A framework for adaptive Monte-Carlo procedures

Transcription

1 A framework for adaptive Monte-Carlo procedures Jérôme Lelong (with B. Lapeyre) Journées MAS Bordeaux Friday 3 September 2010 J. Lelong (ENSIMAG LJK) Journées MAS 2010 Bordeaux 1 / 26

2 Outline Randomly truncated algorithm: Chen s technique Averaging 4 The Basic idea Numerical implementation J. Lelong (ENSIMAG LJK) Journées MAS 2010 Bordeaux 2 / 26

3 Monte-Carlo framework Compute E(Z) using a Monte-Carlo method Suppose we know that E(Z) = E[H(θ,X)],θ R d Aim: use the previous representation to reduce the variance. Find a way to minimize the variance. θ s.t. θv(θ) v(θ ). v(θ) = Var(H(θ,X)) = E [ H(θ,X) 2] E[Z] 2 J. Lelong (ENSIMAG LJK) Journées MAS 2010 Bordeaux 3 / 26

4 The algorithms I Algorithm (Non adaptive importance sampling, Fu and Su (2002), Arouna (2003)) 1 Draw (X 1,...,X n ) to compute θ n an estimator of θ. 2 Draw (X 1,...,X n ) independent of (X 1,...,X n ) and compute ξ n = 1 n n H(θ n,x i ). i=1 J. Lelong (ENSIMAG LJK) Journées MAS 2010 Bordeaux 4 / 26

5 The algorithms II Algorithm (Adaptive Importance Sampling, Arouna (2004)) Draw (X 1,...,X n ) and do for i = 1 : n 1 Compute an estimator θ i of θ using (X 1,...,X i ) 2 Update ξ i Remarks: ξ i+1 = i i + 1 ξ i + 1 i + 1 H(θ i,x i+1 ), with ξ 0 = 0. No need to store the whole sequence (X 1,...,X n ) for computing ξ n ξ n = 1 n n i=1 H(θ i 1,X i ) J. Lelong (ENSIMAG LJK) Journées MAS 2010 Bordeaux 5 / 26

6 Common frameworks I Importance sampling framework in a Gaussian setting If G N (0,I d ). For all θ R d, E [ f (G) ] ] θ 2 θ G = E [e 2 f (G + θ), ] θ 2 θ G+ v(θ) = E [e 2 f 2 (G) E[f (G)] 2. v is strongly convex if ε > 0 s.t. E[ f (G) 2+ε ] >. Arouna (2004 and 2005), Lemaire and Pagès (2008), Jourdain and L. (2009) J. Lelong (ENSIMAG LJK) Journées MAS 2010 Bordeaux 6 / 26

7 Common frameworks II Escher transform Let X be a r.v. in R d with density p and θ R d. p θ (x) = p(x)e θ x ψ(θ) with ψ(θ) = loge [ e θ X ]. Let X (θ) have p θ as a density, then [ ] E[f (X)] = E f (X (θ) ) p(x (θ) ) p θ (X (θ), ) [ v(θ) = E f (X) 2 p(x) ] E[f (X)] 2. p θ (X) v is strongly convex if ε > 0 s.t. E[ f (G) 2+ε ] > and lim θ p θ (x) = 0 for all x. Kawai (2007 and 2008), Lemaire and Pagès (2008). J. Lelong (ENSIMAG LJK) Journées MAS 2010 Bordeaux 7 / 26

8 1 2 3 Randomly truncated algorithm: Chen s technique Averaging 4 The Basic idea Numerical implementation J. Lelong (ENSIMAG LJK) Journées MAS 2010 Bordeaux 8 / 26

9 An adaptive strong law Assume E[Z] = E[H(θ,X)] for all θ and v(θ) = Var(H(θ,X)) is strongly convex. Let (X n,n 0) be i.i.d X. F n = σ(x 1,...,X n ). Theorem 1 Let (θ n ) n 0 be a (F n ) adapted sequence with values in R d, s. t. for all n 0, θ < p.s. (H1) For any compact subset K R d, sup θ K E[ H(θ,X) 2 ] < 1 n (H2) inf v(θ) > 0 and v(θ i ) < θ R d n i=0 Then, ξ n = 1 n n H(θ i 1,X i ) i=1 a.s n E(Z). J. Lelong (ENSIMAG LJK) Journées MAS 2010 Bordeaux 9 / 26

10 Remarks on the assumptions (H1) : No need of E [ sup θ K H(θ,X) 2] < by the use of locally square integrable martingales. If θ E[ H(θ,X) 2 ] is continuous, (H1) is equivalent to E[ H(θ,X) 2 ] < for all θ R d. (H2) : is clearly true when θ n converges to a deterministic constant θ and v is continuous at θ. No assumption to be checked along the path (θ n ) n. J. Lelong (ENSIMAG LJK) Journées MAS 2010 Bordeaux 10 / 26

