Towards efficient option pricing in incomplete markets

Transcription

1 Towards efficient option pricing in incomplete markets GPU TECHNOLOGY CONFERENCE 2016 Shih-Hau Tan Marie Curie Research Project STRIKE 2 University of Greenwich Apr. 6, 2016 (University of Greenwich) Towards efficient option pricing Apr. 6, / 25

2 Outline 1 Motivation 2 Newton-based solver 3 GPU Computing Implementation 4 Conclusion (University of Greenwich) Towards efficient option pricing Apr. 6, / 25

3 Motivation Incomplete Market (University of Greenwich) Towards efficient option pricing Apr. 6, / 25

4 Motivation Nonlinear Option Pricing ˆ Advantages ˆ More reasonable and accurate option price ˆ Easier to do model calibration ˆ Can be used to design better hedging strategies ˆ Types of problems ˆ Nonlinear Black-Scholes Equation ˆ Hamilton-Jacobi-Bellman (HJB) Equation ˆ Backward Stochastic Differential Equation ˆ Challenge ˆ Complicated to solve a single problem (University of Greenwich) Towards efficient option pricing Apr. 6, / 25

5 Motivation Nonlinear Black-Scholes Equation with V t σ2 S 2 2 V V + rs rv = 0, S > 0, t [0, T ) S2 S ˆ σ = σ 0 (1 + Le sign(v SS )) 1/2 (Leland model (1985)) ˆ σ = φ(x) (Barles and Soner model (1998)) ˆ σ = σ 0 (1 ρsv SS ) 1 (Frey-Patie model (2002)) ˆ σ [σ min, σ max ] (Uncertain volatility model (1995)) where V SS = 2 V S 2 (University of Greenwich) Towards efficient option pricing Apr. 6, / 25

6 Motivation Large-Scale Problems (University of Greenwich) Towards efficient option pricing Apr. 6, / 25

7 Motivation Objective ˆ Single nonlinear option pricing problem ˆ Large-scale nonlinear option pricing problems (University of Greenwich) Towards efficient option pricing Apr. 6, / 25

8 Newton-based solver Finite Difference Method V t σ2 S 2 2 V V + rs rv = 0, S > 0, t [0, T ) (1) S2 S with σ = σ(v t, V S, V SS, V ). Apply finite difference implicit scheme on equation (1) V n+1 i t V n i (σn+1 i ) 2 Si 2 which can be simplified as V n+1 i+1 n+1 2Vi + Vi 1 n+1 V n+1 + rs ( S) 2 i i+1 V i 1 n+1 2 S rv n+1 i = 0, a i V n+1 i 1 + b i V n+1 i + c i V n+1 i+1 = V n i, (2) where i represents the spatial discretization and n represents the temporal discretization. (University of Greenwich) Towards efficient option pricing Apr. 6, / 25

9 Newton-based solver Root-finding Problem Equation (2) can be simplified as H(V n+1 )V n+1 = V n, where H is a tridiagonal matrix. Introduce G(V n+1 ) = H(V n+1 )V n+1 V n = 0, then the problem becomes to use Newton s method to solve the root-finding problem of the function G. (University of Greenwich) Towards efficient option pricing Apr. 6, / 25

10 Newton-based solver Features ˆ Can be applied to different cases with different schemes ˆ Complexity = #iterations (linear model) + evaluation of Jacobian matrix ˆ Quadratic convergent rate (University of Greenwich) Towards efficient option pricing Apr. 6, / 25

11 Newton-based solver Complexity Analysis G(V n+1 ) = H(V n+1 )V n+1 V n = 0 ˆ Evaluate a i, b i, c i for H at each iteration and time step ˆ Consider H(V n+1 ) = Σ n+1 H 1 + H 2, Σ n+1 = Diag((σi n+1 ) 2 ), then Jac(G(V n+1 )) = [H(V n+1 )V n+1 ] V n+1 = H(V n+1 ) + Diag(H 1 V n+1 ) (Σ n+1 ) ˆ Tridiagonal solver for (Jac(G)) 1 (University of Greenwich) Towards efficient option pricing Apr. 6, / 25

12 Newton-based solver Parallel Computing At each time step, we have the Newton s iteration as: (University of Greenwich) Towards efficient option pricing Apr. 6, / 25

13 Newton-based solver Batch Operation ˆ Different option pricing problems such as different r, T, K, σ 0 ˆ Combine all problems together (University of Greenwich) Towards efficient option pricing Apr. 6, / 25

14 GPU Computing Implementation Implementation ˆ OpenACC ˆ easier to start ˆ need to be careful on data construct and clauses ˆ CUDA libraries ˆ different libraries for different applications ˆ enable users to get good performance without writing too many codes ˆ Kernel functions ˆ for specific applications ˆ more flexibility to modify ˆ need to write the codes and do memory allocation (University of Greenwich) Towards efficient option pricing Apr. 6, / 25

