Numerical Solution of Jump Diffusion Models
Master's Thesis in Financial Mathematics, Halmstad University, Sweden
In this thesis we discuss the proper numerical treatment of
partial integro differential equations (PIDE) that appear as
pricing equations for options on assets that follow jump-diffusion processes.
Usually these PIDEs need a high computational effort due to the involved
Although Fast Fourier Transform (FFT) methods (pioneered by Andersen and Andreasen )
can be applied to evaluate these jump-integrals, the usage of the FFT induces
additional errors such that the jump integral must be discretized on a much finer grid
than the one for the differential operators (cf. Y. d'Halluin et al.).
Moreover, usually implicit time stepping methods are considered and hence
the FFT is inside some iterative solution procedure (i.e. a micro-iteration)
which makes the benefits using the FFT relatively small.
Consequently, in this thesis we will present alternative approaches without
using a FFT.
Option Pricing, Stochastic Volatility, Lévy Processes, Bates Model,
operator splitting technique, Jump-diffusion process,
nonlinear partial integro differential equation (PIDE)
- A. Almendral and C.W. Oosterlee,
Numerical valuation of options with jumps in the underlying,
Appl. Numer. Math. 53 (2005), 1-18.
- A. Anderson and J. Andresen,
Jump diffusion process: Volatility smile fitting and numerical methods for option pricing,
Rev. Derivatives Res. 4 (2000), 231-262.
- D. Bates,
Jumps and stochastic volatility: the exchange rate processes implicit in Deutschemark options,
Rev. Fin. Studies 9 (1996), 69-107.
- G. Cheang, C. Chiarella and A. Ziogas,
The representation of American options prices under stochastic volatility and jump-diffusion dynamics,
Research Paper Series 256, Quantitative Finance Research Centre,
University of Technology, Sydney, 2009.
- C. Chiarella, B. Kang, G.H. Meyer and A. Ziogas,
The Evaluation of American Option Prices Under Stochastic Volatility and Jump-Diffusion Dynamics Using the Method of Lines,
Int. J. of Theor. and Appl. Finance 12 (2009), 393-425.
- S.S. Clift and P.A. Forsyth,
Numerical Solution of Two Asset Jump Diffusion Models for Option Valuation,
Appl. Numer. Math. 58 (2008), 743-782.
- R. Cont and P. Tankov,
Financial Modelling with Jumps,
- R. Cont and E. Voltchkova,
A Finite Difference Scheme for Option Pricing in Jump-Diffusion and Exponential Levy Models,
SIAM J. Numer. Anal. 43 (2005), 1596-1626.
- Y. d'Halluin, P.A. Forsyth and K.R. Vetzal,
Robust Numerical Methods for Contingent Claims under Jump Diffusion Processes,
IMA J. on Num. Anal. 25 (2005), 87-112.
- W.H. Hundsdorfer and J.G. Verwer,
Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations,
- S. Ikonen and J. Toivanen,
Componentwise splitting methods for pricing American options under stochastic volatility,
Int. J. Theor. Appl. Fin. 10 (2007), 331-361.
- R. Lord, F. Fang, F. Bervoets and C.W. Oosterlee,
A Fast and Accurate FFT-Based Method for Pricing Early-Exercise Options under Levy Processes,
SIAM J. Sci. Comp. 30 (2008), 1678-1705.
- A. Mayo,
Methods for the rapid solution of the pricing PIDEs in exponential and Merton models,
J. Comput. Appl. Math. 222 (2008) 128-143.
- S. McKee, D.P. Wall and S.K. Wilson,
An Alternating Direction Implicit Scheme for Parabolic Equations with Mixed Derivative and Convective Terms,
J. Comput. Phys. 126 (1996), 64-76.
- E. Miglio, C. Sgarra,
A Finite Element framework for Option Pricing with the Bates Model,
Preprint. Presented at the 3rd AMAMEF Conference, Vienna, 17-22 September 2007.
- V. Surkov,
Option Pricing using Fourier Space time-stepping Framework,
Ph.D. thesis, University of Toronto, 2009.
- J. Toivanen,
A Componentwise Splitting Method for Pricing American Options under the Bates Model,
Applied and Numerical Partial Differential Equations: Scientific Computing, Simulation, Optimization and Control in a Multidisciplinary Context, W. Fitzgibbon, Yu. Kuznetsov, P. Neittaanmäki, J. Periaux, O. Pironneau (eds.), Computational and Methods in Applied Sciences 15 (2010), Springer, pp. 213-227.