Ninomiya-Victoir scheme: strong convergence, antithetic version and - PowerPoint PPT Presentation

Ninomiya-Victoir scheme: strong convergence, antithetic version and application to multilevel estimators CERMICS ´ Ecole des Ponts ParisTech project team ENPC-INRIA-UPEM Mathrisk April 18, 2016 Joint work with Benjamin Jourdain and Emmanuelle Cl´ ement

Outline Introduction 1 Monte Carlo Methods 2 The Standard Monte Carlo Method The Multilevel Monte Carlo Numerical Schemes 3 The Ninomiya-Victoir Scheme An antithetic version of the Ninomiya-Victoir scheme Application to the Heston Model 4 The Heston model The Ninomiya-Victoir scheme in the Heston model

Goal We are interested in the computation, by Monte Carlo methods, of the expectation Y = E [ f ( X T )], where X = ( X t ) t ∈ [0 , T ] is the solution to a multidimensional stochastic differential equation (SDE) and � f ( X T ) 2 � f : R n �→ R a given function such that E < + ∞ . We will focus on minimizing the computational complexity subject to a given target error ǫ ∈ R ∗ + . To measure the accuracy of an estimator ˆ Y , we will consider the root mean squared error: �� 2 � � � � 1 ˆ � Y − ˆ RMSE Y ; Y = E Y . (1) 2 � � �

Itˆ o-type SDE We consider a general Itˆ o-type SDE of the form d  σ j ( X t ) dW j � dX t = b ( X t ) dt +  t (2) j =1  X 0 = x where: x ∈ R n , ( X t ) t ∈ [0 , T ] is a n − dimensional stochastic process, � W 1 , . . . , W d � W = is a d − dimensional standard Brownian motion, b , σ 1 , . . . , σ d : R n → R n are Lipschitz continuous.

Outline Introduction 1 Monte Carlo Methods 2 The Standard Monte Carlo Method The Multilevel Monte Carlo Numerical Schemes 3 The Ninomiya-Victoir Scheme An antithetic version of the Ninomiya-Victoir scheme Application to the Heston Model 4 The Heston model The Ninomiya-Victoir scheme in the Heston model

Standard Monte Carlo Method The standard Monte Carlo method consists in: discretizing the SDE, using a numerical scheme X N , with N ∈ N ∗ steps, approximating the expectation using M ∈ N ∗ independent path simulations. To be clear, the crude Monte Carlo estimator is given by M Y CMC = 1 � � X N , k ˆ � (3) f T M k =1 where X N , k are independent copies of a numerical scheme X N .

Complexity analysis Bias � � � � � � �� ˆ ˆ X N B Y CMC ; Y = E Y CMC − Y = E f − E [ f ( X T )] . (4) T The bias is related to the weak error of the scheme: � 1 � = c 1 � � � � X N E f − f ( X T ) N α + o . (5) T N α Variance = 1 � � � � �� ˆ X N (6) V Y CMC M V f . T Cost � ǫ − ( 2+ 1 α ) � C CMC = C × M × N = O (7) .

Outline Introduction 1 Monte Carlo Methods 2 The Standard Monte Carlo Method The Multilevel Monte Carlo Numerical Schemes 3 The Ninomiya-Victoir Scheme An antithetic version of the Ninomiya-Victoir scheme Application to the Heston Model 4 The Heston model The Ninomiya-Victoir scheme in the Heston model

The Multilevel Monte Carlo The main idea of this technique is to use the following telescopic summation to control the bias: L � � �� � � � � �� X 2 L X 2 l X 2 l − 1 X 1 � � � �� f = E f + f − f . E E T T T T l =1 Then, a generalized multilevel Monte Carlo estimator is built as follows: L M l 1 ˆ � � Z l Y MLMC = (8) k M l l =0 k =1 � Z l � where 0 ≤ l ≤ L , 1 ≤ k ≤ M l are independent random variables such that: k Z 0 � X 1 � � � �� = E f (9) E T and: � Z l � � � � � �� X 2 l X 2 l − 1 ∀ l ∈ { 1 , . . . , L } , E = E − f (10) f . T T

Bias and variance Bias � � � � � � �� X 2 L ˆ ˆ B Y MLMC ; Y = E Y MLMC − Y = E f − E [ f ( X T )] . (11) T The bias is related to the weak error of the scheme: � 1 � = c 1 � � � � X 2 L − f ( X T ) 2 α L + o (12) E f . T 2 α L Variance L 1 � � � Z l � ˆ � = (13) V Y MLMC V . M l l =0

Cost and canonical exemple Cost For a given discretization level l ∈ { 0 , . . . , L } , the computational cost of simulating one sample Z l is C λ l 2 l , where: C ∈ R + is a constant, depending only on the discretization scheme, ∀ l ∈ N , λ l ∈ Q ∗ + is a weight, depending only on l , L � M l λ l 2 l . C MLMC = C (14) l =0 Natural choice for Z l , l ∈ { 0 , . . . , L } Z 0 = f X 1 � � (15) T Z l = f � � � � X 2 l X 2 l − 1 − f , ∀ l ∈ { 1 , . . . , L } . (16) T T For this canonical choice, it is natural to take λ 0 = 1 and λ l = 3 2 .

