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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
GPU TECHNOLOGY CONFERENCE 2016 Shih-Hau Tan 1 2
1Marie Curie Research Project ”STRIKE” 2University of Greenwich
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1 Motivation 2 Newton-based solver 3 GPU Computing Implementation 4 Conclusion
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Motivation
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Motivation
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
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Motivation
∂V ∂t + 1 2σ2S2 ∂2V ∂S2 + rS ∂V ∂S − rV = 0, S > 0, t ∈ [0,T) with
❼ σ = σ0(1 + Le sign(VSS))1/2 (Leland model (1985)) ❼ σ = φ(x) (Barles and Soner model (1998)) ❼ σ = σ0(1 − ρSVSS)−1 (Frey-Patie model (2002)) ❼ σ ∈ [σmin,σmax] (Uncertain volatility model (1995))
where VSS = ∂2V ∂S2
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Motivation
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Motivation
❼ Single nonlinear option pricing problem ❼ Large-scale nonlinear option pricing problems
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Newton-based solver
∂V ∂t + 1 2σ2S2 ∂2V ∂S2 + rS ∂V ∂S − rV = 0, S > 0, t ∈ [0,T) (1) with σ = σ(Vt,VS,VSS,V ). Apply finite difference implicit scheme on equation (1)
V n+1
i
− V n
i
∆t + 1 2(σn+1
i
)2S2
i
V n+1
i+1 − 2V n+1 i
+ V n+1
i−1
(∆S)2 + rSi V n+1
i+1 − V n+1 i−1
2∆S − rV n+1
i
= 0, which can be simplified as
aiV n+1
i−1 + biV n+1 i
+ ciV n+1
i+1 = V n i ,
(2) where i represents the spatial discretization and n represents the temporal discretization.
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Newton-based solver
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.
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Newton-based solver
❼ Can be applied to different cases with different schemes ❼ Complexity = #iterations×(linear model) + evaluation of Jacobian
matrix
❼ Quadratic convergent rate
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Newton-based solver
G(V n+1) = H(V n+1)V n+1 − V n = 0
❼ Evaluate ai,bi,ci for H at each iteration and time step ❼ Consider H(V n+1) = Σn+1H1 + H2, Σn+1 = Diag((σn+1
i
)2), then
Jac(G(V n+1)) = ∂[H(V n+1)V n+1] ∂V n+1 = H(V n+1) + Diag(H1V n+1)∇(Σn+1) ❼ Tridiagonal solver for (Jac(G))−1
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Newton-based solver
At each time step, we have the Newton’s iteration as:
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Newton-based solver
❼ Different option pricing problems such as different r,T,K,σ0 ❼ Combine all problems together
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GPU Computing 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
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GPU Computing Implementation
❼ Find parallelism of the algorithm
❼ #pragma acc kernels
{ Jacobian() }
❼ #pragma acc parallel loop
❼ Data movement
❼ copyin(), copyout(), present()
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GPU Computing Implementation
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
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GPU Computing Implementation
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(H1V n+1)∇(Σn+1) ❼ contains 2 matrix constructions and 2 level-2 function evaluations
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GPU Computing Implementation
Consider Frey-Patie model with nonlinear volatility σ = σ0(1 − ρSVSS)−1. Parameters are chosen to be S ∈ [0,300], K = 100, T = 1/12,σ0 = 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)
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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
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GPU Computing Implementation (University of Greenwich) Towards efficient option pricing
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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
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GPU Computing Implementation (University of Greenwich) Towards efficient option pricing
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Conclusion
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
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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.
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Conclusion
Shih-Hau Tan email: shihhau.tan@gmail.com web: http://staffweb.cms.gre.ac.uk/∼ts73/
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