On the impact of correlation on option prices: a Malliavin Calculus - PowerPoint PPT Presentation

On the impact of correlation on option prices: a Malliavin Calculus approach

RESULTS FROM E. Alòs (2006): A generalization of the Hull and White formula with applications to option pricing approximation. Finance and Stochastics 10 (3), 353-365. E. Alòs, J. A. León and J. Vives (2007): On the short-time behaviour of the implied volatility for stochastic volatility models with jumps. Finance and Stochastics 11 (4), 571-589

STOCHASTIC VOLATILITY MODELS Stochastic volatility models allow us to describe the smiles and skews observed in real market data: ( )  −  1 2 = σ + σ ρ + − ρ dX r dt dW * 2 dB *   1 t t t t t  2  Log-price Volatility (stochastic, adapted to the filtration generated by W) ρ ≠ 0 Implied volatility skew ρ = 0 Implied volatility smile

SOME QUESTIONS AND MOTIVATION How to quantify the impact of correlation on option prices? What about the term structure? We will develop a formula of the form Option price =option price in the uncorrelated case (classical Hull and White formula) + correction due by correlation This result will allow us to describe the impact of the correlation on the option prices. As an application, we can use it to construct option pricing approximation formulas, or to study the short-time behaviour of the implied volatility for stochastic volatility models with jumps.

SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (I) Here our pourpose is to present the basic concepts on Malliavin calculus that have been used up to now in financial applications. Basically, we will see how to: Calculate the Malliavin derivative of a diffusion process Use the duality relationship between the Malliavin derivative and the Skorohod integral to develop adequate change-of-variable formulas for anticipating processes MAIN IDEA: THE FUTURE INTEGRATED VOLATILITY IS AN ANTICIPATING PROCESS

SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (II) Malliavin derivative : definition { } [ ] ( ) ( ) ∈ W h h L 2 T , 0 , Gaussian process ( ( ) ( ) ( ) ) = F f W h W h W h , ..., n 1 2 [ ] ( ) × Ω L 2 T random variable in 0 , ∂ f ( ( ) ( ) ( ) ) ( ) = ∑ D F W h W h W h h t , ..., t 1 2 n i ∂ x i [ ] ( ) × Ω 2 L T Malliavin Derivative in 0 , (closable operator)

SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (III) Malliavin derivative: examples W = D W t 1 , ] ) ( [ t s s 0     σ 2   = + σ S t W exp r - (Black - Scholes)     t t 2     W = σ D S S r 1 ( ) [ ] r t t t 0 , ( ) t ( ) ∫ − α − α − = + − t + t r Y m Y m e c e dW t r 0 0 − Ornstein Uhlenbeck ( ) ( ) W − α − = t r D Y ce r 1 ( ) [ ] r t t 0 ,

SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (IV) Skorohod integral: definition It is the adjoint of the Malliavin derivative operator: ( ) ( ) ( ) T ∫ δ W = W ∈ E u F E D F u ds F S , for all s s 0 Example: [ ] ( ) ( ) ( ) ∈ ⇒ δ = W h L 2 T h W h 0 ,

SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (V) Skorohod integral: properties The Skorohod integral of a process multiplied by a random variable ( ) T T T ∫ ∫ ∫ = + W Fu dW F u dW D F u ds s s s s s s 0 0 0 The Skorohod integral is an extension of the classical Itô integral

SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (VI) THE ANTICIPATING ITÔ’S FORMULA t t ∫ ∫ = + + X X u dW v ds t s s s 0 0 0 Non necessarily adapted ( ) ( ) ( ) t ∫ = + F X F X F X u dW ' t s s s 0 0 1 ( ) ( )( ) t t ∫ ∫ + + ∇ F X v ds F X u u ds ' ' ' , s s s s s 2 0 0 ( ) s s ∫ ∫ ∇ = + W + W u u D u dW D v dr where : 2 2 s s s r r s r 0 0

SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (VI) Proof (sketch) ≡ v For the sake of simplicity we assume 0 . We proceed as in the proof of the classical Itô’s formula ( )   ( ) ( ) ∑ t ∫ + = + i 1 F X F X F X  u dW  ' t t  s s  0 i t i ( ) 1 2   t ∑ ∫ + i + 1 F X  u dW  ' ' t  s s  2 i t i 1 t ∫ u s ds 2 2 0

SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (VII) ( )   t ∑ ∫ + i 1 F X  u dW  ' t  s s  i t i ( ) ( ) t t ∑∫ ∑∫ + + = i 1 − i 1 W F X u dW D F X u ds ' ' t s s s t s t i t i i i 1 t ∫ F X u dW ' ( ) s s s 2 0   t s ∫ ∫ W F X  D u dW  u ds ' ' ( ) s  s r r  s 0 0 [ ] ( ) t ∫ − W D F X u ds ' s s s 0

