Towards efficient option pricing in incomplete markets GPU TECHNOLOGY CONFERENCE 2016 Shih-Hau Tan 1 2 1 Marie Curie Research Project ”STRIKE” 2 University of Greenwich Apr. 6, 2016 (University of Greenwich) Towards efficient option pricing Apr. 6, 2016 1 / 25
Outline 1 Motivation 2 Newton-based solver 3 GPU Computing Implementation 4 Conclusion (University of Greenwich) Towards efficient option pricing Apr. 6, 2016 2 / 25
Motivation Incomplete Market (University of Greenwich) Towards efficient option pricing Apr. 6, 2016 3 / 25
Motivation Nonlinear Option Pricing ❼ Advantages ❼ More reasonable and accurate option price ❼ Easier to do model calibration ❼ Can be used to design better hedging strategies ❼ Types of problems ❼ Nonlinear Black-Scholes Equation ❼ Hamilton-Jacobi-Bellman (HJB) Equation ❼ Backward Stochastic Differential Equation ❼ Challenge ❼ Complicated to solve a single problem (University of Greenwich) Towards efficient option pricing Apr. 6, 2016 4 / 25
Motivation Nonlinear Black-Scholes Equation 2 σ 2 S 2 ∂ 2 V ∂V ∂t + 1 ∂S 2 + rS ∂V ∂S − rV = 0 , S > 0 , t ∈ [ 0 ,T ) with ❼ σ = σ 0 ( 1 + Le sign ( V SS )) 1 / 2 (Leland model (1985)) ❼ σ = φ ( x ) (Barles and Soner model (1998)) ❼ σ = σ 0 ( 1 − ρSV SS ) − 1 (Frey-Patie model (2002)) ❼ 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, 2016 5 / 25
Motivation Large-Scale Problems (University of Greenwich) Towards efficient option pricing Apr. 6, 2016 6 / 25
Motivation Objective ❼ Single nonlinear option pricing problem ❼ Large-scale nonlinear option pricing problems (University of Greenwich) Towards efficient option pricing Apr. 6, 2016 7 / 25
Newton-based solver Finite Difference Method 2 σ 2 S 2 ∂ 2 V ∂V ∂t + 1 ∂S 2 + rS ∂V ∂S − rV = 0 , S > 0 , t ∈ [ 0 ,T ) (1) with σ = σ ( V t ,V S ,V SS ,V ) . Apply finite difference implicit scheme on equation (1) V n + 1 − V n V n + 1 i + 1 − 2 V n + 1 + V n + 1 V n + 1 i + 1 − V n + 1 + 1 i i 2 ( σ n + 1 ) 2 S 2 i i − 1 i − 1 − rV n + 1 + rS i = 0 , i i i ( ∆ S ) 2 ∆ t 2∆ S which can be simplified as a i V n + 1 i − 1 + b i V n + 1 + c i V n + 1 i + 1 = V n i , (2) i where i represents the spatial discretization and n represents the temporal discretization. (University of Greenwich) Towards efficient option pricing Apr. 6, 2016 8 / 25
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, 2016 9 / 25
Newton-based solver Features ❼ Can be applied to different cases with different schemes ❼ Complexity = # iterations × (linear model) + evaluation of Jacobian matrix ❼ Quadratic convergent rate (University of Greenwich) Towards efficient option pricing Apr. 6, 2016 10 / 25
Newton-based solver Complexity Analysis G ( V n + 1 ) = H ( V n + 1 ) V n + 1 − V n = 0 ❼ Evaluate a i ,b i ,c i for H at each iteration and time step ❼ Consider H ( V n + 1 ) = Σ n + 1 H 1 + H 2 , Σ n + 1 = Diag (( σ n + 1 ) 2 ) , then i Jac ( G ( V n + 1 )) = ∂ [ H ( V n + 1 ) V n + 1 ] = H ( V n + 1 ) + Diag ( H 1 V n + 1 ) ∇ ( Σ n + 1 ) ∂V n + 1 ❼ Tridiagonal solver for ( Jac ( G )) − 1 (University of Greenwich) Towards efficient option pricing Apr. 6, 2016 11 / 25
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, 2016 12 / 25
Newton-based solver Batch Operation ❼ Different option pricing problems such as different r,T,K,σ 0 ❼ Combine all problems together (University of Greenwich) Towards efficient option pricing Apr. 6, 2016 13 / 25
GPU Computing Implementation Implementation ❼ OpenACC ❼ easier to start ❼ need to be careful on data construct and clauses ❼ CUDA libraries ❼ different libraries for different applications ❼ enable users to get good performance without writing too many codes ❼ Kernel functions ❼ for specific applications ❼ more flexibility to modify ❼ need to write the codes and do memory allocation (University of Greenwich) Towards efficient option pricing Apr. 6, 2016 14 / 25
