Affine Option Pricing Model in Discrete Time Eric Renault 1 Stanislav Khrapov 2 1 Brown University Providence, RI 2 New Economic School Moscow, Russia Third International Moscow Finance Conference November 8, 2013
Introduction
Introduction The Model Option Pricing Estimation Conclusion Continuous vs Discrete Continuous time affine models with stochastic volatility: Cox, Ingersoll, and Ross (1985, Econometrica) Heston (1993, RFS) Duffie, Pan, and Singleton (2000, Econometrica)
Introduction The Model Option Pricing Estimation Conclusion Continuous vs Discrete Popular because of computational convenience ... ... with historical probability measure (P): Robustness to temporal aggregation: Meddahi and Renault (2004, JoE) Robustness to cross-sectional aggregation (portfolio)
Introduction The Model Option Pricing Estimation Conclusion Continuous vs Discrete Popular because of computational convenience ... ... with historical probability measure (P): Robustness to temporal aggregation: Meddahi and Renault (2004, JoE) Robustness to cross-sectional aggregation (portfolio) ... with risk neutral probability measure (Q): Structure preserving change of measure (affine structure preserved) Analytical tractability of computing derivative prices (inverse Fourier transform)
Introduction The Model Option Pricing Estimation Conclusion Discrete Time Extension Volatility model: Darolles, Gourieroux, and Jasiak (2006, JTSA) Gourieroux and Jasiak (2006, JoF) Option pricing model with conditional skewness: Feunou and Tedongap (2012, JBES)
Introduction The Model Option Pricing Estimation Conclusion Discrete Time Extension Advantages of discrete time: Computational/Statistical tractability More flexibility for higher order moments
Introduction The Model Option Pricing Estimation Conclusion Discrete Time Extension Advantages of discrete time: Computational/Statistical tractability More flexibility for higher order moments Challenges of discrete time: Accommodating leverage effect Keeping the advantage of structure preserving change of measure historical/risk-neutral
Affine Stochastic Volatility Model
Introduction The Model Option Pricing Estimation Conclusion The Model CAR Volatility: Darolles, Gourieroux, and Jasiak (2006, JTSA) � � − u σ 2 �� � � − a ( u ) σ 2 � = exp t − b ( u ) E exp � I t t + 1
Introduction The Model Option Pricing Estimation Conclusion The Model CAR Volatility: Darolles, Gourieroux, and Jasiak (2006, JTSA) � � − u σ 2 �� � � − a ( u ) σ 2 � = exp t − b ( u ) E exp � I t t + 1 Log Excess Return � � I t ∪ σ 2 − α ( v ) σ 2 t + 1 − β ( v ) σ 2 � � � � exp {− vr t + 1 } = exp t − γ ( v ) E t + 1
Introduction The Model Option Pricing Estimation Conclusion The Model Joint Return and Volatility − u σ 2 �� − l ( u , v ) σ 2 � � � � � t + 1 − vr t + 1 = exp t − g ( u , v ) E exp � I t
Introduction The Model Option Pricing Estimation Conclusion The Model Joint Return and Volatility − u σ 2 �� − l ( u , v ) σ 2 � � � � � t + 1 − vr t + 1 = exp t − g ( u , v ) E exp � I t l ( u , v ) = a [ u + α ( v )]+ β ( v ) g ( u , v ) = b [ u + α ( v )]+ γ ( v )
Introduction The Model Option Pricing Estimation Conclusion The Model Joint Return and Volatility − u σ 2 �� − l ( u , v ) σ 2 � � � � � t + 1 − vr t + 1 = exp t − g ( u , v ) E exp � I t l ( u , v ) = a [ u + α ( v )]+ β ( v ) g ( u , v ) = b [ u + α ( v )]+ γ ( v ) α ( v ) � = 0 ⇐ ⇒ leverage!
Introduction The Model Option Pricing Estimation Conclusion Risk-Neutral Distribution The affine structure is kept from P to Q when Pricing Kernel = Exponential Affine
Introduction The Model Option Pricing Estimation Conclusion Risk-Neutral Distribution The affine structure is kept from P to Q when Pricing Kernel = Exponential Affine Stochastic Discount Factor (SDF): m 0 ( θ )+ m 1 ( θ ) σ 2 t − θ 1 σ 2 � � M t , t + 1 ( θ ) = exp ( − r f , t ) exp t + 1 − θ 2 r t + 1 risk prices θ 1 ≤ 0 and θ 2 ≥ 0 m 0 ( θ ) , m 1 ( θ ) : bonds and stocks are priced correctly
Introduction The Model Option Pricing Estimation Conclusion Risk-Neutral Distribution Risk-neutral pricing: E Q � r t + 1 , σ 2 �� r t + 1 , σ 2 �� � � � � � = exp ( r f , t ) E M t , t + 1 ( θ ) H H � I t � I t t + 1 t + 1 for any function H