Optimal complexity Theorem (Complexity theorem (Giles)) � 2 such that ∀ l ∈ N : + × R ∗ and ∃ ( β, c 2 ) ∈ Assume that ∃ ( α, c 1 ) ∈ R ∗ R ∗ � + � 1 − Y = c 1 � � � �� X 2 l 2 α l + o (17) E f T 2 α l and � 1 � = c 2 � Z l � 2 β l + o . (18) V 2 β l Then, the optimal complexity is given by: C ∗ ǫ − 2 �  � MLMC = O if β > 1 ,    � �� 2 �  � � 1  C ∗ ǫ − 2  MLMC = O log if β = 1 , (19) ǫ    � ǫ − 2+ β − 1 �  C ∗  MLMC = O if β < 1 . α 

Optimal parameters Optimal parameters � √ �  2 | c 1 |  log 2 L ∗ = ǫ (20)   α      � � �� L ∗ 2 V [ Z 0 ] M ∗ � � λ 0 V [ Z 0 ] + c 2 λ l 2 l (1 − β ) 0 = (21)   ǫ 2 λ 0   l =1   � �� L ∗ �� 2 � c 2 � ∀ l ∈ { 1 , . . . , L ∗ } , M ∗ � λ 0 V [ Z 0 ] + c 2 λ l 2 l (1 − β ) l = . (22) ǫ 2 λ l 2 l ( β +1) l =1 Regression One can estimate ( α, β, c 1 , c 2 ) by using a regression: ∼ c 2 � Z l � (23) V 2 β l ∼ c 1 (1 − 2 α ) � Z l � (24) E . 2 α l

Outline Introduction 1 Monte Carlo Methods 2 The Standard Monte Carlo Method The Multilevel Monte Carlo Numerical Schemes 3 The Ninomiya-Victoir Scheme An antithetic version of the Ninomiya-Victoir scheme Application to the Heston Model 4 The Heston model The Ninomiya-Victoir scheme in the Heston model

Stratonovich form Assuming C 1 regularity for diffusion coefficients σ 1 , . . . , σ d , the Itˆ o-type SDE can be written in Stratonovich form: d  σ j ( X t ) ◦ dW j dX t = σ 0 ( X t ) dt + �  t (25) j =1  X 0 = x d where σ 0 = b − 1 ∂σ j σ j and ∂σ j is the Jacobian matrix of σ j defined � 2 j =1 as follows ∂σ j = ∂σ j � ∂ x k σ ij � �� � � ] = ] . (26) ik i , k ∈ [ [1; n ] i , k ∈ [ [1; n ]

The Ninomiya-Victoir scheme Notations � t k = k T � ] is the subdivision of [0 , T ]. N k ∈ [ [0; N ] η N = ( η 1 , . . . , η N ) is a sequence of independent, identically distributed Rademacher random variables independent of W . ∀ j ∈ { 1 , . . . , d } , ∆ W j t k +1 = W j t k +1 − W j t k . For j ∈ { 0 , . . . , d } and x 0 ∈ R d , let (exp( t σ j ) x 0 ) t ∈ R solve the ODE � dx ( t ) = σ j ( x ( t )) dt x (0) = x 0 . Scheme If η k +1 = 1 � T � T � � X NV , N ,η N � t k +1 σ d � � t k +1 σ 1 � X NV , N ,η N 2 N σ 0 ∆ W 1 2 N σ 0 ∆ W d = exp exp . . . exp exp t k +1 t k and if η k +1 = − 1 � T � T � � X NV , N ,η N � t k +1 σ d � � t k +1 σ 1 � X NV , N ,η N 2 N σ 0 ∆ W 1 ∆ W d 2 N σ 0 = exp exp . . . exp exp . t k +1 t k

Order 2 of weak convergence Denoting by ( X x t ) t ≥ 0 the solution to the SDE starting from X x 0 = x ∈ R n , for f : R n → R n smooth, u ( t , x ) = E [ f ( X x t )] solves the Feynman-Kac PDE � ∂ u ∂ t ( t , x ) = Lu ( t , x ) , ( t , x ) ∈ [0 , ∞ ) × R n u (0 , x ) = f ( x ) , x ∈ R n = σ 0 + 1 � d with L = b . ∇ x + 1 ( σ 1 , . . . , σ d )( σ 1 , . . . , σ d ) ∗ ∇ 2 j =1 ( σ j ) 2 � � 2 Tr x 2 the infinitesimal generator. ∂ 2 u ∂ t 2 = ∂ ∂ t Lu = L ∂ ∂ t u = L 2 u and u ( t 1 , x ) = f ( x ) + t 1 Lf ( x ) + t 2 2 L 2 f ( x ) + O ( t 3 1 1 ) . Ninomiya and Victoir have designed their scheme so that )] = f ( x ) + t 1 Lf ( x ) + t 2 E [ f ( X NV , N ,η N 2 L 2 f ( x ) + O ( t 3 1 1 ) . t 1 N steps One step error O ( 1 → O ( 1 N 3 ) − N 2 ) global error.

Order 1 / 2 of strong convergence Theorem (Strong convergence) Assume that the vector fields, b , ∀ j ∈ { 1 , . . . , d } , σ j and ∂σ j σ j are Lipschitz continuous functions. Then: ∀ p ≥ 1 , ∃ C NV ∈ R ∗ + , ∀ N ∈ N ∗ � � � 2 p ≤ C NV � � X t k − X NV , N ,η N � � max (27) E � η N p . � � � t k � � 0 ≤ k ≤ N

Outline Introduction 1 Monte Carlo Methods 2 The Standard Monte Carlo Method The Multilevel Monte Carlo Numerical Schemes 3 The Ninomiya-Victoir Scheme An antithetic version of the Ninomiya-Victoir scheme Application to the Heston Model 4 The Heston model The Ninomiya-Victoir scheme in the Heston model