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VALUATION OF VULNERABLE OPTIONS UNDER THE DOUBLE EXPONENTIAL JUMP MODEL WITH STOCHASTIC VOLATILITY

Published online by Cambridge University Press:  14 February 2018

Xingyu Han
Xingyu Han*
Affiliation:
Institute for Financial Studies and School of Mathematics, Shandong University, Jinan 250100, People's Republic of China E-mail: xyhan91@foxmail.com
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Abstract

In this paper, we extend the framework of Klein [15] [Journal of Banking & Finance 20: 1211–1229] to a general model under the double exponential jump model with stochastic volatility on the underlying asset and the assets of the counterparty. Firstly, we derive the closed-form characteristic functions for this dynamic. Using the Fourier-cosine expansion technique, we get numerical solutions for vulnerable European put options based on the characteristic functions. The inverse fast Fourier transform method provides a fast numerical algorithm for the twice-exercisable vulnerable Bermuda put options. By virtue of the modified Geske and Johnson method, we obtain an approximate pricing formula of vulnerable American put options. Numerical simulations are made for investigating the impact of stochastic volatility on vulnerable options.

Keywords

double exponential jumpFourier-cosine expansionGeske–Johnson schemeinverse fast Fourier transformstochastic volatilityvulnerable American options
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 33 , Issue 1 , January 2019 , pp. 81 - 104
Copyright
Copyright © Cambridge University Press 2018 

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References

1.Ballestra, L.V. & Cecere, L. (2016). A fast numerical method to price American options under the bates model. Computers & Mathematics with Applications 72(5): 1305–1319.Google Scholar
2.Bates, D.S. (1996). Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options. Review of Financial Studies 9(1): 69–107.Google Scholar
3.Black, F. & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy 81(3): 637–654.Google Scholar
4.Boyd, J.P. (2001). Chebyshev and Fourier spectral methods. Mineola, NY: Dover Publications.Google Scholar
5.Bunch, D.S. & Johnson, H. (1992). A simple and numerically efficient valuation method for American puts using a modified Geske-Johnson approach. The Journal of Finance 47(2): 809–816.Google Scholar
6.Chang, C.C., Lin, J.B., Tsai, W.C. & Wang, Y.H. (2012). Using richardson extrapolation techniques to price American options with alternative stochastic processes. Review of Quantitative Finance and Accounting 39(3): 383–406.Google Scholar
7.Chang, L.F. & Hung, M.W. (2006). Valuation of vulnerable American options with correlated credit risk. Review of Derivatives Research 9(2): 137–165.Google Scholar
8.Fang, F. & Oosterlee, C.W. (2008). A novel pricing method for european options based on fourier-cosine series expansions. Siam Journal on Scientific Computing 31(2): 826–848.Google Scholar
9.Geske, R. & Johnson, H.E. (1984). The American put option valued analytically. The Journal of Finance 39(5): 1511–1524.Google Scholar
10.Wang, G., Wang, X. & Liu, Z. (2017). Pricing vulnerable American put options under jump-diffusion processes. Probability in the Engineering and Informational Sciences 31(2): 121–138.Google Scholar
11.Heston, S.L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6(2): 327–343.Google Scholar
12.Hung, M.-W. & Liu, Y.-H. (2005). Pricing vulnerable options in incomplete markets. Journal of Futures Markets 25(2): 135–170.Google Scholar
13.Jeon, J., Yoon, J.-H. & Kang, M. (2017). Pricing vulnerable path-dependent options using integral transforms. Journal of Computational and Applied Mathematics 313: 259–272.Google Scholar
14.Johnson, H. & Stulz, R. (1987). The pricing of options with default risk. The Journal of Finance 42(2): 267–280.Google Scholar
15.Klein, P. (1996). Pricing black-scholes options with correlated credit risk. Journal of Banking & Finance 20(7): 1211–1229.Google Scholar
16.Klein, P. & Yang, J. (2010). Vulnerable American options. Managerial Finance 36(5): 414–430.Google Scholar
17.Kou, S.G. (2007). Jump-diffusion models for asset pricing in financial engineering. Handbooks in Operations Research and Management Science 15: 73–116.Google Scholar
18.Kou, S.G. (2002). A jump-diffusion model for option pricing. Management Science 48(8): 1086–1101.Google Scholar
19.Kou, S.G. & Wang, H. (2004). Option pricing under a double exponential jump diffusion model. Management Science 50(9): 1178–1192.Google Scholar
20.Lee, M.-K., Yang, S.-J. & Kim, J.-H. (2016). A closed form solution for vulnerable options with Heston's stochastic volatility. Chaos, Solitons & Fractals 86: 23–27.Google Scholar
21.Lord, R., Fang, F., Bervoets, F. & Oosterlee, C.W. (2008). A fast and accurate FFT-based method for pricing early-exercise options under lévy processes. Ssrn Electronic Journal 30(1952): 1678–1705.Google Scholar
22.Ma, Y., Shrestha, K. & Xu, W. (2017). Pricing vulnerable options with jump clustering. Journal of Futures Markets 37(12): 1155–1178.Google Scholar
23.Merton, R.C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics 3(1-2): 125–144.Google Scholar
24.Ruijter, M.J. & Oosterlee, C.W. (2012). Two-dimensional fourier cosine series expansion method for pricing financial options. Siam Journal on Scientific Computing 34(5): B642–B671.Google Scholar
25.Tian, L., Wang, G., Wang, X. & Wang, Y. (2014). Pricing vulnerable options with correlated credit risk under jump-diffusion processes. Journal of Futures Markets 34(10): 957–979.Google Scholar
26.Wang, G., Wang, X. & Zhou, K. (2017). Pricing vulnerable options with stochastic volatility. Physica A: Statistical Mechanics and its Applications 485: 91–103.Google Scholar
27.Wang, X. (2017). Analytical valuation of vulnerable options in a discrete-time framework. Probability in the Engineering and Informational Sciences 31(1): 100–120.Google Scholar
28.Wang, X. (2017). Differences in the prices of vulnerable options with different counterparties. Journal of Futures Markets 37(2): 148–163.Google Scholar
29.Wang, X. (2017). Pricing vulnerable european options with stochastic correlation. Probability in the Engineering and Informational Sciences 32(1): 67–95.Google Scholar
30.Yang, S.-J., Lee, M.-K. & Kim, J.-H. (2014). Pricing vulnerable options under a stochastic volatility model. Applied Mathematics Letters 34: 7–12.Google Scholar
31.Yoon, J.-H. & Kim, J.-H. (2015). The pricing of vulnerable options with double mellin transforms. Journal of Mathematical Analysis and Applications 422(2): 838–857.Google Scholar
32.Zhu, X. & Chen, S. (2007). Richardson extrapolation techniques for the pricing of American-style options. Journal of Futures Markets 27(8): 791–817.Google Scholar