Stochastic Grid Bundling Method

Transcription

1 Stochastic Grid Bundling Method GPU Acceleration Delft University of Technology - Centrum Wiskunde & Informatica Álvaro Leitao Rodríguez and Cornelis W. Oosterlee London - December 17, 2015 A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

2 Outline 1 Introduction 2 Basket Bermudan Options 3 Stochastic Grid Bundling Method 4 Continuation value computation: new approach 5 Parallel GPU Implementation 6 Results 7 Conclusions A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

3 Introduction and Motivation Originally introduced by Jain and Oosterlee in Multi-dimensional early-exercise option contracts. Increase the dimensionality. SGBM becomes very expensive. Solution: parallelization of the method. GPU computing (GPGPU). A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

4 Basket Bermudan Options Right to exercise at a set of times: t [t 0 = 0,..., t m,..., t M = T ]. d-dimensional underlying process: S t = (S 1 t,..., S d t ) R d. Intrinsic value of the option: h t := h(s t ). The value of the option at the terminal time T: V T (S T ) = max(h(s T ), 0). The conditional continuation value Q tm, i.e. the discounted expected payoff at time t m : Q tm (S tm ) = D tm E [ V tm+1 (S tm+1 ) S tm ]. The Bermudan option value at time t m and state S tm : V tm (S tm ) = max(h(s tm ), Q tm (S tm )). Value of the option at the initial state S t0, i.e. V t0 (S t0 ). A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

5 Basket Bermudan options scheme Figure: d-dimensional Bermudan option A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

6 Stochastic Grid Bundling Method Simulation and regression-based method. Forward in time: Monte Carlo simulation. Backward in time: Early-exercise policy computation. Step I: Generation of stochastic grid points {S t0 (n),..., S tm (n)}, n = 1,..., N. Step II: Option value at terminal time t M = T V tm (S tm ) = max(h(s tm ), 0). A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

7 Stochastic Grid Bundling Method (II) Backward in time, t m, m M,: Step III: Bundling into ν non-overlapping sets or partitions B tm 1 (1),..., B tm 1 (ν) Step IV: Parameterizing the option values Z(S tm, α β t m ) V tm (S tm ). Step V: Computing the continuation and option values at t m 1 Q tm 1 (S tm 1 (n)) = E[Z(S tm, α β t m ) S tm 1 (n)]. The option value is then given by: V tm 1 (S tm 1 (n)) = max(h(s tm 1 (n)), Q tm 1 (S tm 1 (n))). A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

8 Bundling Original: Iterative process (K-means clustering). Problems: Too expensive (time and memory) and distribution. New technique: Equal-partitioning. Efficient for parallelization. Two stages: sorting and splitting. SORT SPLIT Figure: Equal partitioning scheme A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

9 Parametrizing the option value Basis functions φ 1, φ 2,..)., φ K. In our case, Z (S tm, αt β m depends on S tm only through φ k (S tm ): ) K Z (S tm, αt β m = αt β m (k)φ k (S tm ). k=1 Computation of αt β m (or α t β m ) by least squares regression. The αt β m determines the early-exercise policy. The continuation value: [( K ) ] Q tm 1 (S tm 1 (n)) = D tm 1 E α t β m (k)φ k (S tm ) S tm 1 k=1 K = D tm 1 α t β m (k)e [ ] φ k (S tm ) S tm 1. k=1 A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

10 Basis functions Choose φ k : the expectations E [ φ k (S tm ) S tm 1 ] should be easy to calculate. The intrinsic value of the option (payoff), h( ), is usually an important and useful basis function. For S t following a GBM: expectations available analytically for geometric and arithmetic basket Bermudan options. Other underlying models: new proposed approach based on the discrete characteristic function. A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

11 Estimating the option value SGBM has been developed as duality-based method. Provide two estimators (confidence interval). Direct estimator (high-biased estimation): ( V tm 1 (S tm 1 (n)) = max h ( S tm 1 (n) ), Q ( tm 1 Stm 1 (n) )), E[ V t0 (S t0 )] = 1 N N V t0 (S t0 (n)). n=1 Path estimator (low-biased estimation): τ (S(n)) = min{t m : h (S tm (n)) Q tm (S tm (n)), m = 1,..., M}, v(n) = h ( S τ (S(n))), 1 V t0 (S t0 ) = lim NL N L N L v(n). n=1 A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

