一类特殊混合跳扩散Black-Scholes模型的欧式回望期权定价

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.

图/表 3

表 1

表 2

图 1

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股价$S$	${\mathscr P}_0$	${\mathscr P}_1$	${\mathscr P}_2$	${\mathscr P}_3$	${\mathscr P}_4$	${\mathscr P}_5$	精确值
10	71.419978523	73.636156468	76.519896751	81.145748979	84.912047501	88.658351421	87.5310
20	63.260586024	65.223576189	67.777862873	71.875233511	75.211250358	78.529556890	77.5310
30	55.101193512	56.810995913	59.035828989	62.604718032	65.510453213	68.400762361	67.5310
40	46.941801092	48.398415621	50.293795111	53.334202565	55.809656065	58.271967823	57.5310
50	38.782408511	39.985835343	41.551761231	44.063687088	46.108858922	48.143173289	47.5310
60	30.623016001	31.573255065	32.809727353	34.793171621	36.408061771	38.014378764	37.5310
70	22.464439443	23.161516031	24.068567676	25.523583232	26.708234734	27.886597117	27.5320
80	14.368445442	14.814301500	15.394459401	16.325099631	17.082812751	17.836503307	17.6097
90	7.057221764	7.276208938	7.561159943	8.018254242	8.390413468	8.760596967	8.6492

股价$S$	$ {\mathscr P}_0$	${\mathscr P}_1$	${\mathscr P}_2$	${\mathscr P}_3$	${\mathscr P}_4$	${\mathscr P}_5$	精确值
10	67.386870055	71.018366151	76.238252122	83.309467972	86.150639011	89.728816161	87.5310
20	59.688240989	62.904855955	67.528394784	73.791757867	76.308338686	79.477726082	77.5310
30	51.989611921	54.791345753	58.818537464	64.274047874	66.466038363	69.226636056	67.5310
40	44.290982862	46.677835544	50.108680141	54.756337743	56.623738038	58.975546033	57.5310
50	36.592353837	38.564325343	41.398822822	45.238627682	46.781437737	48.724456004	47.5310
60	28.893724747	30.450815141	32.688965565	35.720917626	36.939137373	38.473365978	37.5310
70	21.195865544	22.338116291	23.979979171	26.204159333	27.097821272	28.223301060	27.5320
80	13.557054821	14.287648062	15.337797447	16.760401881	17.331995611	18.051862002	17.6097
90	6.658698249	7.017537244	7.533329792	8.232057784	8.512802401	8.866372785	8.6492

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