一类特殊混合跳扩散Black-Scholes模型的欧式回望期权定价

数学物理学报 ›› 2019, Vol. 39 ›› Issue (6): 1514-1531.

一类特殊混合跳扩散Black-Scholes模型的欧式回望期权定价

杨朝强^1,²()

¹ 兰州财经大学图书馆兰州 730101
² 兰州财经大学统计学院兰州 730101

收稿日期:2018-04-02 出版日期:2019-12-26 发布日期:2019-12-28
作者简介:杨朝强, E-mail: woyuyanjiang@163.com, yangzq@lzufe.edu.cn
基金资助:
兰州财经大学科研基金项目(Lzufe2019C-009);兰州财经大学科研基金项目(Lzufe2017C-09)

Pricing European Lookback Option in a Special Kind of Mixed Jump-Diffusion Black-Scholes Model

Zhaoqiang Yang^1,²()

¹ Library of Lanzhou University of Finance and Economics, Lanzhou 730101
² School of Statistics, Lanzhou University of Finance and Economics, Lanzhou 730101

Received:2018-04-02 Online:2019-12-26 Published:2019-12-28
Supported by:
the Research Project of Lanzhou University of Finance and Economics(Lzufe2019C-009);the Research Project of Lanzhou University of Finance and Economics(Lzufe2017C-09)

摘要/Abstract

摘要：

在混合跳扩散Black-Scholes（B-S）模型下研究了欧式固定履约价的回望期权定价问题.结合Merton假设条件以及风险资产所满足的随机微分方程的Cauchy初值问题，利用多尺度参数摄动方法求解了欧式回望期权所适合的抛物型随机偏微分方程，给出了欧式固定履约价的回望期权的近似定价公式，并利用Feynman-Kac公式分析了近似公式的误差估计.数值模拟研究表明，当混合跳扩散模型的波动率为常数时，欧式回望期权是有精确解的，并且随着模拟阶数的增大，回望期权价格的近似解逐渐地逼近期权价格的精确解.

关键词: 混合跳扩散分数布朗运动, 多尺度参数摄动方法, 欧式固定履约回望期权, Feynman-Kac公式, 误差估计

Abstract:

This article considers the pricing problem of European fixed strike lookback options under the environment of mixed jump-diffusion fractional Brownian motion. Under the conditions of Merton assumptions, we analyze the Cauchy initial problem of stochastic parabolic partial differential equations which the risky asset satisfied, by using the perturbation method of multiscale-parameter, the approximate pricing formulae of European lookback options are given by solving stochastic parabolic partial differential equations. Then the error estimates of the approximate solutions are given by using Feynman-Kac formula. Numerical simulation illustrate that the European lookback options have exact solutions when the volatilities are constant, and as the order of simulation increases, the approximate solutions are gradually approximates the exact solutions.

Key words: Mixed jump-diffusion fractional Brownian motion, Perturbation method of multiscale-parameter, European fixed strike lookback options, Feynman-Kac formula, Error estimates

中图分类号:

O211.6

杨朝强. 一类特殊混合跳扩散Black-Scholes模型的欧式回望期权定价[J]. 数学物理学报, 2019, 39(6): 1514-1531.

Zhaoqiang Yang. Pricing European Lookback Option in a Special Kind of Mixed Jump-Diffusion Black-Scholes Model[J]. Acta mathematica scientia,Series A, 2019, 39(6): 1514-1531.

