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A NEW STOPPING PROBLEM AND THE CRITICAL EXERCISE PRICE FOR AMERICAN FRACTIONAL LOOKBACK OPTION IN A SPECIAL MIXED JUMP-DIFFUSION MODEL

Published online by Cambridge University Press:  21 September 2018

Zhaoqiang Yang Open the ORCID record for Zhaoqiang Yang [Opens in a new window]
Zhaoqiang Yang*
Affiliation:
Library and School of Statistics, Lanzhou University of Finance and Economics Lanzhou 730101, China E-mail: [email protected] or [email protected]
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Abstract

A new stopping problem and the critical exercise price of American fractional lookback option are developed in the case where the stock price follows a special mixed jump diffusion fractional Brownian motion. By using Itô formula and Wick-Itô-Skorohod integral a new market pricing model is built, and the fundamental solutions of stochastic parabolic partial differential equations are deduced under the condition of Merton assumptions. With an optimal stopping problem and the exercise boundary, the explicit integral representation of early exercise premium and the critical exercise price are also derived. Numerical simulation illustrates the asymptotic behavior of this critical boundary.

Keywords

asymptotic behaviorfundamental solutionsmixed jump-diffusion fractional brownian motionoptimal stopping problemwick-itô-skorohod integral
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 34 , Issue 1 , January 2020 , pp. 27 - 52
Copyright
Copyright © Cambridge University Press 2018 

