Pricing Discrete Barrier Options Under the Jump-Diffusion Model with Stochastic Volatility and Stochastic Intensity

Pingtao Duan, Yuting Liu, Zhiming Ma

Communications in Mathematics and Statistics ›› 2022, Vol. 12 ›› Issue (2) : 239-263.

Communications in Mathematics and Statistics ›› 2022, Vol. 12 ›› Issue (2) : 239-263. DOI: 10.1007/s40304-022-00287-6
Article

Pricing Discrete Barrier Options Under the Jump-Diffusion Model with Stochastic Volatility and Stochastic Intensity

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Abstract

This paper considers the problem of numerically evaluating discrete barrier option prices when the underlying asset follows the jump-diffusion model with stochastic volatility and stochastic intensity. We derive the three-dimensional characteristic function of the log-asset price, the volatility and the jump intensity. We also provide the approximate formula of the discrete barrier option prices by the three-dimensional Fourier cosine series expansion (3D-COS) method. Numerical results show that the 3D-COS method is rather correct, fast and competent for pricing the discrete barrier options.

Keywords

Option pricing / Discrete barrier options / Jump-diffusion model / Stochastic volatility / Stochastic intensity

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Pingtao Duan, Yuting Liu, Zhiming Ma. Pricing Discrete Barrier Options Under the Jump-Diffusion Model with Stochastic Volatility and Stochastic Intensity. Communications in Mathematics and Statistics, 2022, 12(2): 239‒263 https://doi.org/10.1007/s40304-022-00287-6

References

[1.]
Bates DS. Jumps and stochastic volatility: exchange rate processes implicit in deutsche mark options. Rev. Financ. Stud.. 1996, 9 1 69-107
CrossRef Google scholar
[2.]
Black F, Scholes MS. The pricing of options and corporate liabilities. J. Polit. Econ.. 1973, 81 3 637-654
CrossRef Google scholar
[3.]
Carr P, Madan DB. Option valuation using the fast Fourier transform. J. Comput. Financ.. 1999, 2 4 61-73
CrossRef Google scholar
[4.]
Cox JC, Ingersoll JE, Ross SA. A theory of the term structure of interest rates. Econometrica. 1985, 53 2 385-407
CrossRef Google scholar
[5.]
Duffie D, Pan J, Singleton KJ. Transform analysis and asset pricing for affine jump-diffusions. Econometrica. 2000, 68 6 1343-1376
CrossRef Google scholar
[6.]
Fang F, Oosterlee CW. A novel pricing method for European options based on Fourier-cosine series expansions. SIAM J. Sci. Comput.. 2008, 31 2 826-848
CrossRef Google scholar
[7.]
Fang F, Oosterlee CW. Pricing early-exercise and discrete barrier options by Fourier-cosine series expansions. Numer. Math.. 2009, 114 1 27-62
CrossRef Google scholar
[8.]
Fang F, Oosterlee CW. A Fourier-based valuation method for Bermudan and barrier options under Heston’s model. SIAM J Financ. Math.. 2011, 2 1 439-463
CrossRef Google scholar
[9.]
Feller W. Two singular diffusion problems. Ann. Math.. 1951, 54 1 173
CrossRef Google scholar
[10.]
Funahashi H, Higuchi T. An analytical approximation for single barrier options under stochastic volatility models. Ann. Oper. Res.. 2018, 266 1 129-157
CrossRef Google scholar
[11.]
Fusai G, Abrahams ID, Sgarra C. An exact analytical solution for discrete barrier options. Finan. Stochast.. 2006, 10 1 1-26
CrossRef Google scholar
[12.]
Heston SL. A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud.. 1993, 6 2 327-343
CrossRef Google scholar
[13.]
Huang J, Zhu W, Ruan X. Fast Fourier transform based power option pricing with stochastic interest rate, volatility, and jump intensity. J. Appl. Math.. 2013
CrossRef Google scholar
[14.]
Huang J, Zhu W, Ruan X. Option pricing using the fast Fourier transform under the double exponential jump model with stochastic volatility and stochastic intensity. J. Comput. Appl. Math.. 2014, 263 1 152-159
CrossRef Google scholar
[15.]
Janek, A., Kluge, T., Weron, R., Wystup, U.: Fx smile in the Heston model. HSC Research Reports 133–162,(2010)
[16.]
Kirkby JL, Nguyen D, Cui Z. A unified approach to Bermudan and barrier options under stochastic volatility models with jumps. J. Econ. Dyn. Control. 2017, 80 75-100
CrossRef Google scholar
[17.]
Kou S. A jump-diffusion model for option pricing. Manag. Sci.. 2002, 48 8 1086-1101
CrossRef Google scholar
[18.]
Kou S, Wang H. Option pricing under a double exponential jump diffusion model. Manag. Sci.. 2004, 50 9 1178-1192
CrossRef Google scholar
[19.]
Lorig, M., Lozanocarbasse, O.: Exponential Lévy-type models with stochastic volatility and stochastic jump-intensity. Presented at the (2012)
[20.]
Merton RC. Option pricing when underlying stock returns are discontinuous. J. Financ. Econ.. 1976, 3 125-144
CrossRef Google scholar
[21.]
Pellegrino T, Sabino P. Pricing and hedging multi-asset spread options by a three-dimensional Fourier cosine series expansion method. J. Energy Markets. 2014, 7 2 71-92
CrossRef Google scholar
[22.]
Ruijter M, Oosterlee CW. Two-dimensional Fourier cosine series expansion method for pricing financial options. SIAM J. Sci. Comput.. 2012, 34 5 B642-B671
CrossRef Google scholar
[23.]
Santaclara P, Yan S. Crashes, volatility, and the equity premium: lessons from S &P 500 options. Rev. Econ. Stat.. 2010, 92 2 435-451
CrossRef Google scholar
[24.]
Shiraya K, Takahashi A, Yamada T. Pricing discrete barrier options under stochastic volatility. Asia-Pacific Finan. Markets.. 2012, 19 3 205-232
CrossRef Google scholar
[25.]
Yang B, Yue J, Wang M, Huang N. Volatility swaps valuation under stochastic volatility with jumps and stochastic intensity. Appl. Math. Comput.. 2019, 355 73-84
[26.]
Zhang S, Geng J. Efficiently pricing continuously monitored barrier options under stochastic volatility model with jumps. Int. J. Comput. Math.. 2017, 94 11 2166-2177
CrossRef Google scholar
[27.]
Zhang S, Wang L. Fast Fourier transform option pricing with stochastic interest rate, stochastic volatility and double jumps. Appl. Math. Comput.. 2013, 219 23 10928-10933
Funding
Fundamental Research Funds for the Central Universities(2018JBM320)

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