12 SGBM - schematic algorithm Data: S t0, X, µ δ, σ δ, ρ i,j, T, N, M Pre-Bundling (only in k-means case). Generation of the grid points (Monte Carlo). Step I. Option value at terminal time t = M. Step II. for Time t = (M 1)... 1 do Bundling. Step III. for Bundle β = 1... ν do Exercise policy (Regression). Step IV. Continuation value. Step V. Direct estimator. Step V. Generation of the grid points (Monte Carlo). Step I. Option value at terminal time t = M. Step II. for Time t = (M 1)... 1 do Bundling. Step III. for Bundle β = 1... ν do Continuation value. Step V. Path estimator. Step V. A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

13 Cont. value computation: new approach More generally applicable. More involved models or options. First discretize, then derive the discrete characteristic function. S 1 t m+1 = S 1 tm + µ 1(S tm ) t + σ 1 (S tm ) W 1 t m+1, S 2 t m+1 = S 2 tm + µ 2(S tm ) t + ρ 1,2 σ 2 (S tm ) W 1 t m+1 + L 2,2 σ 2 (S tm ) W 2 t m+1, S d t m+1 = S d tm + µ d (S tm ) t + ρ 1,d σ d (S tm ) W 1 t m+1 + L 2,d σ d (S tm ) W 2 t m L d,d σ d (S tm ) W d t m+1, By definition, the d-variate discrete characteristic function: ( ) d ψ S u1, u t 2,..., u d S tm = E exp iu j S j m+1 t S tm m+1 j=1 d j = E exp iu j S j tm + µ j (S tm ) t + σ j (S tm ) L k,j W k t S tm m+1 j=1 k=1 d = exp iu j (S ) t) d d j tm + µ j (S tm E exp iu j L k,j σ j (S tm ) W k t m+1 j=1 k=1 j=k d = exp iu j (S ) t) d d j tm + µ j (S tm ψ N (0, t) u j L k,j σ j (S tm ), j=1 k=1 j=k A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

14 Cont. value computation: new approach Joint moments of the product: ) M Stm+1 = E [(S 1 c1 ) tm+1 (S 2 c2 ) tm+1 (S d cd ] tm+1 Stm [ ] = ( i) c 1+c 2 + +c c 1 +c 2 + +c d ψ Stm+1 (u S tm ) d u c 1 1 uc 2 2 uc d So, if the basis functions are the product of asset processes: ( d ) k 1 φ k (S tm ) = St δ m, k = 1,..., K, δ=1 However, this approx. is, in general, worse than analytic. Feasible thank to the GPU implementation: time steps. d u=0, A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

15 Parallel SGBM on GPU NVIDIA CUDA platform. Parallel strategy: two parallelization stages: Forward: Monte Carlo simulation. Backward: Bundles Oportunity of parallelization. Novelty in early-exercise option pricing methods. Two implementations K-means vs. Equal-partitioning: K-means: sequential parts. K-means: transfers between CPU and GPU cannot be avoided. K-means: all data need to be stored. K-means: Load-balancing. Equal-partitioning: fully parallelizable. Equal-partitioning: No transfers. Equal-partitioning: efficient memory use. A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

16 Parallel SGBM on GPU - Forward in time One GPU thread per Monte Carlo simulation. Random numbers on the fly : curand library. Compute intermediate results: Expectations. Intrinsic value of the option. Equal-partitioning: sorting criterion calculations. Intermediate results in the registers: fast memory access. Original bundling: all the data still necessary. A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

17 Parallel SGBM on GPU - Forward in time Figure: SGBM Monte Carlo A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

18 Parallel SGBM on GPU - Backward in time One parallelization stage per exercise time step. Sort w.r.t bundles: efficient memory access. Parallelization in bundles. Each bundle calculations (option value and early-exercise policy) in parallel. All GPU threads collaborate in order to compute the continuation value. Path estimator: One GPU thread per path (the early-exercise policy is already computed). Final reduction: Thrust library. A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

19 Parallel SGBM on GPU - Backward in time Figure: SGBM backward stage A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

20 Parallel SGBM on GPU - Backward in time Figure: SGBM backward stage A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

21 Results Accelerator Island system of Cartesius Supercomputer. Intel Xeon E v2. NVIDIA Tesla K40m. C-compiler: GCC CUDA version: 5.5. Geometric and arithmetic basket Bermudan put options: S t0 = (40,..., 40) R d, X = 40, r t = 0.06, σ = (0.2,..., 0.2) R d, ρ ij = 0.25, T = 1 and M = 10. Basis functions: K = 3. Multi-dimensional Geometric Brownian Motion. Euler discretization, δt = T /M. A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

