Robust Numerical Scheme for Pricing American Options under Jump Diffusion Models

The goal of option pricing theory is to help the investors
to manage their money, enhance returns and control their financial
future by theoretically valuing their options. However, most of the
option pricing models have no analytical solution. Furthermore,
not all the numerical methods are efficient to solve these models
because they have nonsmoothing payoffs or discontinuous derivatives
at the exercise price. In this paper, we solve the American option
under jump diffusion models by using efficient time-dependent
numerical methods. several techniques are integrated to reduced
the overcome the computational complexity. Fast Fourier Transform
(FFT) algorithm is used as a matrix-vector multiplication solver,
which reduces the complexity from O(M2) into O(M logM).
Partial fraction decomposition technique is applied to rational
approximation schemes to overcome the complexity of inverting
polynomial of matrices. The proposed method is easy to implement
on serial or parallel versions. Numerical results are presented to prove
the accuracy and efficiency of the proposed method.

Authors:

• Salah Alrabeei
• Mohammad Yousuf

Keywords:

• Integral differential equations
• American options
• jump–diffusion model
• rational approximation.

References:

[1] F. Black and M. Scholes, “The pricing of options and corporate
liabilities,” The journal of political economy, pp. 637–654, 1973.
[2] R. C. Merton, “Option pricing when underlying stock returns are
discontinuous,” Journal of financial economics, vol. 3, no. 1-2,
pp. 125–144, 1976.
[3] S. G. Kou, “A jump-diffusion model for option pricing,” Management
science, vol. 48, no. 8, pp. 1086–1101, 2002.
[4] L. Andersen and J. Andreasen, “Jump-diffusion processes: Volatility
smile fitting and numerical methods for option pricing,” Review of
Derivatives Research, vol. 4, no. 3, pp. 231–262, 2000.
[5] K. I. Amin, “Jump diffusion option valuation in discrete time,” The
journal of finance, vol. 48, no. 5, pp. 1833–1863, 1993.
[6] S. Hussain and N. Rehman, “Regularity of the american option value
function in jump-diffusion model.,” Journal of Computational Analysis
and Applications, vol. 22, no. 2, pp. 286–297, 2017.
[7] Y. Chen, W. Wang, and A. Xiao, “An efficient algorithm for
options under merton’s jump-diffusion model on nonuniform grids,”
Computational Economics, vol. 53, no. 4, pp. 1565–1591, 2019.
[8] X. Gan, Y. Yang, and K. Zhang, “A robust numerical method for pricing
american options under kou’s jump-diffusion models based on penalty
method,” Journal of Applied Mathematics and Computing, vol. 62,
no. 1-2, pp. 1–21, 2020.
[9] D. Tangman, A. Gopaul, and M. Bhuruth, “Exponential time integration
and chebychev discretisation schemes for fast pricing of options,”
Applied Numerical Mathematics, vol. 58, no. 9, pp. 1309–1319, 2008.
[10] L. Boen and K. J. in’t Hout, “Operator splitting schemes for american
options under the two-asset merton jump-diffusion model,” Applied
Numerical Mathematics, 2020.
[11] S. Salmi and J. Toivanen, “Imex schemes for pricing options under
jump–diffusion models,” Applied Numerical Mathematics, vol. 84,
pp. 33–45, 2014.
[12] M. K. Kadalbajoo, L. P. Tripathi, and A. Kumar, “Second order accurate
imex methods for option pricing under merton and kou jump-diffusion
models,” Journal of Scientific Computing, vol. 65, no. 3, pp. 979–1024,
2015.
[13] R. Mollapourasl, A. Fereshtian, H. Li, and X. Lu, “Rbf-pu method
for pricing options under the jump–diffusion model with local
volatility,” Journal of Computational and Applied Mathematics, vol. 337,
pp. 98–118, 2018.
[14] K. Kazmi, “An imex predictor–corrector method for pricing options
under regime-switching jump-diffusion models,” International Journal
of Computer Mathematics, vol. 96, no. 6, pp. 1137–1157, 2019.
[15] A. Khaliq, B. Wade, M. Yousuf, and J. Vigo-Aguiar, “High
order smoothing schemes for inhomogeneous parabolic problems
with applications in option pricing,” Numerical Methods for Partial
Differential Equations, vol. 23, no. 5, pp. 1249–1276, 2007.
[16] H. P. Bhatt and A.-Q. M. Khaliq, “Fourth-order compact schemes for the
numerical simulation of coupled burgers’ equation,” Computer Physics
Communications, vol. 200, pp. 117–138, 2016.
[17] H. P. Bhatt and A. Khaliq, “A compact fourth-order l-stable scheme
for reaction–diffusion systems with nonsmooth data,” Journal of
Computational and Applied Mathematics, vol. 299, pp. 176–193, 2016.
[18] R. Zvan, P. Forsyth, and K. Vetzal, “Penalty methods for american
options with stochastic volatility,” Journal of Computational and Applied
Mathematics, vol. 91, no. 2, pp. 199–218, 1998.
[19] Y. dHalluin, P. A. Forsyth, and G. Labahn, “A penalty method
for american options with jump diffusion processes,” Numerische
Mathematik, vol. 97, no. 2, pp. 321–352, 2004.
[20] L. N. Trefethen, Spectral methods in MATLAB, vol. 10. Siam, 2000.
[21] S. Cox and P. Matthews, “Exponential time differencing for stiff
systems,” Journal of Computational Physics, vol. 176, no. 2,
pp. 430–455, 2002.
[22] A.-K. Kassam and L. N. Trefethen, “Fourth-order time-stepping for
stiff pdes,” SIAM Journal on Scientific Computing, vol. 26, no. 4,
pp. 1214–1233, 2005.
[23] A. Khaliq, J. Martin-Vaquero, B. Wade, and M. Yousuf, “Smoothing
schemes for reaction-diffusion systems with nonsmooth data,” Journal of
Computational and Applied Mathematics, vol. 223, no. 1, pp. 374–386,
2009.
[24] V. Thom´ee, Galerkin finite element methods for parabolic problems,
vol. 1054. Springer, 1984.
[25] E. Gallopoulos and Y. Saad, “On the parallel solution of parabolic
equations,” in Proceedings of the 3rd international conference on
Supercomputing, pp. 17–28, ACM, 1989.
[26] A. Khaliq, E. Twizell, and D. Voss, “On parallel algorithms
for semidiscretized parabolic partial differential equations based on
subdiagonal pad´e approximations,” Numerical Methods for Partial
Differential Equations, vol. 9, no. 2, pp. 107–116, 1993.
[27] M. Yousuf, A. Khaliq, and B. Kleefeld, “The numerical approximation
of nonlinear black–scholes model for exotic path-dependent american
options with transaction cost,” International Journal of Computer
Mathematics, vol. 89, no. 9, pp. 1239–1254, 2012.
[28] C. R. Vogel, Computational methods for inverse problems, vol. 23. Siam,
2002.
[29] A. Almendral and C. W. Oosterlee, “Numerical valuation of options
with jumps in the underlying,” Applied Numerical Mathematics, vol. 53,
no. 1, pp. 1–18, 2005.