15 GPU Computing Implementation OpenACC ˆ Find parallelism of the algorithm ˆ #pragma acc kernels { Jacobian() } ˆ #pragma acc parallel loop ˆ Data movement ˆ copyin(), copyout(), present() (University of Greenwich) Towards efficient option pricing Apr. 6, / 25

16 GPU Computing Implementation CUDA Libraries ˆ cublas ˆ cublassgbmv for matrix-vector calculation ˆ cublassnrm2 for Euclidean norm ˆ cusparse ˆ cusparsesgtsv for tridiagonal solver ˆ has problem on cudafree and cudamalloc (University of Greenwich) Towards efficient option pricing Apr. 6, / 25

17 GPU Computing Implementation Kernel Functions ˆ Tridiagonal matrix construction ˆ evaluate a[i], b[i], c[i] ˆ Sparse matrix calculation ˆ evaluate A + B, A b where A, B are tridiagonal matrices ˆ Jacobian matrix ˆ Jac(G(V n+1 )) = H(V n+1 ) + Diag(H 1 V n+1 ) (Σ n+1 ) ˆ contains 2 matrix constructions and 2 level-2 function evaluations (University of Greenwich) Towards efficient option pricing Apr. 6, / 25

18 GPU Computing Implementation Numerical Experiment Consider Frey-Patie model with nonlinear volatility σ = σ 0 (1 ρsv SS ) 1. Parameters are chosen to be S [0, 300], K = 100, T = 1/12, σ 0 = 0.4, ρ = 0.01, r = 0.03, and tolerance for Newton s iteration is tol = 10 3 for single precision and tol = 10 8 for double precision. The grid points for space and time are M = N = We calculate 64, 128, 256 option pricing problems. System information: Processor: Intel(R) Core(TM) i GHz Memory: 4096MB RAM Compiler: gcc 4.7, CUDA 7.0, PGI 15.9 Graphic card : Quadro K2100M (Kepler microarchitecture, compute capability 3.0) (University of Greenwich) Towards efficient option pricing Apr. 6, / 25

19 GPU Computing Implementation Table: Computation time (s) for single precision. means only estimate the time for calling cudafree and cudamalloc once. #Options CPU OpenACC Library Library Kernel Kernel n = n = n = Table: Speed up for single precision. #Options CPU OpenACC Library Library Kernel Kernel n = x 2.5x 2.8x 7.5x 2.0x 4.6x n = x 1.9x 4.0x 2.9x 3.7x 9.2x n = x 2.6x 4.9x 10x 5.5x 17x (University of Greenwich) Towards efficient option pricing Apr. 6, / 25

20 GPU Computing Implementation (University of Greenwich) Towards efficient option pricing Apr. 6, / 25

21 GPU Computing Implementation Table: Computation time (s) for double precision. means only estimate the time for calling cudafree and cudamalloc once. #Options CPU OpenACC Library Library Kernel Kernel n = n = n = Table: Speed up for double precision. #Options CPU OpenACC Library Library Kernel Kernel n = x 1.8x 2.4x 8.2x 2.2x 5.9x n = x 1.9x 3.3x 7.6x 3.8x 10x n = x 1.8x 5.1x 12x 7.2x 23x (University of Greenwich) Towards efficient option pricing Apr. 6, / 25

22 GPU Computing Implementation (University of Greenwich) Towards efficient option pricing Apr. 6, / 25

23 Conclusion Summarize ˆ Newton-based solver for nonlinear option pricing ˆ Batch operation for dealing with large-scale problems ˆ Comparison of using different implementations of doing GPU Computing ˆ Work in progress ˆ multi-asset problems with Alternating Direction Implicit (ADI) method ˆ Asian option pricing problem with semi-lagrangian scheme ˆ Multiple GPUs computing (University of Greenwich) Towards efficient option pricing Apr. 6, / 25

24 Conclusion ˆ Reference [1] M. Ehrhardt (edt): Nonlinear Models in Mathematical Finance, Nova Science Publishers, Inc. New York, [2] J. Guyon, P. Henry-Labordère, Nonlinear Option Pricing, CRC Press, [3] M.B. Giles, E. Laszlo, I. Reguly, J. Appleyard, J. Demouth, GPU implementation of finite difference solvers, Seventh Workshop on High Performance Computational Finance (WHPCF 14), IEEE, ˆ Acknowledgement Special thanks to ˆ Mr. Lung-Sheng Chien from NVIDIA, USA. ˆ Mr. Alvaro Leitao from CWI, the Netherlands. (University of Greenwich) Towards efficient option pricing Apr. 6, / 25

25 Conclusion Shih-Hau Tan web: ts73/ Thank you very much! (University of Greenwich) Towards efficient option pricing Apr. 6, / 25