22 Equal-partitioning: convergence test d Reference price 5d Direct estimator 5d Path estimator 10d Reference price 10d Direct estimator 10d Path estimator 15d Reference price 15d Direct estimator 15d Path estimator d Direct estimator 5d Path estimator 10d Direct estimator 10d Path estimator 15d Direct estimator 15d Path estimator Vt0 (St0) 1.3 Vt0 (St0) Bundles ν (a) Geometric basket put option Bundles ν (b) Arithmetic basket put option Figure: Convergence with equal-partitioning bundling technique. Test configuration: N = 2 18 and t = T /M. A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

23 Speedup Geometric basket Bermudan option k-means equal-partitioning d = 5 d = 10 d = 15 d = 5 d = 10 d = 15 C CUDA Speedup Arithmetic basket Bermudan option k-means equal-partitioning d = 5 d = 10 d = 15 d = 5 d = 10 d = 15 C CUDA Speedup Table: SGBM total time (s) for the C and CUDA versions. Test configuration: N = 2 22, t = T /M and ν = A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

24 Speedup - High dimensions Geometric basket Bermudan option ν = 2 10 ν = 2 14 d = 30 d = 40 d = 50 d = 30 d = 40 d = 50 C CUDA Speedup Arithmetic basket Bermudan option ν = 2 10 ν = 2 14 d = 30 d = 40 d = 50 d = 30 d = 40 d = 50 C CUDA Speedup Table: SGBM total time (s) for a high-dimensional problem with equal-partitioning. Test configuration: N = 2 20 and t = T /M. A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

25 Cont. value computation: New approach Reference price Direct estimator Path estimator Reference price Direct estimator Path estimator 1.26 Vt0 (St0) Vt0 (St0) MC Steps = T/ t (a) Geometric basket put option MC Steps = T/ t (b) Arithmetic basket put option Figure: CEV model convergence, γ = 1.0. Test configuration: N = 2 16, ν = 2 10 and d = 5. A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

26 Cont. value computation: New approach Geometric basket Bermudan option γ = 0.25 γ = 0.5 γ = 0.75 γ = 1.0 SGBM DE SGBM PE Arithmetic basket Bermudan option γ = 0.25 γ = 0.5 γ = 0.75 γ = 1.0 SGBM DE SGBM PE Table: CEV option pricing. Test configuration: N = 2 16, t = T /4000, ν = 2 10 and d = 5. A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

27 Conclusions Efficient parallel GPU implementation. Extend the SGBM s applicability: Increasing dimensionality. New bundling technique. More general approach to compute the continuation value. Future work: Explore the new CUDA 7 features: cusolver (QR factorization). CVA calculations. A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

28 References Shashi Jain and Cornelis W. Oosterlee. The Stochastic Grid Bundling Method: Efficient pricing of Bermudan options and their Greeks. Applied Mathematics and Computation, 269: , Álvaro Leitao and Cornelis W. Oosterlee. GPU Acceleration of the Stochastic Grid Bundling Method for Early-Exercise options. International Journal of Computer Mathematics, 92(12): , A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

29 Acknowledgements Thank you for your attention A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

30 Appendix Geo. basket Bermudan option - Basis functions: φ k (S tm ) = ( ( ) k 1 d St δ m ) 1 d, k = 1,..., K, δ=1 The expectation can directly be computed as: E [ φ k (S tm ) S tm 1 (n) ] = ( P tm 1 (n)e ( ) ) k 1 µ+ (k 1) σ2 t 2, where, ( d P tm 1 (n) = St δ m 1 (n) δ=1 ) 1 d ( d, µ = 1 d δ=1 r q δ σ2 δ 2 ), σ 2 = 1 d 2 2 d d Cpq 2. p=1 q=1 A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31

31 Appendix Arith. basket Bermudan option - Basis functions: φ k (S tm ) = ( 1 d ) k 1 d St δ m, k = 1,..., K., δ=1 The summation can be expressed as a linear combination of the products: ( d ) k St δ m = δ=1 k 1 +k 2 + +k d =k ( k k 1, k 2,..., k d ) 1 δ d ( S δ t m ) kδ, And the expression for Geometric basket option can be applied. A. Leitao & Kees Oosterlee (TUD & CWI) SGBM on GPU London - December 17, / 31