GPU Computing Implementation OpenACC ❼ Find parallelism of the algorithm ❼ # pragma acc kernels { Jacobian() } ❼ # pragma acc parallel loop ❼ Data movement ❼ copyin(), copyout(), present() (University of Greenwich) Towards efficient option pricing Apr. 6, 2016 15 / 25
GPU Computing Implementation CUDA Libraries ❼ cuBLAS ❼ cublasSgbmv for matrix-vector calculation ❼ cublasSnrm2 for Euclidean norm ❼ cuSPARSE ❼ cusparseSgtsv for tridiagonal solver ❼ has problem on cudaFree and cudaMalloc (University of Greenwich) Towards efficient option pricing Apr. 6, 2016 16 / 25
GPU Computing Implementation Kernel Functions ❼ Tridiagonal matrix construction ❼ evaluate a [ i ] ,b [ i ] ,c [ i ] ❼ Sparse matrix calculation ❼ evaluate A + B,A × b where A,B are tridiagonal matrices ❼ Jacobian matrix ❼ Jac ( G ( V n + 1 )) = H ( V n + 1 ) + Diag ( H 1 V n + 1 ) ∇ ( Σ n + 1 ) ❼ contains 2 matrix constructions and 2 level-2 function evaluations (University of Greenwich) Towards efficient option pricing Apr. 6, 2016 17 / 25
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 = 1000 . We calculate 64 , 128 , 256 option pricing problems. System information: Processor: Intel(R) Core(TM) i5-2500 CPU @ 3.30GHz 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, 2016 18 / 25
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 = 64 16.0 6.29 5.70 2.11 7.66 3.42 n = 128 32.5 16.3 7.99 3.61 8.77 3.52 n = 256 64.8 24.8 13.2 6.46 11.6 3.78 Table: Speed up for single precision. # Options CPU OpenACC Library Library ∗ Kernel Kernel ∗ n = 64 1.0x 2.5x 2.8x 7.5x 2.0x 4.6x n = 128 1.0x 1.9x 4.0x 2.9x 3.7x 9.2x n = 256 1.0x 2.6x 4.9x 10x 5.5x 17x (University of Greenwich) Towards efficient option pricing Apr. 6, 2016 19 / 25
GPU Computing Implementation (University of Greenwich) Towards efficient option pricing Apr. 6, 2016 20 / 25
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 = 64 24.8 13.5 10.2 3.03 10.9 4.19 n = 128 49.7 25.5 14.7 6.49 13.0 4.55 n = 256 122 66.5 23.9 9.56 17.0 5.15 Table: Speed up for double precision. # Options CPU OpenACC Library Library ∗ Kernel Kernel ∗ n = 64 1.0x 1.8x 2.4x 8.2x 2.2x 5.9x n = 128 1.0x 1.9x 3.3x 7.6x 3.8x 10x n = 256 1.0x 1.8x 5.1x 12x 7.2x 23x (University of Greenwich) Towards efficient option pricing Apr. 6, 2016 21 / 25
GPU Computing Implementation (University of Greenwich) Towards efficient option pricing Apr. 6, 2016 22 / 25
Conclusion Summarize ❼ Newton-based solver for nonlinear option pricing ❼ Batch operation for dealing with large-scale problems ❼ Comparison of using different implementations of doing GPU Computing ❼ Work in progress ❼ multi-asset problems with Alternating Direction Implicit (ADI) method ❼ Asian option pricing problem with semi-Lagrangian scheme ❼ Multiple GPUs computing (University of Greenwich) Towards efficient option pricing Apr. 6, 2016 23 / 25
Conclusion ❼ Reference [1] M. Ehrhardt (edt): Nonlinear Models in Mathematical Finance , Nova Science Publishers, Inc. New York, 2008. [2] J. Guyon, P. Henry-Labord` ere, Nonlinear Option Pricing , CRC Press, 2013. [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, 2014. ❼ Acknowledgement Special thanks to ❼ Mr. Lung-Sheng Chien from NVIDIA, USA. ❼ Mr. Alvaro Leitao from CWI, the Netherlands. (University of Greenwich) Towards efficient option pricing Apr. 6, 2016 24 / 25
Conclusion Shih-Hau Tan email: shihhau.tan@gmail.com web: http://staffweb.cms.gre.ac.uk/ ∼ ts73/ Thank you very much! (University of Greenwich) Towards efficient option pricing Apr. 6, 2016 25 / 25
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