AN EXTENSION OF THE HULL AND WHITE FORMULA (I) Basic idea The Black-Scholes function final condition implies that ( ) = ν V BS T X , ; T T T   T 1 Black-Scholes  ∫  = σ v 2 2 ds Log-price   t s − T t pricing formula   t where [ ] = − r T − t * ( ) V E e H F t t option price payoff

AN EXTENSION OF THE HULL AND WHITE FORMULA (II) Then ( ) = − r T − t ( ) * V e E V F t T t [ ] ( ) = ν E * BS T X F , ; T T t compare The classical Hull and White term ( ) ( ) ν * E BS t X F , ; t t t is the option price in the uncorrelated case

AN EXTENSION OF THE HULL AND WHITE FORMULA (III) Then we want to evaluate the difference ( ) ( ) ν − ν BS T X BS t X , ; , ; T T t t ν is an anticipati ng process t We need to construct an adequate anticipating Itô’s formula

AN EXTENSION OF THE HULL AND WHITE FORMULA (IV) Anticipating Itô’s formula ∂ F ( ) ( ) ( ) t ∫ = + F t X Y F X Y s X Y ds , , 0 , , , , t t 0 0 s s ∂ s 0 ∂ ∂ F F ( ) ( ) t t ∫ ∫ + + s X Y dX s X Y dY , , , , s s s s s s ∂ ∂ x y 0 0 ∂ ) ( ) 2 F t ( additional ∫ + − s X Y D Y u ds , , s s s s ∂ ∂ term x y 0 ∂ 2 F ( ) t ∫ + s X Y u 2 ds , , s s s ∂ x 2 0 T ∫ ( ) = θ Y ds T ∫ − = θ W D Y D dr : t s t s s r s

AN EXTENSION OF THE HULL AND WHITE FORMULA (V) Main result: the extension of the Hull and White formula We apply the above Itô’s anticipating formula to the process   ( ) 1   − rt ν = − rt e BS t X e BS t X Y , ; , ;   t t t t − T t   and we obtain

AN EXTENSION OF THE HULL AND WHITE FORMULA (VI) ( ) ( ) − − − rT = rT ν = rt ν e V e BS T X e BS t X , , , , T T T t t     ∂ ∂ ( ) 2 1 ( ) ( ) T ∫ −   + rs ν + σ − ν − ν e L 2 2 BS s X ds , ,     BS s s s s s ∂ ∂ 2 x x 2 t     ) ( ) ∂ BS T ( ∫ + − ν σ ρ + − ρ rs * 2 * e s X dW dZ , , 1 s s s s t ∂ x t ρ ∂ 2 ) ( ) BS 1 ( ) t ∫ + − rs ν − σ e s X D Y ds , , ( s s s s ∂ ∂ σ ν − x T s 2 0 s ) ( ) σ − ν ∂ 2 2 BS 1 ( T ∫ − − rs ν e s X s s ds , , ( ) s s ∂ σ ν − T s 2 t s Black-Scholes differential operator Cancel Zero expectation

AN EXTENSION OF THE HULL AND WHITE FORMULA (VII) ( ) ( ) ( ) − − rT = rt ν e E * V F e E * BS t X F , , T t t t t ρ ∂ 2 ( ) BS 1 t ( ) ∫ + − ν − σ rs e s X D Y ds , , s s s s ∂ ∂ σ ν − x T s 2 ( ) 0 s   ∂ 3 ∂ 2   ( ) ( ) T ∫ *   Λ = σ σ − ν = ν W 2 s X H s X  D dr  , , : , , :   s s s s s  s r  s ∂ ∂ x 3 x 2   s ( ) ( ) − − = r T t V e E * V F t T t ( ) ( ) = ν E * BS t X F , , t t t ρ   ( ) t ( ) ∫ − − + r s t ν Λ E * e H s X ds F , ,   s s s t   2 0

AN EXTENSION OF THE HULL AND WHITE FORMULA (VIII) The above arguments do not requiere the volatility to be Markovian. The main contribution of this formula is to describe the effect of the correlation as the term ρ   t ( ) ( ) ∫ − − r s t ν Λ E * e H s X ds F  , ,  s s s t   2 0

APPLICATIONS TO OPTION PRICING APPROXIMATION (I) Consider the approximation ( ) = ν V BS t X * , , aprox t t ρ ( )   T ∫ + ν Λ H t X * E * ds F , , ,   t t s t   2 t ( ) 1 T ∫ ν = σ * * 2 E F ds t s t − T t t