Pricing European Options under Jump Diffusion Models with Fast L-stable Padé Scheme
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. Modeling
option pricing by Black-School models with jumps guarantees to
consider the market movement. However, only numerical methods
can solve this model. 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, the exponential time differencing (ETD) method is applied
for solving partial integrodifferential equations arising in pricing
European options under Merton’s and Kou’s jump-diffusion models.
Fast Fourier Transform (FFT) algorithm is used as a matrix-vector
multiplication solver, which reduces the complexity from O(M2)
into O(M logM). A partial fraction form of Pad`e schemes is used
to overcome the complexity of inverting polynomial of matrices.
These two tools guarantee to get efficient and accurate numerical
solutions. We construct a parallel and easy to implement a version
of the numerical scheme. Numerical experiments are given to show
how fast and accurate is our scheme.
[1] R. C. Merton, “Option pricing when underlying stock returns are
discontinuous,” Journal of financial economics, vol. 3, no. 1-2,
pp. 125–144, 1976.
[2] S. G. Kou, “A jump-diffusion model for option pricing,” Management
science, vol. 48, no. 8, pp. 1086–1101, 2002.
[3] 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.
[4] K. J. in’t Hout and J. Toivanen, “Adi schemes for valuing european
options under the bates model,” Applied Numerical Mathematics,
vol. 130, pp. 143–156, 2018.
[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. Zhang, “Radial basis functions method for valuing options: A
multinomial tree approach,” Journal of Computational and Applied
Mathematics, vol. 319, pp. 97–107, 2017.
[7] N. Cantarutti and J. Guerra, “Multinomial method for option
pricing under variance gamma,” International Journal of Computer
Mathematics, vol. 96, no. 6, pp. 1087–1106, 2019.
[8] 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.
[9] A. Itkin, “Efficient solution of backward jump-diffusion partial
integro-differential equations with splitting and matrix exponentials,”
Journal of Computational Finance, vol. 19, no. 3, 2016.
[10] L. Boen et al., “Operator splitting schemes for the two-asset
merton jump–diffusion model,” Journal of Computational and Applied
Mathematics, 2019.
[11] 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.
[12] 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.
[13] S. Cox and P. Matthews, “Exponential time differencing for stiff
systems,” Journal of Computational Physics, vol. 176, no. 2,
pp. 430–455, 2002.
[14] 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.
[15] 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.
[16] 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.
[17] 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.
[18] G. Beylkin, J. M. Keiser, and L. Vozovoi, “A new class of time
discretization schemes for the solution of nonlinear pdes,” Journal of
computational physics, vol. 147, no. 2, pp. 362–387, 1998.
[19] C. R. Vogel, Computational methods for inverse problems, vol. 23. Siam,
2002.
[20] N. Rambeerich, D. Tangman, A. Gopaul, and M. Bhuruth, “Exponential
time integration for fast finite element solutions of some financial
engineering problems,” Journal of Computational and Applied
Mathematics, vol. 224, no. 2, pp. 668–678, 2009.
[1] R. C. Merton, “Option pricing when underlying stock returns are
discontinuous,” Journal of financial economics, vol. 3, no. 1-2,
pp. 125–144, 1976.
[2] S. G. Kou, “A jump-diffusion model for option pricing,” Management
science, vol. 48, no. 8, pp. 1086–1101, 2002.
[3] 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.
[4] K. J. in’t Hout and J. Toivanen, “Adi schemes for valuing european
options under the bates model,” Applied Numerical Mathematics,
vol. 130, pp. 143–156, 2018.
[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. Zhang, “Radial basis functions method for valuing options: A
multinomial tree approach,” Journal of Computational and Applied
Mathematics, vol. 319, pp. 97–107, 2017.
[7] N. Cantarutti and J. Guerra, “Multinomial method for option
pricing under variance gamma,” International Journal of Computer
Mathematics, vol. 96, no. 6, pp. 1087–1106, 2019.
[8] 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.
[9] A. Itkin, “Efficient solution of backward jump-diffusion partial
integro-differential equations with splitting and matrix exponentials,”
Journal of Computational Finance, vol. 19, no. 3, 2016.
[10] L. Boen et al., “Operator splitting schemes for the two-asset
merton jump–diffusion model,” Journal of Computational and Applied
Mathematics, 2019.
[11] 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.
[12] 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.
[13] S. Cox and P. Matthews, “Exponential time differencing for stiff
systems,” Journal of Computational Physics, vol. 176, no. 2,
pp. 430–455, 2002.
[14] 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.
[15] 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.
[16] 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.
[17] 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.
[18] G. Beylkin, J. M. Keiser, and L. Vozovoi, “A new class of time
discretization schemes for the solution of nonlinear pdes,” Journal of
computational physics, vol. 147, no. 2, pp. 362–387, 1998.
[19] C. R. Vogel, Computational methods for inverse problems, vol. 23. Siam,
2002.
[20] N. Rambeerich, D. Tangman, A. Gopaul, and M. Bhuruth, “Exponential
time integration for fast finite element solutions of some financial
engineering problems,” Journal of Computational and Applied
Mathematics, vol. 224, no. 2, pp. 668–678, 2009.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:79969", author = "Salah Alrabeei and Mohammad Yousuf", title = "Pricing European Options under Jump Diffusion Models with Fast L-stable Padé Scheme", abstract = "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. Modeling
option pricing by Black-School models with jumps guarantees to
consider the market movement. However, only numerical methods
can solve this model. 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, the exponential time differencing (ETD) method is applied
for solving partial integrodifferential equations arising in pricing
European options under Merton’s and Kou’s jump-diffusion models.
Fast Fourier Transform (FFT) algorithm is used as a matrix-vector
multiplication solver, which reduces the complexity from O(M2)
into O(M logM). A partial fraction form of Pad`e schemes is used
to overcome the complexity of inverting polynomial of matrices.
These two tools guarantee to get efficient and accurate numerical
solutions. We construct a parallel and easy to implement a version
of the numerical scheme. Numerical experiments are given to show
how fast and accurate is our scheme.", keywords = "Integral differential equations, L-stable methods,
pricing European options, Jump–diffusion model.", volume = "14", number = "11", pages = "111-5", }
