Hostname: page-component-78857b5c4d-76nq5 Total loading time: 0 Render date: 2023-08-03T01:52:13.833Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "coreDisableSocialShare": false, "coreDisableEcommerceForArticlePurchase": false, "coreDisableEcommerceForBookPurchase": false, "coreDisableEcommerceForElementPurchase": false, "coreUseNewShare": false, "useRatesEcommerce": true } hasContentIssue false

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*
Affiliation:
Library and School of Statistics, Lanzhou University of Finance and Economics Lanzhou 730101, China E-mail: woyuyanjiang@163.com or yangzq@lzufe.edu.cn

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.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Alvaro, C. & Diego, C.N. (2007). Fractional diffusion models of option prices in markets with jumps. Physica A 374(2): 749763.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): 127149.CrossRefGoogle 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): 412437.CrossRefGoogle Scholar
5.Cai, N. & Kou, S.G. (2011). Option pricing under a mixed-exponential jump diffusion model. Management Science 57(11): 20672081.CrossRefGoogle 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): 669695.Google Scholar
7.Cheridito, P. (2001). Mixed fractional Brownian motion. Bernoulli 7(6): 913934.CrossRefGoogle Scholar
8.Cheridito, P. (2003). Arbitrage in fractional Brownian motion models. Finance and Stochastics 7(4): 533553.CrossRefGoogle Scholar
9.Conze, A. & Viswanathan, G. (1991). Path dependent options: the case of lookback options. Journal of Finance 46(5): 18931907.CrossRefGoogle 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): 111120.CrossRefGoogle Scholar
12.Eberlein, E. & Papapantoleon, A. (2005). Equivalence of floating and fixed strike Asian and lookback options. Stochastic Processes and their Applications 115(1): 3140.CrossRefGoogle Scholar
13.Elliott, R.J. & Hoek, J.V.D. (2003). A general fractional white noise theory and applications to finance. Mathematical Finance 13(2): 301330.CrossRefGoogle 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): 501529.CrossRefGoogle 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): 27022713.CrossRefGoogle 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): 2633.CrossRefGoogle 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): 132.CrossRefGoogle 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): 51405154.CrossRefGoogle Scholar
19.Kou, S.G. (2002). A jump diffusion model for option pricing. Management Science 48(8): 10861101.CrossRefGoogle Scholar
20.Lai, T.L. & Lim, T.W. (2004). Exercise regions and efficient valuation of American lookback options. Mathematical Finance 14(2): 249269.CrossRefGoogle Scholar
21.Leung, K.S. (2013). An analytic pricing formula for lookback options under stochastic volatility. Applied Mathematics Letters 26(1): 145149.CrossRefGoogle Scholar
22.Merton, R.C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics 3(1–2): 125144.CrossRefGoogle Scholar
23.Mounir, Z.L. (2006). On the mixed fractional Brownian motion. Journal of Applied Mathematics and Stochastic Analysis 2006(1): 19.Google Scholar
24.Necula, C. (2002). Option pricing in a fractional Brownian motion environment. Mathematical Reports 2(3): 259273.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): 25372543.CrossRefGoogle Scholar
26.Rama, C. & Peter, T. (2002). Non-parametric calibration of jump-diffusion option pricing models. Journal of Computational Finance 7(3): 149.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): 589596.CrossRefGoogle Scholar
28.Sun, L. (2002). Pricing currency options in the mixed fractional Brownian motion. Physica A 392(16): 34413458.CrossRefGoogle Scholar
29.Sun, X.C. & Yan, L.T. (2012). Mixed-fractional models to credit risk pricing. Journal of Statistical and Econometric Methods 1(3): 7996.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): 18.CrossRefGoogle Scholar
32.Tomas, B. & Henrik, H. (2005). A note on Wick products and the fractional Black-Scholes model. Finance and Stochastics 9(2): 197209.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): 121138.CrossRefGoogle 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): 935942.CrossRefGoogle 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): 64186431.CrossRefGoogle 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.CrossRefGoogle 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): 45914602.CrossRefGoogle Scholar