Option Pricing under Delay Geometric Brownian Motion with Regime Switching

Tianyao Fang; Liangjian Hu; Yun Xin

doi:doi:10.11648/j.sjams.20160406.13

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Option Pricing under Delay Geometric Brownian Motion with Regime Switching

Tianyao Fang, Liangjian Hu, Yun Xin

Published in Science Journal of Applied Mathematics and Statistics (Volume 4, Issue 6)

Received: 18 October 2016 Published: 19 October 2016

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Abstract

We investigate the option pricing problem when the price dynamics of the underlying risky assets are driven by delay geometric Brownian motions with regime switching. That is, the market interest rate, the appreciation rate and the volatility of the risky assets depend on the past stock prices and the unobservable states of the economy which are modulated by a continuous-time Markov chain. The market described by the model is incomplete, the martingale measure is not unique and the Esscher transform is employed to determine an equivalent martingale measure. We proved the model has a unique positive solution and the price of the contingent claims under the model can be computable numerically if not analytically.

Published in	Science Journal of Applied Mathematics and Statistics (Volume 4, Issue 6)
DOI	10.11648/j.sjams.20160406.13
Page(s)	263-268
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Option Pricing, Regime Switching, Esscher Transform, Itô Formula, Euler-Maruyama

References

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[2]	M. Arriojas, Y. Hu, S. Mohammed and G. Pap, “A Delayed Black and Scholes Formula,” Stochastic Analysis and Applications, 2007, 25, pp. 471-492.
[3]	L. Yan and Q. Zhan, “Successive Approximation of SFDEs with Finite Delay Driven by G-Brownian Motion,” Abstract and Applied Analysis, 2013, 6, pp. 189-206.
[4]	X. Mao and S. Sabanis, “Delay Geometric Brownian Motion in Financial Option Valuation,” Journal of Probability and Stochastic Processes, 2013, 85(2), pp. 295-320.
[5]	J. Hamilton, “A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle,” Ecomometrica, 1989, 57, pp. 357-384.
[6]	J. Elliott, L. Chan and T. Siu, “Option Pricing and Esscher Transform under Regime Switching,” Annals of Finance, 2005, pp. 423-432.
[7]	J. Elliott, L. Chan and T. Siu, “Option Valuation under a Regime-Switching Constant Elasticity of Variance Process,” Applied Mathematics and Computation, 2013, 219(9), pp. 4434-4443.
[8]	J. Elliott, L. Chan and T. Siu, “On Pricing Barrier Options with Regime Switching,” Journal of Computational and Applied Mathematics, 2014, 256(1), pp. 196-210.
[9]	N. Ratanov, “Option Pricing under Jump-Diffusion Processes with Regime Switching,” Methodology and Computing in Applied Probability, 2016, 18, pp. 829-845.
[10]	J. Ma and Z. Zhou, “Moving Mesh Methods for Pricing Asian Options with Regime Switching,” Journal of Computational and Applied Mathematics, 2016, 298, pp. 211-221.
[11]	X. Mao, “Stochastic Differential Equations and Applications,” Horwood Publishing, Chichester, 2007.
[12]	X. Mao and C. Yuan, “Stochastic Differential Equations with Markovian Switching,” Imperial College Press, 2006.
[13]	C. Yuan and W. Glover, “Approximate Solutions of Stochastic Differential Delay Equations with Markovian Switching,” Journal of Computational and Applied Mathematics, 2006, 194, pp. 207-226.
[14]	K. Fan, Y. Shen, T. Siu and R. Wang, “On a Markov Chain Approximation Method for Option Pricing with Regime Switching,” Journal of Industrial and Management Optimization, 2016, 12, pp. 529-541.
[15]	Z. Jin and L. Qian, “Lookback Option Pricing for Regime Switching Jump Diffusion Models,” Mathematical Control and Related Fields, 2015, 5, pp. 237-258.
[16]	R. Merton, “The Theory of Rational Option Pricing,” Bell Journal of Economics and Management Science, 1971, 4(4), pp. 141-183.

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APA Style

Tianyao Fang, Liangjian Hu, Yun Xin. (2016). Option Pricing under Delay Geometric Brownian Motion with Regime Switching. Science Journal of Applied Mathematics and Statistics, 4(6), 263-268. https://doi.org/10.11648/j.sjams.20160406.13

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ACS Style

Tianyao Fang; Liangjian Hu; Yun Xin. Option Pricing under Delay Geometric Brownian Motion with Regime Switching. Sci. J. Appl. Math. Stat. 2016, 4(6), 263-268. doi: 10.11648/j.sjams.20160406.13

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AMA Style

Tianyao Fang, Liangjian Hu, Yun Xin. Option Pricing under Delay Geometric Brownian Motion with Regime Switching. Sci J Appl Math Stat. 2016;4(6):263-268. doi: 10.11648/j.sjams.20160406.13