11 A Central Limit Theorem Theorem 2 Let the sequence (θ n,n 0) be adapted to F n. Assume (H1), (H2) and θ n θ p.s. η > 0 s.t. θ E[ H(θ,X) 2+η ] is continuous at θ and finite θ R d. v is continuous at θ and v(θ ) > 0 Then, Moreover, assume that ( ) 1 n n ξ D n E[Z] N n (0,v(θ )). η > 0 s.t. θ E[ H(θ,X) 4+η ] is continuous at θ and finite θ R d. Then, n σ n (ξ n E[Z]) D N (0,1) with n σ2 n = 1 n n H(θ i 1,X i ) 2 ξ 2 n J. Lelong (ENSIMAG LJK) Journées MAS 2010 Bordeaux 11 / 26 i=1

12 1 2 3 Randomly truncated algorithm: Chen s technique Averaging 4 The Basic idea Numerical implementation J. Lelong (ENSIMAG LJK) Journées MAS 2010 Bordeaux 12 / 26

13 Randomly truncated algorithm: Chen s technique Variance minimisation v is strongly convex v is infinitely differentiable and θ v(θ) = θ E [ (H(θ,X) 2 ) ] = E[U(θ,X)] Minimizing v is equivalent to finding θ s.t. E [ U(θ,X) ] = 0 J. Lelong (ENSIMAG LJK) Journées MAS 2010 Bordeaux 13 / 26

14 Randomly truncated algorithm: Chen s technique Randomly truncated procedure Let (γ n ) n 0 s.t. n γ n = + and n γ 2 n < +. For θ 0 K 0 and α 0 = 0, we define if θ n+ 1 2 if θ n+ 1 2 θ n+ 1 = θ n γ n+1 U(θ n,x n+1 ), 2 α n+1 = α n, K αn θ n+1 = θ n+ 1 2 K αn θ n+1 = θ 0 α n+1 = α n + 1. α n = number of truncations up to time n. θ n+1 = T Kαn ( θn γ n+1 U(θ n,x n+1 ) ) θ n is F n measurable and X n+1 is independent of F n. Introduced by Chen and Zhu (1986). J. Lelong (ENSIMAG LJK) Journées MAS 2010 Bordeaux 14 / 26

15 Randomly truncated algorithm: Chen s technique a.s convergence Let u(θ) = E[U(θ,X)]. Theorem 3 (L., 2008) Assume (H3) u is continuous and! θ R d,u(θ ) = 0 and θ R d, θ θ, (θ θ u(θ)) > 0. For all q > 0, sup θ q E[ U(θ,X) 2 ] <. Then, the sequence (θ n ) n converges a.s. to θ and the sequence (α n ) n is a.s. finite. Previous results from Chen and Zhu (1986), and Chen, Gao and Guo (1988). (H3) satisfied if U is the gradient of a strictly convex function J. Lelong (ENSIMAG LJK) Journées MAS 2010 Bordeaux 15 / 26

16 Averaging Moving window average Assume γ n = γ (n+1) α with 1/2 < α < 1. For all τ > 0, we set ˆθ n (τ) = γ p τ p+ τ/γ p i=p θ i with p = sup{k 1 : k + τ/γ k n} n. Averaging smooths the convergence. Averaging from a strictly positive rank reduces the impact of the initial condition. If (θ n ) n converges, so does ( ˆθ n (τ)) n for all τ > 0. True Césaro averaging : Polyak and Juditsky (1992), Pelletier (2000), Andrieu and Moulines (2006). J. Lelong (ENSIMAG LJK) Journées MAS 2010 Bordeaux 16 / 26

17 1 2 3 Randomly truncated algorithm: Chen s technique Averaging 4 The Basic idea Numerical implementation J. Lelong (ENSIMAG LJK) Journées MAS 2010 Bordeaux 17 / 26

18 The Basic idea General problem I Generalised multidimensional Black-Scholes Model ds t = S t (µ(t,s t )dt + σ(t,s t ) dw t ), S 0 = x Payoff ˆψ(S t,t T), price V 0 = E[e rt ˆψ(S t,t T)] Can be approximated by ˆV 0 = E[e rt ˆψ(S T1,...,S Td )] With G N (0,I d ) ˆV 0 = E[ψ(G)] = E [ψ(g + Aθ)e Aθ G Aθ 2 2 ] with θ R p and A R d p, p << d. J. Lelong (ENSIMAG LJK) Journées MAS 2010 Bordeaux 18 / 26

19 The Basic idea General Problem II [ [ Minimise v(θ) = E ψ(g + Aθ) 2 e 2Aθ G Aθ 2] = E ψ(g) 2 e Aθ 2 Aθ G+ 2 ]. v(θ) = E [A (Aθ G)ψ(G) 2 e Aθ G+ Aθ 2 2 v(θ) = E [ A Gψ(G + Aθ) 2 e 2Aθ G+Aθ2] ] U 1 (θ,g) = A (Aθ G)φ(G) 2 Aθ 2 Aθ G+ e 2 U 2 (θ,g) = A Gφ(G + Aθ) 2 e 2Aθ G+ Aθ 2 we can write v(θ) = E[U 2 (θ,g)] = E[U 1 (θ,g)] to construct two estimators of θ : (θ 1 n ) n and (θ 2 n ) n J. Lelong (ENSIMAG LJK) Journées MAS 2010 Bordeaux 19 / 26

