A probabillistic Numerical Method for Fully Nonlinear PDEs Nizar - PowerPoint PPT Presentation

Motivation : American options The Probabilistic Scheme Numerical results A probabillistic Numerical Method for Fully Nonlinear PDEs Nizar TOUZI Ecole Polytechnique Paris Joint work with Xavier WARIN and Arash FAHIM Computational Methos in Finance Fields Institute, Toronto March 22-24, 2010 Nizar TOUZI Nonlinear Monte Carlo

Motivation : American options The Probabilistic Scheme Numerical results Outline 1 Motivation : American options 2 The Probabilistic Scheme A natural MC-FD scheme The semilinear case The fully nonlinear case 3 Numerical results On the choice of µ and σ Portfolio optimization, 2 state variables Portfolio optimization, 5 state variables Nizar TOUZI Nonlinear Monte Carlo

Motivation : American options The Probabilistic Scheme Numerical results Pricing and Hedging US Options • (Ω , F , P ) , P : risk-neutral measure, complete market • Consider an American option defined by the payoff process { G t , t ≥ 0 } • Then, pricing and hedging reduce to : V 0 = sup { E [ G τ ] : τ stopping time ≤ T } which can be approximated by the (discrete-time) Snell envelop( t k := kT / n ) : � � �� V n V n V n T := G T and t k := max G t k , E t k + 1 |F t k Standard numerical scheme ! Nizar TOUZI Nonlinear Monte Carlo

Motivation : American options The Probabilistic Scheme Numerical results Approximation of conditional expectations Main observation : the latter conditional expectations are regressions : � � � � Y n Y n t i + 1 |F t i = t i + 1 | X t i E E = ⇒ Classical methods from statistics : • Kernel regression <Carrière> • Projection on subspaces of L 2 ( P ) <Longstaff-Schwarz, Gobet-Lemor-Warin AAP05> from numerical probabilistic methods • quantization... <Bally-Pagès SPA03> Stochastic mesh <Broadie-Glasserman> Integration by parts <Lions-Reigner 00, Bouchard-T. SPA04> Nizar TOUZI Nonlinear Monte Carlo

Motivation : American options The Probabilistic Scheme Numerical results Objective : from US option to nonlinear PDEs • Suggest a Monte Carlo type of scheme for nonlinear PDEs • Numerical complexity reduces to the same problem as US options... • Nonlinear PDEs appear in many problems in finance Continuous-time portfolio optimization Algorithmic trading under market impact Hedging in illiquid markets ... Nizar TOUZI Nonlinear Monte Carlo

Motivation : American options A natural MC-FD scheme The Probabilistic Scheme The semilinear case Numerical results The fully nonlinear case Outline 1 Motivation : American options 2 The Probabilistic Scheme A natural MC-FD scheme The semilinear case The fully nonlinear case 3 Numerical results On the choice of µ and σ Portfolio optimization, 2 state variables Portfolio optimization, 5 state variables Nizar TOUZI Nonlinear Monte Carlo

Motivation : American options A natural MC-FD scheme The Probabilistic Scheme The semilinear case Numerical results The fully nonlinear case The Monte Carlo component • Consider the fully nonlinear PDE : D v := ( v , Dv , D 2 v ) 0 = − v t ( t , x ) − F ( t , x , D v ( t , x )) , • Isolate a diffusion part in the equation : − v t ( t , x ) − 1 0 = 21 ∆ v ( t , x ) − f ( t , x , D v ( t , x )) • Let X s = x + 1 W s − t + h , s ≥ t − h , evaluate at ( s , X s ) , and take expectations : �� t � � t − ( v t + 1 0 = E 2 ∆ v )( s , X s ) ds − f ( ., D v ) ( s , X s ) ds t − h t − h � � � t = v ( t − h , x ) − E v ( t , X t ) + f ( ., D v ) ( s , X s ) ds t − h Nizar TOUZI Nonlinear Monte Carlo

Motivation : American options A natural MC-FD scheme The Probabilistic Scheme The semilinear case Numerical results The fully nonlinear case The Finite-Differences component • From the previous slide : ˆ v ( t − h , x ) = E [ˆ v ( t , X t )] + h f ( ., E [ D ˆ v ( t , X t )]) v and D 2 ˆ • Need to avoid the calculation of D ˆ v at each time step = ⇒ Integration by parts � � � � v ( t , X t ) W 2 h − h v ( t , X t ) W h , E [ D 2 ˆ E [ D ˆ v ( t , X t )] = E v ( t , X t )] = E h 2 h yields the numerical scheme : ˆ v ( t − h , x ) � �� � � � v ( t , X t ) W 2 h − h v ( t , X t ) W h = E [ˆ v ( t , X t )] + h f x , E [ˆ v ( t , X t )] , E ˆ , E ˆ h h 2 Nizar TOUZI Nonlinear Monte Carlo

Motivation : American options A natural MC-FD scheme The Probabilistic Scheme The semilinear case Numerical results The fully nonlinear case The Finite-Differences component • From the previous slide : ˆ v ( t − h , x ) = E [ˆ v ( t , X t )] + h f ( ., E [ D ˆ v ( t , X t )]) v and D 2 ˆ • Need to avoid the calculation of D ˆ v at each time step = ⇒ Integration by parts � � � � v ( t , X t ) W 2 h − h v ( t , X t ) W h , E [ D 2 ˆ E [ D ˆ v ( t , X t )] = E v ( t , X t )] = E h 2 h yields the numerical scheme : ˆ v ( t − h , x ) � �� � � � v ( t , X t ) W 2 h − h v ( t , X t ) W h = E [ˆ v ( t , X t )] + h f x , E [ˆ v ( t , X t )] , E ˆ , E ˆ h h 2 Nizar TOUZI Nonlinear Monte Carlo

