P e z s, S s, Z s )dws obtained by computing a P a P, using that P does not depend on the variable y and the notation Ws = ρ W s ) + ρ 1 W s 1) + 1 ρ ρ 1 W s ) 9

30 Implementation of Control Variates P MC = 1 N ˆP i C = 1 N N k=1 N k=1 e rt HS k) T ) M k) i P) λ i = 1 i = 1, where the superscript corresponds to the kth realization. Since PC 1 = P C =, we deduce P CV = P MC ˆP 1 C ˆP C The variance of this estimator is given by the sum of the quadratic variations of the martingales M 1 P P) and M P P). It involves only the gradient of P P in contrast with the importance sampling method where division by P is required. 3

31 Numerical Results: One-Factor Model r m 1 m ν 1 ν ρ 1 ρ ρ 1 Λ 1 λ fy, z) 1% expy) $S Y Z $K T years α V MC /V IS P) V MC /V CV P BS )

32 Numerical Results: Two-Factor Model r m 1 m ν 1 ν ρ 1 ρ ρ 1 Λ 1 λ fy, z) 1% expy + z) $S Y Z $K T years α δ V MC /V IS P) V MC /V CV P BS )

33 Conclusions and Ongoing Research Monte Carlo simulations play an important role in computational finance Variance reduction techniques are essential Approximations obtained by perturbation methods can be used Control variates are more efficient than importance sampling Path dependent options remain challenging problems 33