20 The Basic idea Bespoke estimators We define θ 1 n+1 = T K αn ( θ 1 n γ n+1 U 1 (θ 1 n,g n+1) ) involves φ(g) θ 2 n+1 = T K αn ( θ 2 n γ n+1 U 2 (θ 2 n,g n+1) ) involves φ(g + Aθ 2 n ) and their averaging versions ( θ 1 n) n and ( θ 2 n) n. For the different estimators of θ, we can define as many approximations of E(φ(G)) ξ 1 n = 1 n n i=1h(θ 1 i 1,G i), ξ 2 n = 1 n H(θ 2 i 1 n,g i), i=1 ξ 1 n = 1 n n i=1h( θ 1 i 1,G i ), ξ2 n = 1 n H( θ 2 i 1,G i ), n i=1 Aθ 2 Aθ G with H(θ,G) = φ(g + Aθ)e 2 and where the sequence (G n ) n has already been used to build the (θ n ) n estimators. J. Lelong (ENSIMAG LJK) Journées MAS 2010 Bordeaux 20 / 26

21 The Basic idea Complexity of the different estimators We assume : complexity number of evaluations of φ. Non-adaptive algorithms : we need 2n samples to achieve a convergence rate v(θ )/n. Complexity: 2n, efficient only when v(θ ) v(0)/2. Adaptive algorithms : we need n samples to achieve a convergence rate v(θ )/n. Estimators ξ 1 2 ξ ξ 1 ξ 2 Complexity 2n n 2n 2n FIG.: Complexities of the different estimators J. Lelong (ENSIMAG LJK) Journées MAS 2010 Bordeaux 21 / 26

22 Numerical implementation Basket Option : ( d i=1 ωi S i T K ) + where (ω 1,...,ω d ) R d ρ K γ Price Var MC Var ξ 2 Var ξ TAB.: Basket option in dimension d = 40 with r = 0.05, T = 1, S i 0 = 50, σi = 0.2, ω i = 1 for all i = 1,...,d and n = d Estimators MC ξ 2 ξ 2 CPU time TAB.: CPU times for the option of Table 1. J. Lelong (ENSIMAG LJK) Journées MAS 2010 Bordeaux 22 / 26

23 Numerical implementation Barrier Basket option: ( D i=1 ωi S i T K ) +1 { i D, j N, S i tj L i } B t1 B t2. B tn 1 B tn t1 I D t1 I D t2 t 1 I D = G tn 1 t N 2 I D 0 t1 t2 t 1 I D... tn 1 t N 2 I D tn t N 1 I D where I D is the identity matrix in dimension D. t1 I D t2 t 1 I D A =. tn t N 1 I D G + Aθ corresponds to (B t1 + θt 1,B t2 + θt 2,...,B tn + θt N ). θ R 5 whereas G R 120. J. Lelong (ENSIMAG LJK) Journées MAS 2010 Bordeaux 23 / 26

24 Numerical implementation Barrier Basket option K γ Price Var MC Var ξ 2 Var ξ 2 Var Var ξ 2 Var ξ 2 Var θ 2 +MC θ2 +MC reduced reduced reduced TAB.: Down and Out Call option in dimension I = 5 with σ = 0.2, S 0 = (50,40,60,30,20), L = (40,30,45,20,10), ρ = 0.3, r = 0.05, T = 2, ω = (0.2,0.2,0.2,0.2,0.2) and n = Estimators MC ξ 2 ξ 2 θ 2 + MC ξ 2 ξ 2 θ 2 + MC reduced reduced reduced CPU time TAB.: CPU times for the option of Table 3. J. Lelong (ENSIMAG LJK) Journées MAS 2010 Bordeaux 24 / 26

25 Numerical implementation e4 2e4 3e4 4e4 5e4 1e4 2e4 3e4 4e4 5e4 6e4 7e4 8e4 9e4 10e4 FIG.: approximation of θ with averaging FIG.: approximation of θ without averaging J. Lelong (ENSIMAG LJK) Journées MAS 2010 Bordeaux 25 / 26

26 Numerical implementation Conclusion It always reduces the variance The extra computational cost can be negligible No regularity assumptions on the payoff Averaging improves the robustness of the algorithm w.r.t the step sequence but adds an extra computational cost To encounter the fine tuning of the algorithm, one can use sample average approximation (Jourdain and L., 2008), but it cannot be implemented in an adaptive manner which increases its computational cost J. Lelong (ENSIMAG LJK) Journées MAS 2010 Bordeaux 26 / 26