On a GPU Acceleration of the Stochastic Grid Bundling Method

Transcription

1 On a GPU Acceleration of the Stochastic Grid Bundling Method Álvaro Leitao Rodríguez and Cornelis W. Oosterlee Technische Universiteit Delft (TU Delft) - Centrum Wiskunde & Informatica (CWI) 13/6/2014 Álvaro Leitao Rodríguez and Cornelis W. Oosterlee On a GPU (Technische Acceleration Universiteit of the Stochastic Delft (TU Delft) Grid Bundling - Centrum Method Wiskunde 13/6/2014 & Informatica 1 /(CWI)

2 Outline 1 Introduction 2 Bermudan Options 3 Stochastic Grid Bundling Method 4 Parallel GPU Implementation 5 Results 6 Conclusions Álvaro Leitao Rodríguez and Cornelis W. Oosterlee On a GPU (Technische Acceleration Universiteit of the Stochastic Delft (TU Delft) Grid Bundling - Centrum Method Wiskunde 13/6/2014 & Informatica 2 /(CWI)

3 Introduction and Motivation Multi-dimensional early-exercise option contracts. Increasing the dimensionality. Counterparty Valuation Adjustment (CVA). SGBM becomes expensive. Solution: parallelization of the method. General-Purpose computing on Graphics Processing Units (GPGPU). Álvaro Leitao Rodríguez and Cornelis W. Oosterlee On a GPU (Technische Acceleration Universiteit of the Stochastic Delft (TU Delft) Grid Bundling - Centrum Method Wiskunde 13/6/2014 & Informatica 3 /(CWI)

4 Bermudan Options Right to exercise at a set number 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 ). Álvaro Leitao Rodríguez and Cornelis W. Oosterlee On a GPU (Technische Acceleration Universiteit of the Stochastic Delft (TU Delft) Grid Bundling - Centrum Method Wiskunde 13/6/2014 & Informatica 4 /(CWI)

5 Bermudan options scheme Figure: d-dimensional Bermudan option Álvaro Leitao Rodríguez and Cornelis W. Oosterlee On a GPU (Technische Acceleration Universiteit of the Stochastic Delft (TU Delft) Grid Bundling - Centrum Method Wiskunde 13/6/2014 & Informatica 5 /(CWI)

6 Stochastic Grid Bundling Method Regression-based method. Forward in time: Monte Carlo simulation. Backward in time: Early-exercise policy by using dynamic programming. 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). Álvaro Leitao Rodríguez and Cornelis W. Oosterlee On a GPU (Technische Acceleration Universiteit of the Stochastic Delft (TU Delft) Grid Bundling - Centrum Method Wiskunde 13/6/2014 & Informatica 6 /(CWI)

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: Mapping high-dimensional state space onto a low-dimensional space (least squares regression) 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))). Álvaro Leitao Rodríguez and Cornelis W. Oosterlee On a GPU (Technische Acceleration Universiteit of the Stochastic Delft (TU Delft) Grid Bundling - Centrum Method Wiskunde 13/6/2014 & Informatica 7 /(CWI)

8 Bundling Original: Iterative process (K-means clustering). Problems: Too expensive (time and memory) and distribution. New technique: Equal-partitioning. Two stages: sorting and splitting. SORT SPLIT Figure: Equal partitioning scheme Álvaro Leitao Rodríguez and Cornelis W. Oosterlee On a GPU (Technische Acceleration Universiteit of the Stochastic Delft (TU Delft) Grid Bundling - Centrum Method Wiskunde 13/6/2014 & Informatica 8 /(CWI)

9 Parallel SGBM on GPU First implementation: efficient C-version. NVIDIA CUDA platform. Two parallelization stages: Forward: Monte Carlo simulation. Backward: Bundles at each time step. Memory transfers to/from GPU: Unified Virtual Addressing (UVA). Asynchronous transfers. Page-locked memory accesses. Álvaro Leitao Rodríguez and Cornelis W. Oosterlee On a GPU (Technische Acceleration Universiteit of the Stochastic Delft (TU Delft) Grid Bundling - Centrum Method Wiskunde 13/6/2014 & Informatica 9 /(CWI)

10 Parallel SGBM on GPU - Forward in time One GPU thread per Monte Carlo simulation. Random numbers on the fly : curand library. Avoiding memory transfers and usage: Compute the intrinsic value of the option. Equal-partitioning: sorting criterion calculations. Original: memory transfers and usage cannot be avoided. Álvaro Leitao Rodríguez and Cornelis W. Oosterlee On a GPU (Technische Acceleration Universiteit of the Stochastic Delft (TU Delft) Grid Bundling - Centrum Method Wiskunde 13/6/2014 & Informatica 10 /(CWI)

11 Monte Carlo implementation scheme Figure: SGBM Monte Carlo Álvaro Leitao Rodríguez and Cornelis W. Oosterlee On a GPU (Technische Acceleration Universiteit of the Stochastic Delft (TU Delft) Grid Bundling - Centrum Method Wiskunde 13/6/2014 & Informatica 11 /(CWI)

