BENCHOP – SLV: the BENCHmarking project in Option Pricing – Stochastic and Local Volatility problems

ABSTRACT In the recent project BENCHOP – the BENCHmarking project in Option Pricing we found that Stochastic and Local Volatility problems were particularly challenging. Here we continue the effort by introducing a set of benchmark problems for this type of problems. Eight different methods targeted for the Stochastic Differential Equation (SDE) formulation and the Partial Differential Equation (PDE) formulation of the problem, as well as Fourier methods making use of the characteristic function, were implemented to solve these problems. Comparisons are made with respect to time to reach a certain error level in the computed solution for the different methods. The implemented Fourier method was superior to all others for the two problems where it was implemented. Generally, methods targeting the PDE formulation of the problem outperformed the methods for the SDE formulation. Among the methods for the PDE formulation the ADI method stood out as the best performing one.

[1]  Erik Lindström,et al.  Sequential calibration of options , 2008, Comput. Stat. Data Anal..

[2]  Colin Rose Computational Statistics , 2011, International Encyclopedia of Statistical Science.

[3]  P. Hagan,et al.  MANAGING SMILE RISK , 2002 .

[4]  Bengt Fornberg,et al.  On the role of polynomials in RBF-FD approximations: II. Numerical solution of elliptic PDEs , 2017, J. Comput. Phys..

[5]  Karel J. in 't Hout,et al.  An adjoint method for the exact calibration of stochastic local volatility models , 2016, J. Comput. Sci..

[6]  Kyoung-Sook Moon EFFICIENT MONTE CARLO ALGORITHM FOR PRICING BARRIER OPTIONS , 2008 .

[7]  W. Feller TWO SINGULAR DIFFUSION PROBLEMS , 1951 .

[8]  Elisabeth Larsson,et al.  A Radial Basis Function Partition of Unity Collocation Method for Convection–Diffusion Equations Arising in Financial Applications , 2015, J. Sci. Comput..

[9]  Bengt Fornberg,et al.  On the role of polynomials in RBF-FD approximations: I. Interpolation and accuracy , 2016, J. Comput. Phys..

[10]  J. Witteveen,et al.  The stochastic collocation Monte Carlo sampler: highly efficient sampling from ‘expensive’ distributions , 2018, Quantitative Finance.

[11]  Mark Broadie,et al.  Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes , 2006, Oper. Res..

[12]  K. I. '. Hout,et al.  ADI finite difference schemes for option pricing in the Heston model with correlation , 2008, 0811.3427.

[13]  Victor Shcherbakov,et al.  Pricing Derivatives under Multiple Stochastic Factors by Localized Radial Basis Function Methods , 2017, ArXiv.

[14]  Roger Lee Option Pricing by Transform Methods: Extensions, Unification, and Error Control , 2004 .

[15]  FornbergBengt,et al.  On the role of polynomials in RBF-FD approximations , 2016 .

[16]  Bruno Welfert,et al.  Unconditional stability of second-order ADI schemes applied to multi-dimensional diffusion equations with mixed derivative terms , 2009 .

[17]  Tinne Haentjens,et al.  Alternating direction implicit finite difference schemes for the Heston-Hull-White partial differential equation , 2012 .

[18]  Johan Tysk,et al.  Boundary conditions for the single-factor term structure equation , 2011, 1101.1149.

[19]  Lina von Sydow,et al.  Radial Basis Function generated Finite Differences for option pricing problems , 2017, Comput. Math. Appl..

[20]  Elisabeth Larsson,et al.  Radial basis function partition of unity methods for pricing vanilla basket options , 2016, Comput. Math. Appl..

[21]  V. Shcherbakov Radial basis function partition of unity operator splitting method for pricing multi-asset American options , 2016 .

[22]  Gene H. Golub,et al.  Calculation of Gauss quadrature rules , 1967, Milestones in Matrix Computation.

[23]  S. Heston A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options , 1993 .

[24]  V. Lucic Boundary conditions for computing densities in hybrid models via PDE methods , 2008 .

[25]  Leif Andersen Simple and efficient simulation of the Heston stochastic volatility model , 2008 .

[26]  Jonas Persson,et al.  BENCHOP – The BENCHmarking project in option pricing† , 2015, Int. J. Comput. Math..

[27]  Geof H. Givens,et al.  Computational Statistics: Givens/Computational Statistics , 2012 .

[28]  Cornelis W. Oosterlee,et al.  The Stochastic Grid Bundling Method: Efficient Pricing of Bermudan Options and their Greeks , 2013, Appl. Math. Comput..

[29]  Alexander Lipton,et al.  Pricing of vanilla and first-generation exotic options in the local stochastic volatility framework: survey and new results , 2013, 1312.5693.

[30]  Cornelis W. Oosterlee,et al.  On an efficient multiple time step Monte Carlo simulation of the SABR model , 2016 .

[31]  Bin Chen,et al.  A LOW-BIAS SIMULATION SCHEME FOR THE SABR STOCHASTIC VOLATILITY MODEL , 2012 .

[32]  Michael B. Giles,et al.  Multilevel Monte Carlo Path Simulation , 2008, Oper. Res..

[33]  Cornelis W. Oosterlee,et al.  Monte Carlo Calculation of Exposure Profiles and Greeks for Bermudan and Barrier Options under the Heston Hull-White Model , 2014 .

[34]  Martin Vohralík,et al.  Algebraic and Discretization Error Estimation by Equilibrated Fluxes for Discontinuous Galerkin Methods on Nonmatching Grids , 2015, J. Sci. Comput..

[35]  Cornelis W. Oosterlee,et al.  On the Heston Model with Stochastic Interest Rates , 2010, SIAM J. Financial Math..