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@article{10.11648/j.sjams.20160406.13,
  author = {Tianyao Fang and Liangjian Hu and Yun Xin},
  title = {Option Pricing under Delay Geometric Brownian Motion with Regime Switching},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {4},
  number = {6},
  pages = {263-268},
  doi = {10.11648/j.sjams.20160406.13},
  url = {https://doi.org/10.11648/j.sjams.20160406.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20160406.13},
  abstract = {We investigate the option pricing problem when the price dynamics of the underlying risky assets are driven by delay geometric Brownian motions with regime switching. That is, the market interest rate, the appreciation rate and the volatility of the risky assets depend on the past stock prices and the unobservable states of the economy which are modulated by a continuous-time Markov chain. The market described by the model is incomplete, the martingale measure is not unique and the Esscher transform is employed to determine an equivalent martingale measure. We proved the model has a unique positive solution and the price of the contingent claims under the model can be computable numerically if not analytically.},
 year = {2016}
}

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TY  - JOUR
T1  - Option Pricing under Delay Geometric Brownian Motion with Regime Switching
AU  - Tianyao Fang
AU  - Liangjian Hu
AU  - Yun Xin
Y1  - 2016/10/19
PY  - 2016
N1  - https://doi.org/10.11648/j.sjams.20160406.13
DO  - 10.11648/j.sjams.20160406.13
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
SP  - 263
EP  - 268
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20160406.13
AB  - We investigate the option pricing problem when the price dynamics of the underlying risky assets are driven by delay geometric Brownian motions with regime switching. That is, the market interest rate, the appreciation rate and the volatility of the risky assets depend on the past stock prices and the unobservable states of the economy which are modulated by a continuous-time Markov chain. The market described by the model is incomplete, the martingale measure is not unique and the Esscher transform is employed to determine an equivalent martingale measure. We proved the model has a unique positive solution and the price of the contingent claims under the model can be computable numerically if not analytically.
VL  - 4
IS  - 6
ER  -

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Author Information

Tianyao Fang

Department of Applied Mathematics, Donghua University, Shanghai, China
Liangjian Hu

Department of Applied Mathematics, Donghua University, Shanghai, China
Yun Xin

Department of Applied Mathematics, Donghua University, Shanghai, China

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APA Style

Tianyao Fang, Liangjian Hu, Yun Xin. (2016). Option Pricing under Delay Geometric Brownian Motion with Regime Switching. Science Journal of Applied Mathematics and Statistics, 4(6), 263-268. https://doi.org/10.11648/j.sjams.20160406.13

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ACS Style

Tianyao Fang; Liangjian Hu; Yun Xin. Option Pricing under Delay Geometric Brownian Motion with Regime Switching. Sci. J. Appl. Math. Stat. 2016, 4(6), 263-268. doi: 10.11648/j.sjams.20160406.13

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AMA Style

Tianyao Fang, Liangjian Hu, Yun Xin. Option Pricing under Delay Geometric Brownian Motion with Regime Switching. Sci J Appl Math Stat. 2016;4(6):263-268. doi: 10.11648/j.sjams.20160406.13

Copy | Download

@article{10.11648/j.sjams.20160406.13,
  author = {Tianyao Fang and Liangjian Hu and Yun Xin},
  title = {Option Pricing under Delay Geometric Brownian Motion with Regime Switching},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {4},
  number = {6},
  pages = {263-268},
  doi = {10.11648/j.sjams.20160406.13},
  url = {https://doi.org/10.11648/j.sjams.20160406.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20160406.13},
  abstract = {We investigate the option pricing problem when the price dynamics of the underlying risky assets are driven by delay geometric Brownian motions with regime switching. That is, the market interest rate, the appreciation rate and the volatility of the risky assets depend on the past stock prices and the unobservable states of the economy which are modulated by a continuous-time Markov chain. The market described by the model is incomplete, the martingale measure is not unique and the Esscher transform is employed to determine an equivalent martingale measure. We proved the model has a unique positive solution and the price of the contingent claims under the model can be computable numerically if not analytically.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - Option Pricing under Delay Geometric Brownian Motion with Regime Switching
AU  - Tianyao Fang
AU  - Liangjian Hu
AU  - Yun Xin
Y1  - 2016/10/19
PY  - 2016
N1  - https://doi.org/10.11648/j.sjams.20160406.13
DO  - 10.11648/j.sjams.20160406.13
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
SP  - 263
EP  - 268
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20160406.13
AB  - We investigate the option pricing problem when the price dynamics of the underlying risky assets are driven by delay geometric Brownian motions with regime switching. That is, the market interest rate, the appreciation rate and the volatility of the risky assets depend on the past stock prices and the unobservable states of the economy which are modulated by a continuous-time Markov chain. The market described by the model is incomplete, the martingale measure is not unique and the Esscher transform is employed to determine an equivalent martingale measure. We proved the model has a unique positive solution and the price of the contingent claims under the model can be computable numerically if not analytically.
VL  - 4
IS  - 6
ER  -

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