12 Parallel SGBM on GPU - Backward in time One GPU thread per bundle. Each bundle calculations (option value and regression) in parallel. All threads collaborate in order to compute the continuation value. Final reduction: Thrust library. Álvaro Leitao Rodríguez and Cornelis W. Oosterlee On a GPU (Technische Acceleration Universiteit of the Stochastic Delft (TU Delft) Grid Bundling - Centrum Method Wiskunde 13/6/2014 & Informatica 12 /(CWI)

13 Backward implementation scheme Figure: SGBM backward stage Álvaro Leitao Rodríguez and Cornelis W. Oosterlee On a GPU (Technische Acceleration Universiteit of the Stochastic Delft (TU Delft) Grid Bundling - Centrum Method Wiskunde 13/6/2014 & Informatica 13 /(CWI)

14 Results Accelerator Island system of Cartesius Supercomputer. Intel Xeon E (Sandy Bridge). NVIDIA Tesla K20m. C-compiler: GCC CUDA version: 5.5. Geometric basket Bermudan put option: S t0 = (40,..., 40) R d, K = 40, r t = 0.06, σ = (0.2,..., 0.2) R d, ρ ij = 0.25, T = 1 and M = 10. Multi-dimensional Geometric Brownian Motion. Euler discretization, δt = T /M. Álvaro Leitao Rodríguez and Cornelis W. Oosterlee On a GPU (Technische Acceleration Universiteit of the Stochastic Delft (TU Delft) Grid Bundling - Centrum Method Wiskunde 13/6/2014 & Informatica 14 /(CWI)

15 Results d True price 5d Equal part. 15d True price 15d Equal part. 50d True price 50d Equal part Vt0 (S0) Bundles ν Figure: Convergence of SGBM with Equal-partitioning technique. Álvaro Leitao Rodríguez and Cornelis W. Oosterlee On a GPU (Technische Acceleration Universiteit of the Stochastic Delft (TU Delft) Grid Bundling - Centrum Method Wiskunde 13/6/2014 & Informatica 15 /(CWI)

16 Results Table: Time (s) for the C and CUDA versions. Test configuration: N = 2 22, δt = T /M and ν = k-means Equal-partitioning 5d 10d 15d 5d 10d 15d C CUDA Speedup Álvaro Leitao Rodríguez and Cornelis W. Oosterlee On a GPU (Technische Acceleration Universiteit of the Stochastic Delft (TU Delft) Grid Bundling - Centrum Method Wiskunde 13/6/2014 & Informatica 16 /(CWI)

17 Results Table: Time (s) for a high-dimensional problem with equal-partitioning. Test configuration: N = 2 22 and δt = T /M. ν = 2 12 ν = d 40d 50d 30d 40d 50d C CUDA Speedup Álvaro Leitao Rodríguez and Cornelis W. Oosterlee On a GPU (Technische Acceleration Universiteit of the Stochastic Delft (TU Delft) Grid Bundling - Centrum Method Wiskunde 13/6/2014 & Informatica 17 /(CWI)

18 Conclusions Efficient parallel GPU implementation. Extend the SGBM s applicability: Increasing dimensionality and amount of bundles. New bundling technique. Future work: American options. CVA calculations. Álvaro Leitao Rodríguez and Cornelis W. Oosterlee On a GPU (Technische Acceleration Universiteit of the Stochastic Delft (TU Delft) Grid Bundling - Centrum Method Wiskunde 13/6/2014 & Informatica 18 /(CWI)

19 References CUDA webpage. home new.html. curand webpage. Shashi Jain and Cornelis W. Oosterlee. The stochastic grid bundling method: Efficient pricing of bermudan options and their greeks, Thrust webpage. Álvaro Leitao Rodríguez and Cornelis W. Oosterlee On a GPU (Technische Acceleration Universiteit of the Stochastic Delft (TU Delft) Grid Bundling - Centrum Method Wiskunde 13/6/2014 & Informatica 19 /(CWI)

20 Questions and/or suggestions Álvaro Leitao Rodríguez and Cornelis W. Oosterlee On a GPU (Technische Acceleration Universiteit of the Stochastic Delft (TU Delft) Grid Bundling - Centrum Method Wiskunde 13/6/2014 & Informatica 20 /(CWI)

21 Acknowledgements Thanks Álvaro Leitao Rodríguez and Cornelis W. Oosterlee On a GPU (Technische Acceleration Universiteit of the Stochastic Delft (TU Delft) Grid Bundling - Centrum Method Wiskunde 13/6/2014 & Informatica 20 /(CWI)

22 Appendix Basis functions: φ k (S tm ) = ( ( ) k 1 d St δ m ) 1 d, k = 1,..., 5, 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 ) 1 d P tm 1 (n) = St δ m 1 (n), µ = 1 d δ=1 d δ=1 ( ) ( r q δ σ2 δ, σ 2 = 1 d d 2 Cpq) 2. 2 d 2 p=1 q=1 Álvaro Leitao Rodríguez and Cornelis W. Oosterlee On a GPU (Technische Acceleration Universiteit of the Stochastic Delft (TU Delft) Grid Bundling - Centrum Method Wiskunde 13/6/2014 & Informatica /(CWI)