Prices and Asymptotics of Variance Swaps Carole Bernard Zhenyu - PowerPoint PPT Presentation

Prices and Asymptotics of Variance Swaps Carole Bernard Zhenyu (Rocky) Cui Beirut, May 2013. Carole Bernard Lebanese Mathematical Society 1/32

Motivation Convex Order Conjecture Variance Swap Numerics Conclusions Outline ◮ Motivation ◮ Convex order conjecture ◮ Discrete variance swaps: prices and asymptotics ◮ Conclusion & Future Directions Carole Bernard Lebanese Mathematical Society 2/32

Motivation Convex Order Conjecture Variance Swap Numerics Conclusions Variance Swap ◮ A variance swap is an OTC contract: � 1 � Notional × T Realized Variance − Strike n − 1 � 2 � S ti +1 ◮ Realized Variance: RV = � ln with S ti i =0 0 = t 0 < t 1 < ... < t n = T . ◮ Quadratic Variation: QV = i =0 , 1 ,..., n − 1 ( t i +1 − t i ) → 0 RV . lim n →∞ , max ◮ In practice, variance swaps are discretely sampled but it is typically easier to compute the continuously sampled in popular stochastic volatility models. ◮ Question: Finding “fair” strikes so that the initial value of the contract is 0. Carole Bernard Lebanese Mathematical Society 3/32

Motivation Convex Order Conjecture Variance Swap Numerics Conclusions Model Setting (1/2) ◮ Under the risk-neutral probability measure Q , rdt + √ V t dW (1) � dS t = t S t ( M ) µ ( V t ) dt + σ ( V t ) dW (2) dV t = t where E [ dW (1) dW (2) ] = ρ dt . t t Carole Bernard Lebanese Mathematical Society 4/32

Motivation Convex Order Conjecture Variance Swap Numerics Conclusions Three Stochastic Volatility Models Assume E [ dW (1) dW (2) ] = ρ dt . t t ◮ The correlated Heston model: = rdt + √ V t dW (1) � dS t , t ( H ) S t = κ ( θ − V t ) dt + γ √ V t dW (2) dV t t ◮ The correlated Hull-White model: = rdt + √ V t dW (1) � dS t , t S t ( HW ) = µ V t dt + σ V t dW (2) dV t t • The correlated Sch¨ obel-Zhu model: � = rdt + V t dW (1) dS t t S t ( SZ ) = κ ( θ − V t ) dt + γ dW (2) dV t t Carole Bernard Lebanese Mathematical Society 5/32

Motivation Convex Order Conjecture Variance Swap Numerics Conclusions Model Setting (2/2) ◮ Under the risk-neutral probability measure Q , rdt + √ V t dW (1) � dS t = t S t ( M ) µ ( V t ) dt + σ ( V t ) dW (2) dV t = t where E [ dW (1) dW (2) ] = ρ dt . t t ◮ The fair strike of the “discrete variance swap” is � n − 1 � 2 � d ( n ) := 1 � ln S t i +1 = 1 K M � T E [ RV ] T E S t i i =0 ◮ The fair strike of the “continuous variance swap” is �� T c := 1 � = 1 K M T E V s ds T E [ QV ] 0 Carole Bernard Lebanese Mathematical Society 6/32

Motivation Convex Order Conjecture Variance Swap Numerics Conclusions Contributions ◮ A general expression for the fair strike of a discrete variance swap in the time-homogeneous stochastic volatility model: ◮ Application in three popular stochastic volatility models (1) Heston model: more explicit than Broadie and Jain (2008). (2) Hull-White model: a new closed-form formula. (3) Sch¨ obel-Zhu model: : a new closed-form formula. ◮ Asymptotic expansion of the fair strike with respect to n , T , vol of vol... ◮ A counter-example to the “ Convex Order Conjecture ”. Carole Bernard Lebanese Mathematical Society 7/32

