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Analytical Pricing Of Double-Barrier Options Under A Double-Exponential Jump Diffusion Process: Applications Of Laplace Transform

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  • ARTUR SEPP

    (Institute of Mathematical Statistics, Faculty of Mathematics and Computer Science, University of Tartu, J. Liivi 2, 50409 Tartu, Estonia)

Abstract

We derive explicit formulas for pricing double (single) barrier and touch options with time-dependent rebates assuming that the asset price follows a double-exponential jump diffusion process. We also consider incorporating time-dependent volatility.Assuming risk-neutrality, the value of a barrier option satisfies the generalized Black–Scholes equation with the appropriate boundary conditions. We take the Laplace transform of this equation in time and solve it explicitly. Option price and risk parameters are computed via the numerical inversion of the corresponding solution. Numerical examples reveal that the pricing formulas are easy to implement and they result in accurate prices and risk parameters.Proposed formulas allow fast computing of smile-consistent prices of barrier and touch options.

Suggested Citation

  • Artur Sepp, 2004. "Analytical Pricing Of Double-Barrier Options Under A Double-Exponential Jump Diffusion Process: Applications Of Laplace Transform," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 7(02), pages 151-175.
    • Handle: RePEc:wsi:ijtafx:v:07:y:2004:i:02:n:s0219024904002402
    DOI: 10.1142/S0219024904002402
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    References listed on IDEAS

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    1. Alexander Lipton, 2001. "Mathematical Methods for Foreign Exchange:A Financial Engineer's Approach," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 4694, February.
    2. S. G. Kou & Hui Wang, 2004. "Option Pricing Under a Double Exponential Jump Diffusion Model," Management Science, INFORMS, vol. 50(9), pages 1178-1192, September.
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    3. Liu, Yu-hong & Jiang, I-Ming & Hsu, Wei-tze, 2018. "Compound option pricing under a double exponential Jump-diffusion model," The North American Journal of Economics and Finance, Elsevier, vol. 43(C), pages 30-53.
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    10. Hann-Shing Ju & Ren-Raw Chen & Shih-Kuo Yeh & Tung-Hsiao Yang, 2015. "Evaluation of conducting capital structure arbitrage using the multi-period extended Geske–Johnson model," Review of Quantitative Finance and Accounting, Springer, vol. 44(1), pages 89-111, January.
    11. Shiyu Song & Yongjin Wang, 2017. "Pricing double barrier options under a volatility regime-switching model with psychological barriers," Review of Derivatives Research, Springer, vol. 20(3), pages 255-280, October.
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    13. Zhaojun Yang & Chunhong Zhang, 2015. "The Pricing of Two Newly Invented Swaps in a Jump-Diffusion Model," Annals of Economics and Finance, Society for AEF, vol. 16(2), pages 371-392, November.
    14. Bo, Lijun & Song, Renming & Tang, Dan & Wang, Yongjin & Yang, Xuewei, 2012. "Lévy risk model with two-sided jumps and a barrier dividend strategy," Insurance: Mathematics and Economics, Elsevier, vol. 50(2), pages 280-291.
    15. Chen, Son-Nan & Hsu, Pao-Peng, 2018. "Pricing and hedging barrier options under a Markov-modulated double exponential jump diffusion-CIR model," International Review of Economics & Finance, Elsevier, vol. 56(C), pages 330-346.
    16. Alexander Lipton & Ioana Savescu, 2012. "Pricing credit default swaps with bilateral value adjustments," Papers 1207.6049, arXiv.org.
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    18. Oleg Kudryavtsev & Sergei Levendorskiǐ, 2009. "Fast and accurate pricing of barrier options under Lévy processes," Finance and Stochastics, Springer, vol. 13(4), pages 531-562, September.
    19. Zhiqiang Zhou & Hongying Wu, 2018. "Laplace Transform Method for Pricing American CEV Strangles Option with Two Free Boundaries," Discrete Dynamics in Nature and Society, Hindawi, vol. 2018, pages 1-12, September.
    20. Hieber, Peter & Scherer, Matthias, 2012. "A note on first-passage times of continuously time-changed Brownian motion," Statistics & Probability Letters, Elsevier, vol. 82(1), pages 165-172.
    21. Gapeev, Pavel V. & Stoev, Yavor I., 2017. "On the Laplace transforms of the first exit times in one-dimensional non-affine jump–diffusion models," Statistics & Probability Letters, Elsevier, vol. 121(C), pages 152-162.
    22. Daniel Hackmann, 2017. "Analytic techniques for option pricing under a hyperexponential L\'{e}vy model," Papers 1705.05934, arXiv.org.

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