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References

1.Alvaro, C. & Diego, C.N. (2007). Fractional diffusion models of option prices in markets with jumps. Physica A 374(2): 749–763.Google Scholar
2.Beckers, S. (1973). A note on estimating the parameters of the diffusion-jump model of stock returns. Journal of Financial and Quantitative Analysis 16(1): 127–149.Google Scholar
3.Biagini, F., Hu, Y.Z., Øksendal, B. & Zhang, T.S. (2004). Stochastic calculus for fractional Brownian motion and applications. London: Cambridge University Press.Google Scholar
4.Cai, N., Chen, N. & Wan, X.W. (2010). Occupation times of jump-diffusion processes with double exponential jumps and the pricing of options. Mathematics of Operations Research 35(2): 412–437.Google Scholar
5.Cai, N. & Kou, S.G. (2011). Option pricing under a mixed-exponential jump diffusion model. Management Science 57(11): 2067–2081.Google Scholar
6.Chang, M.A., Chinhyung, C. & Keenwan, P. (2001). The price of foreign currency options under jump-diffusion processes. Journal of Futures Markets 27(7): 669–695.Google Scholar
7.Cheridito, P. (2001). Mixed fractional Brownian motion. Bernoulli 7(6): 913–934.Google Scholar
8.Cheridito, P. (2003). Arbitrage in fractional Brownian motion models. Finance and Stochastics 7(4): 533–553.Google Scholar
9.Conze, A. & Viswanathan, G. (1991). Path dependent options: the case of lookback options. Journal of Finance 46(5): 1893–1907.Google Scholar
10.David, A. (2004). Levy processes and stochastic calculus. London: Cambridge University Press.Google Scholar
11.Ei-Nouty, C. (2003). The fractional mixed fractional Brownian motion. Statistics Proba- bility and Letters 65(2): 111–120.Google Scholar
12.Eberlein, E. & Papapantoleon, A. (2005). Equivalence of floating and fixed strike Asian and lookback options. Stochastic Processes and their Applications 115(1): 31–40.Google Scholar
13.Elliott, R.J. & Hoek, J.V.D. (2003). A general fractional white noise theory and applications to finance. Mathematical Finance 13(2): 301–330.Google Scholar
14.Feng, L.M. & Linetsky, V. (2009). Computing exponential moments of the discrete maximum of a Lévy process and lookback options. Finance and Stochastics 13(4): 501–529.Google Scholar
15.Fuh, C.D. & Luo, S.F. (2013). Pricing discrete path-dependent options under a double exponential jump-diffusion model. Journal of Banking and Finance 37(8): 2702–2713.Google Scholar
16.He, X.J. & Chen, W.T. (2014). The pricing of credit default swaps under a generalized mixed fractional Brownian motion. Physica A 404(36): 26–33.Google Scholar
17.Hu, Y.Z. & Øksendal, B. (2003). Fractional white noise calculus and applications to finance. Infinite Dimensional Analysis, Quantum Probability and Related Topics 6(1): 1–32.Google Scholar
18.Kim, K.I., Park, H.S. & Qian, X.S. (2011). A mathematical modeling for the lookback option with jump-diffusion using binomial tree method. Journal of Computational and Applied Mathematics 235(1): 5140–5154.Google Scholar
19.Kou, S.G. (2002). A jump diffusion model for option pricing. Management Science 48(8): 1086–1101.Google Scholar
20.Lai, T.L. & Lim, T.W. (2004). Exercise regions and efficient valuation of American lookback options. Mathematical Finance 14(2): 249–269.Google Scholar
21.Leung, K.S. (2013). An analytic pricing formula for lookback options under stochastic volatility. Applied Mathematics Letters 26(1): 145–149.Google Scholar
22.Merton, R.C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics 3(1–2): 125–144.Google Scholar
23.Mounir, Z.L. (2006). On the mixed fractional Brownian motion. Journal of Applied Mathematics and Stochastic Analysis 2006(1): 1–9.Google Scholar
24.Necula, C. (2002). Option pricing in a fractional Brownian motion environment. Mathematical Reports 2(3): 259–273.Google Scholar
25.Park, H.S. & Kim, J.H. (2013). An semi-analytic pricing formula for lookback options under a general stochastic volatility model. Statistics and Probability Letters 83(11): 2537–2543.Google Scholar
26.Rama, C. & Peter, T. (2002). Non-parametric calibration of jump-diffusion option pricing models. Journal of Computational Finance 7(3): 1–49.Google Scholar
27.Rao, B.L.S.P. (2015). Option pricing for processes driven by mixed fractional Brownian motion with superimposed jumps. Probability in the Engineering and Informational Sciences 29(4): 589–596.Google Scholar
28.Sun, L. (2002). Pricing currency options in the mixed fractional Brownian motion. Physica A 392(16): 3441–3458.Google Scholar
29.Sun, X.C. & Yan, L.T. (2012). Mixed-fractional models to credit risk pricing. Journal of Statistical and Econometric Methods 1(3): 79–96.Google Scholar
30.Shokrollahi, F. & Kılıçman, A. (2014). Pricing currency option in a mixed fractional Brownian motion with jumps environment. Mathematical Problems in Engineering Article number 858210.Google Scholar
31.Shokrollahi, F. & Kılıçman, A. (2015). Actuarial approach in a mixed fractional Brownian motion with jumps environment for pricing currency option. Advances in Difference Equations 2015(1): 1–8.Google Scholar
32.Tomas, B. & Henrik, H. (2005). A note on Wick products and the fractional Black-Scholes model. Finance and Stochastics 9(2): 197–209.Google Scholar
33.Wang, G.Y., Wang, X.C. & Liu, Z.Y. (2015). Pricing vulnerable American put options under jump-diffusion processes. Probability in the Engineering and Informational Sciences 31(2): 121–138.Google Scholar
34.Xiao, W.L., Zhang, W.G., Zhang, X.L. & Wang, Y.L. (2010). Pricing currency options in a fractional Brownian motion with jumps. Economic Modelling 27(5): 935–942.Google Scholar
35.Xiao, W.L., Zhang, W.G. & Zhang, X.L. (2012). Pricing model for equity warrants in a mixed fractional Brownian environment and its algorithm. Physica A 391(24): 6418–6431.Google Scholar
36.Yang, Z.Q. (2017). Optimal exercise boundary of American fractional lookback option in a mixed jump-diffusion fractional Brownian motion environment. Mathematical Problems in Engineering Article number 5904125.Google Scholar
37.Yu, H., Kwok, Y.K. & Wu, L.X. (2001). Early exercise policies of American floating strike and fixed strike lookback options. Nonlinear Analysis 47(7): 4591–4602.Google Scholar