Motivation Convex Order Conjecture Variance Swap Numerics Conclusions Convex Order Conjecture ◮ Notations: (1) RV = � n − 1 i =0 (log( S t i +1 / S t i )) 2 : discrete realized variance for a partition of [0 , T ] with n + 1 points; � T (2) QV = 0 V s ds : continuous quadratic variation . ◮ Usual practice: approximate E [ f ( RV )] with E [ f ( QV )], see Jarrow et al (2012). ◮ B¨ ulher (2006): “while the approximation of realized variance via quadratic variation works very well for variance swaps, it is not sufficient for non-linear payoffs with short maturities ”. ◮ Call option on RV: ( RV − K ) + ; Call option on QV: ( QV − K ) + . Carole Bernard Lebanese Mathematical Society 8/32

Motivation Convex Order Conjecture Variance Swap Numerics Conclusions Convex Order Conjecture (Cont’d) ◮ The convex-order conjecture (Keller-Ressel (2011)): “ The price of a call option on realized variance is higher than the price of a call option on quadratic variation ” ◮ Equivalently, E [ f ( RV )] � E [ f ( QV )] where f is convex. ◮ When f ( x ) = x , our closed-form expression shows that when the correlation between the underlying and its variance is positive, it is possible to observe K M d ( n ) < K M (Illustrated by c examples in Heston, Hull-White and Sch¨ obel-Zhu models ( M )). Carole Bernard Lebanese Mathematical Society 11/32

Motivation Convex Order Conjecture Variance Swap Numerics Conclusions Conditional Black-Scholes Representation ◮ Recall rdt + √ V t dW (1) � dS t = t S t ( M ) µ ( V t ) dt + σ ( V t ) dW (2) dV t = t ◮ Cholesky decomposition: dW (1) = ρ dW (2) 1 − ρ 2 dW (3) � + . t t t ◮ Key representation of the log stock price � T ln( S T ) = ln( S 0 ) + rT − 1 V t dt 2 0 � T � T � � V t dW (3) � � 1 − ρ 2 + ρ f ( V T ) − f ( V 0 ) − h ( V t ) dt + t 0 0 √ z � v σ ( z ) dz , h ( v ) = µ ( v ) f ′ ( v ) + 1 2 σ 2 ( v ) f ′′ ( v ) . where f ( v ) = 0 Carole Bernard Lebanese Mathematical Society 12/32

Motivation Convex Order Conjecture Variance Swap Numerics Conclusions Proposition Under some technical conditions, ( ∆ = T n ):, � t +∆ �� � 2 � ln S t +∆ = r 2 ∆ 2 − r ∆ E [ V s ] ds E S t t ��� t +∆ � t +∆ � 2 � + 1 + (1 − ρ 2 ) 4 E V s ds E [ V s ] ds t t ��� t +∆ � 2 � � ( f ( V t +∆ ) − f ( V t )) 2 � + ρ 2 E + ρ 2 E h ( V s ) ds t �� t +∆ � t +∆ � + ρ E h ( V s ) ds V s ds t t � t +∆ � � − ρ E ( f ( V t +∆ ) − f ( V t )) (2 ρ h ( V s ) + V s ) ds . t Carole Bernard Lebanese Mathematical Society 13/32

Motivation Convex Order Conjecture Variance Swap Numerics Conclusions Sensitivity w.r.t. interest rate Proposition (Sensitivity to r ) The fair strike of the discrete variance swap: d ( n ) = b M ( n ) − T c r + T n r 2 , K M n K M where b M ( n ) does not depend on r. dK M d ( n ) = T n (2 r − K M c ) dr d ( r ) reaches minimum when r ∗ = K M K M 2 . c Carole Bernard Lebanese Mathematical Society 14/32