Motivation : American options A natural MC-FD scheme The Probabilistic Scheme The semilinear case Numerical results The fully nonlinear case Intuition From Greeks Calculation • Using the approximation f ′ ( x ) ∼ h = 0 E [ f ′ ( x + W h )] : � f ′ ( x + y ) e − y 2 / ( 2 h ) f ′ ( x ) ∼ √ dy 2 π � e − y 2 / ( 2 h ) f ( x + y ) y √ = dy h 2 π � � f ( x + W h ) W h = E h • Similarly, by an additional integration by parts : � f ( x + y ) y 2 − h e − y 2 / ( 2 h ) f ′′ ( x ) = √ dy h 2 2 π � � W 2 �� h − h = f ( x + W h ) E h 2 Nizar TOUZI Nonlinear Monte Carlo

Motivation : American options A natural MC-FD scheme The Probabilistic Scheme The semilinear case Numerical results The fully nonlinear case A probabilistic numerical scheme for fully nonlinear PDEs This suggests the following discretization : � � Y n X n = g , t n t n � � � � Y n E n Y n X n t i − 1 , Y n t i − 1 , Z n t i − 1 , Γ n = + f ∆ t i , 1 ≤ i ≤ n , t i − 1 i − 1 t i t i − 1 � � ∆ W t i Z n E n Y n = t i − 1 i − 1 t i ∆ t i � � | ∆ W t i | 2 − ∆ t i Γ n E n Y n = t i − 1 i − 1 t i | ∆ t i | 2 Nizar TOUZI Nonlinear Monte Carlo

Motivation : American options A natural MC-FD scheme The Probabilistic Scheme The semilinear case Numerical results The fully nonlinear case Connection with Finite Differences : X h := x + W h • Consider the binomial approximation of the Brownian motion � 1 � ′′ √ 2 δ { 1 } + 1 “ W h ∼ h 2 δ {− 1 } Then : √ √ � � � � ψ ( X h ) W h ∼ ψ ( x + h ) − ψ ( x − h ) ψ ′ ( X h ) √ = E E h 2 h • With the trinomial approximation of the Brownian motion � 1 � ′′ √ 6 δ { 1 } + 2 3 δ { 0 } + 1 “ W h ∼ 3 h 6 δ {− 1 } Then : √ √ � � ψ ( X h ) W 2 � � h − h ∼ ψ ( x + 3 h ) − 2 ψ ( x ) + ψ ( x − 3 h ) ψ ′′ ( X h ) = E E h 2 3 h Nizar TOUZI Nonlinear Monte Carlo

Motivation : American options A natural MC-FD scheme The Probabilistic Scheme The semilinear case Numerical results The fully nonlinear case Description of the scheme 1. Simulate trajectories of the forward process X (well understood) 2. Backward algorithm : � � � ˆ � Y n X n = g � t n t n � � � � � ˆ ˆ t i − 1 , ˆ t i − 1 , ˆ Y n E n � Y n X n Y n Z n � = + f ∆ t i , 1 ≤ i ≤ n , t i − 1 t i − 1 t i t i − 1 � � � � ∆ W t i � ˆ ˆ Z n E n � Y n = � t i − 1 t i − 1 t i ∆ t i Question : what kind of objet are we simulating ? Nizar TOUZI Nonlinear Monte Carlo

Motivation : American options A natural MC-FD scheme The Probabilistic Scheme The semilinear case Numerical results The fully nonlinear case Backward SDE : Definition Find an F W − adapted ( Y , Z ) satisfying : � T � T Y t = G + F r ( Y r , Z r ) dr − Z r · dW r t t i.e. dY t = − F t ( Y t , Z t ) dt + Z t · dW t and Y T = G where the generator F : Ω × [ 0 , T ] × R × R d − → R , and { F t ( y , z ) , t ∈ [ 0 , T ] } is F W − adapted If F is Lipschitz in ( y , z ) uniformly in ( ω, t ) , and G ∈ L 2 ( P ) , then there is a unique solution satisfying � T | Y t | 2 + E | Z t | 2 dt E sup < ∞ t ≤ T 0 Nizar TOUZI Nonlinear Monte Carlo

Motivation : American options A natural MC-FD scheme The Probabilistic Scheme The semilinear case Numerical results The fully nonlinear case Markov BSDE’s Let X . be defined by the (forward) SDE dX t = b ( t , X t ) dt + σ ( t , X t ) dW t : [ 0 , T ] × R d × R × R d − and F t ( y , z ) = f ( t , X t , y , z ) , f → R g : R d − G = g ( X T ) ∈ L 2 ( P ) , → R If f continuous, Lipschitz in ( x , y , z ) uniformly in t , then there is a unique solution to the BSDE dY t = − f ( t , X t , Y t , Z t ) dt + Z t · σ ( t , X t ) dW t , Y T = g ( X T ) Moreover, there exists a measurable function V : Y t = V ( t , X t ) , 0 ≤ t ≤ T Nizar TOUZI Nonlinear Monte Carlo