参考文献

1	Elliott J , Hoek J . A general fractional white noise theory and applications to finance. Mathematical Finance, 2003, 13 (2): 301- 330
2	Hu Y Z , ?sendal B . Fractional White noise calculus and applications to finance, infinite dimensional analysis. Quantum Probability and Related Topics, 2003, 6 (1): 1- 32
3	Mishura Y . Stochastic Calculus Fractional Brownian Motions and Related Processes. Berlin: Springer, 2008
4	Tomas B , Henrik H . A note on Wick products and the fractional Black-Scholes model. Finance and Stochastics, 2005, 9 (2): 197- 209
5	Xiao W L , Zhang W G , Zhang X L , Zhang X . Pricing model for equity warrants in a mixed fractional Brownian environment and its algorithm. Physica A, 2012, 391 (24): 6418- 6431
6	Sun L . Pricing currency options in the mixed fractional Brownian motion. Physica A, 2013, 392 (16): 3441- 3458
7	He X J , Chen W T . The pricing of credit default swaps under a generalized mixed fractional Brownian motion. Physica A, 2014, 404 (36): 26- 33
8	Cheridito P . Mixed fractional Brownian motion. Bernoulli, 2001, 7 (6): 913- 934
9	Zili M. On the mixed fractional Brownian motion. Journal of Applied Mathematics and Stochastic Analysis, 2006, 1-9: Art ID: 32435
10	Bender C , Sottinen T , Valkeila E . Pricing by hedging and no-arbitrage beyond semimartingales. Finance and Stochastics, 2008, 12 (4): 441- 468
11	Foad S , Adem K . Pricing currency option in a mixed fractional Brownian motion with jumps environment. Mathematical Problems in Engineering, 2014, 1 (1): 1- 13
12	Foad S , Adem K . Actuarial approach in a mixed fractional Brownian motion with jumps environment for pricing currency option. Advances in Difference Equations, 2015, 257 (1): 1- 8
13	Rao B L S P . Option pricing for processes driven by mixed fractional Brownian motion with superimposed jumps. Probability in the Engineering and Informational Sciences, 2015, 29 (4): 589- 596
14	Miao J , Yang X . Pricing model for convertible bonds:A mixed fractional Brownian motion with jumps. East Asian Journal on Applied Mathematics, 2015, 5 (3): 222- 237
15	Yang Z Q . Optimal exercise boundary of American fractional lookback option in a mixed jump-diffusion fractional Brownian motion environment. Mathematical Problems in Engineering, 2017, 3 (1): 1- 17
16	张伟江. 奇异摄动导论. 北京: 科学出版社, 2014
16	Zhang W J . Introduction to Singular Perturbation. Beijing: Science Press, 2014
17	Nesterov A V , Shuliko O V . Asymptotics of the solution to a singularly perturbed system of parabolic equations in the critical case. Computational Mathematics and Mathematical Physics, 2010, 50 (2): 256- 263
18	Ma Y S , Li Y . A uniform asymptotic expansion for stochastic volatility model in pricing multi-asset European options. Applied Stochastic Models in Business and Industry, 2012, 28 (4): 324- 341
19	Butuzov V F , Bychkov A I . Asymptotics of the solution of an initial-boundary value problem for a singularly perturbed parabolic equation in the case of double root of the degenerate equation. Computational Mathematics and Mathematical Physics, 2013, 49 (10): 1261- 1273
20	Lai T L , Lim T W . Exercise regions and efficient valuation of American lookback options. Mathematical Finance, 2004, 14 (2): 249- 269
21	Eberlein E , Papapantoleon A . Equivalence of floating and fixed strike Asian and lookback options. Stochastic Processes and their Applications, 2005, 115 (1): 31- 40
22	Leung K S . An analytic pricing formula for lookback options under stochastic volatility. Applied Mathematics Letters, 2013, 26 (1): 145- 149
23	Park S H , Kim J H . An semi-analytic pricing formula for lookback options under a general stochastic volatility model. Statistics and Probability Letters, 2013, 83 (11): 2537- 2543
24	Fuh C D , Luo S F , Yen J F . Pricing discrete path-dependent options under a double exponential jump-diffusion model. Journal of Banking and Finance, 2013, 37 (8): 2702- 2713
25	杨朝强. 一类特殊混合跳-扩散模型的欧式回望期权定价. 华东师范大学学报(自然科学版), 2017, 194 (4): 1- 17
25	Yang Z Q . Pricing European lookback option by a special kind of mixed jump-diffusion model. Journal of East China Normal University (Natural Science), 2017, 194 (4): 1- 17
26	杨朝强. 混合跳-扩散模型下一类基金公司的金融债券定价与违约概率研究. 系统工程, 2018, 36 (2): 16- 28
26	Yang Z Q . Pricing of fund corporate bonds and default probability study under mixed jump-diffusion model. Systems Engineering, 2018, 36 (2): 16- 28

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