Motivation Convex Order Conjecture Variance Swap Numerics Conclusions Three Stochastic Volatility Models Assume E [ dW (1) dW (2) ] = ρ dt . t t ◮ The correlated Heston model: = rdt + √ V t dW (1) � dS t , t ( H ) S t = κ ( θ − V t ) dt + γ √ V t dW (2) dV t t ◮ The correlated Hull-White model: = rdt + √ V t dW (1) � dS t , t S t ( HW ) = µ V t dt + σ V t dW (2) dV t t • The correlated Sch¨ obel-Zhu model: � = rdt + V t dW (1) dS t t S t ( SZ ) = κ ( θ − V t ) dt + γ dW (2) dV t t Carole Bernard Lebanese Mathematical Society 15/32

Motivation Convex Order Conjecture Variance Swap Numerics Conclusions Heston Model The fair strike of the discrete variance swap is 1 κ 2 T ( θ − 2 r ) 2 + n θ � � 4 κ 2 − 4 ρκγ + γ 2 �� K H � d ( n ) = 2 κ T 8 n κ 3 T � 1 − e κ T � γ 2 ( θ − 2 V 0 ) + 2 κ ( V 0 − θ ) 2 � � e − 2 κ T − 1 n + n κ T 1 + e n 2 κ 2 + γ 2 − 2 ρκγ � � 1 − e − κ T � + κ 2 T ( θ − 2 r ) � � � +4 ( V 0 − θ ) n + 4 ( V 0 − θ ) κ T γ ( γ − 2 ρκ ) 1 − e − κ T � � � 1 − e − κ T − 2 n 2 θγ ( γ − 4 ρκ ) n κ T 1 − e n The fair strike of the continuous variance swap is �� T c = 1 � = θ + (1 − e − κ T ) V 0 − θ K H T E V s ds . κ T 0 Carole Bernard Lebanese Mathematical Society 16/32

Motivation Convex Order Conjecture Variance Swap Numerics Conclusions Hull-White Model The fair strike of the discrete variance swap is ( n ) = r 2 T + V 0 � 1 − rT � ( e µ T − 1) K HW d n µ T n � e ( 2 µ + σ 2 ) T − 1 � � µ T � � e ( 2 µ + σ 2 ) T − 1 � V 2 V 2 e n − 1 0 0 � + − 2 T (2 µ + σ 2 )( µ + σ 2 ) � ( 2 µ + σ 2 ) T 2 T µ ( µ + σ 2 ) e − 1 n � 3(4 µ + σ 2) T � � 3(4 µ + σ 2) T � µ T V 03 / 2 σ ( e V 03 / 2 σ n − 1) 8 ρ e − 1 64 ρ e − 1 8 8 + − � � 3 T (4 µ + σ 2 ) (4 µ + 3 σ 2 ) 3(4 µ + σ 2) T µ T (4 µ + 3 σ 2 ) e − 1 8 n The fair strike of the continuous variance swap is �� T = 1 � = V 0 T µ ( e µ T − 1) . K HW T E V s ds c 0 Carole Bernard Lebanese Mathematical Society 17/32

Motivation Convex Order Conjecture Variance Swap Numerics Conclusions Sch¨ obel-Zhu Model The fair strike of the discrete variance swap is explicit but too complicated to appear on a slide. The fair strike of the continuous variance swap is = γ 2 � ( V 0 − θ ) 2 γ 2 � 2 κ + θ 2 + K SZ (1 − e − 2 κ T ) − c 4 κ 2 T 2 κ T + 2 θ ( V 0 − θ ) (1 − e − κ T ) . κ T Carole Bernard Lebanese Mathematical Society 18/32

Motivation Convex Order Conjecture Variance Swap Numerics Conclusions Heston model: Expansion w.r.t n � 1 c + a H � K H d ( n )= K H 1 n + O . n 2 where a H 1 is a linear and decreasing function of ρ : 1 a H ρ � ρ H 1 � 0 ⇐ ⇒ 0 where � 4 + θγ 2 � θ 2 r 2 T − rK H c T + T + c 1 8 κ ρ H 0 = . � � γ ( θ − V 0 ) (1 − e − κ T ) − θγ T − 2 κ 2 1 Explicit expression of a H 1 is in Proposition 5 . 1, Bernard and Cui (2012). Carole Bernard Lebanese Mathematical Society 19/32