Introduction The Model Option Pricing Estimation Conclusion Risk-Neutral Distribution Risk-neutral pricing: E Q � r t + 1 , σ 2 �� r t + 1 , σ 2 �� � � � � � = exp ( r f , t ) E M t , t + 1 ( θ ) H H � I t � I t t + 1 t + 1 for any function H Risk-neutral distribution: − l ∗ ( u , v ) σ 2 t − g ∗ ( u , v ) E Q � − u σ 2 � �� � � � t + 1 − vr t + 1 = exp exp � I t with l ∗ ( u , v ) = l ( θ 1 + u , θ 2 + v ) − l ( θ 1 , θ 2 ) g ∗ ( u , v ) g ( θ 1 + u , θ 2 + v ) − g ( θ 1 , θ 2 ) =
Introduction The Model Option Pricing Estimation Conclusion Affine Moments Volatility moments: a ′ ( 0 ) σ 2 t + b ′ ( 0 ) σ 2 � � � = E � I t t + 1 − a ′′ ( 0 ) σ 2 t − b ′′ ( 0 ) � σ 2 � � = V � I t t + 1
Introduction The Model Option Pricing Estimation Conclusion Affine Moments Volatility moments: a ′ ( 0 ) σ 2 t + b ′ ( 0 ) σ 2 � � � = E � I t t + 1 − a ′′ ( 0 ) σ 2 t − b ′′ ( 0 ) � σ 2 � � = V � I t t + 1 Return expectation: t ] = α ′ ( 0 ) σ 2 t + 1 + β ′ ( 0 ) σ 2 t + γ ′ ( 0 ) E [ r t + 1 | I σ
Introduction The Model Option Pricing Estimation Conclusion Affine Moments Volatility moments: a ′ ( 0 ) σ 2 t + b ′ ( 0 ) σ 2 � � � = E � I t t + 1 − a ′′ ( 0 ) σ 2 t − b ′′ ( 0 ) � σ 2 � � = V � I t t + 1 Return expectation: t ] = α ′ ( 0 ) σ 2 t + 1 + β ′ ( 0 ) σ 2 t + γ ′ ( 0 ) E [ r t + 1 | I σ Leverage effect: � 1 / 2 � � σ 2 � � V � I t = α ′ ( 0 ) t + 1 � r t + 1 , σ 2 � � φ ≈ Corr � I t t + 1 V [ r t + 1 | I t ] α ( v ) � = 0 ⇐ ⇒ leverage!
Option Pricing
Introduction The Model Option Pricing Estimation Conclusion Generalized Black-Scholes Assume α ( v ) , β ( v ) , γ ( v ) are quadratic, then r t + 1 | I σ t ∼ N ( E [ r t + 1 | I σ t ] , V [ r t + 1 | I σ t ])
Introduction The Model Option Pricing Estimation Conclusion Generalized Black-Scholes Assume α ( v ) , β ( v ) , γ ( v ) are quadratic, then r t + 1 | I σ t ∼ N ( E [ r t + 1 | I σ t ] , V [ r t + 1 | I σ t ]) Option price: C t ( x t , φ ) = E Q � � � 1 − φ 2 � σ 2 �� S t ξ t , t + 1 ( φ ) , t + 1 , K BS t where x t = log ( K / S t ) is the moneyness and t ]+ 1 log ξ t , t + 1 ( φ ) = E Q [ r t + 1 | I σ 2 V Q [ r t + 1 | I σ t ] is price distortion
Introduction The Model Option Pricing Estimation Conclusion Leverage and Volatility Smirk Two effects of φ : Price distortion S t ξ t , t + 1 ( φ ) � 1 − φ 2 � σ 2 Volatility t + 1
Introduction The Model Option Pricing Estimation Conclusion Leverage and Volatility Smirk Two effects of φ : Price distortion S t ξ t , t + 1 ( φ ) � 1 − φ 2 � σ 2 Volatility t + 1 Around φ = 0 the first order effect is through volatility: C t ( x t , φ ) ≈ C t ( x t , 0 )+ k φ · Cov Q � σ 2 � � t + 1 , Φ( d ) � I t with d = 1 x t 2 σ t + 1 − σ t + 1
Introduction The Model Option Pricing Estimation Conclusion Leverage and Volatility Smirk Two effects of φ : Price distortion S t ξ t , t + 1 ( φ ) � 1 − φ 2 � σ 2 Volatility t + 1 Around φ = 0 the first order effect is through volatility: C t ( x t , φ ) ≈ C t ( x t , 0 )+ k φ · Cov Q � σ 2 � � t + 1 , Φ( d ) � I t with d = 1 x t 2 σ t + 1 − σ t + 1 Cov() more positive out of the money = ⇒ the smile is pushed down on the out of the money side
Introduction The Model Option Pricing Estimation Conclusion Leverage and Volatility Smirk 0.32 0.32 φ T 10 0.0 0.30 -0.1 0.30 20 -0.2 30 0.28 40 -0.3 0.28 50 -0.4 -0.5 60 0.26 Implied vol Implied vol 0.26 0.24 0.24 0.22 0.22 0.20 0.18 0.20 −0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 0.20 −0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 0.20 Log-moneyness, log(K/S) Log-moneyness, log(K/S)
Estimation
Introduction The Model Option Pricing Estimation Conclusion Maximum Likelihood Joint likelihood r t + 1 , σ 2 � � σ 2 � � σ 2 t + 1 , σ 2 � � � � t ; c , ρ , δ , φ , θ 2 = f t ; φ , θ 2 f r t + 1 t + 1 σ 2 � � σ 2 � t ; c , ρ , δ � × f t + 1
Introduction The Model Option Pricing Estimation Conclusion Maximum Likelihood Joint likelihood r t + 1 , σ 2 � � σ 2 � � σ 2 t + 1 , σ 2 � � � � t ; c , ρ , δ , φ , θ 2 = f t ; φ , θ 2 f r t + 1 t + 1 σ 2 � σ 2 � � t ; c , ρ , δ � × f t + 1 where � σ 2 t + 1 , σ 2 � � � t ; φ , θ 2 ∼ f r t + 1 Normal σ 2 � � σ 2 � t ; c , ρ , δ � ∼ nc − Gamma f t + 1 σ 2 t + 1 is ARG(1) from Gourieroux and Jasiak (2006, JoF)
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