一类特殊混合跳扩散Black-Scholes模型的欧式回望期权定价

Abstract

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

CLC Number:

O211.6

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.

Trendmd

Figures/Tables 3

References 26

1	Elliott J , Hoek J . A general fractional white noise theory and applications to finance. Mathematical Finance, 2003, 13 (2): 301- 330 doi: 10.1111/1467-9965.00018
2	Hu Y Z , Øsendal B . Fractional White noise calculus and applications to finance, infinite dimensional analysis. Quantum Probability and Related Topics, 2003, 6 (1): 1- 32 doi: 10.1142/S0219025703001110
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 doi: 10.1007/s00780-004-0144-5
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 doi: 10.1016/j.physa.2012.07.041
6	Sun L . Pricing currency options in the mixed fractional Brownian motion. Physica A, 2013, 392 (16): 3441- 3458 doi: 10.1016/j.physa.2013.03.055
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 doi: 10.2307/3318626
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 doi: 10.1007/s00780-008-0074-8
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 doi: 10.1017/S0269964815000200
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 doi: 10.4208/eajam.221214.240415a
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
	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 doi: 10.1134/S0965542510020077
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 doi: 10.1002/asmb.880
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 doi: 10.1111/j.0960-1627.2004.00191.x
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 doi: 10.1016/j.spa.2004.07.003
22	Leung K S . An analytic pricing formula for lookback options under stochastic volatility. Applied Mathematics Letters, 2013, 26 (1): 145- 149 doi: 10.1016/j.aml.2012.07.008
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 doi: 10.1016/j.spl.2013.08.002
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 doi: 10.1016/j.jbankfin.2013.03.023
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 doi: 10.3969/j.issn.1000-5641.2017.04.001
	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 doi: 10.3969/j.issn.1000-5641.2017.04.001
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 doi: 10.3969/j.issn.1001-2362.2018.02.007
	Yang Z Q . Pricing of fund corporate bonds and default probability study under mixed jump-diffusion model. Systems Engineering, 2018, 36 (2): 16- 28 doi: 10.3969/j.issn.1001-2362.2018.02.007

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

[1]	CHEN Chuan-Jun, ZHAO Xin. Analysis of a Two-grid Method for Characteristic Finite Element Solutions of Nonlinear Convection-diffusion Equations [J]. Acta mathematica scientia,Series A, 2014, 34(3): 643-654.
[2]	SHI Dong-Yang, YU Zhi-Yun. Superclose and Superconvergence Analysis of a Low Order Nonconforming Mixed Finite Element Method for Stationary Stokes Equations with Damping [J]. Acta mathematica scientia,Series A, 2013, 33(4): 735-745.
[3]	SHI Dong-Yang, FEI Li-Fang, XU Chao. Anisotropic Nonconforming Finite Element Analysis for Electromagnetic Wave Propagation in Double Negative Meta-materials [J]. Acta mathematica scientia,Series A, 2012, 32(5): 982-995.
[4]	ZHAO Zhong-Jian, CHEN Shao-Chun. New Approch of the Error Estimate for Nonconforming Plate Element [J]. Acta mathematica scientia,Series A, 2011, 31(4): 1091-1096.
[5]	FU Hong-Fei, RUI Hong-Xing. Error Estimates for the Finite Element Approximation of Nonlinear Parabolic Optimal Control Problems [J]. Acta mathematica scientia,Series A, 2010, 30(6): 1542-1554.
[6]	AN Rong, LI Kai-Tai. Mixed Finite Element Approximation for the Plate Contact Problem [J]. Acta mathematica scientia,Series A, 2010, 30(3): 666-676.
[7]	SHI Dong-Yang, GUO Cheng, WANG Hai-Hong. Nonconforming Finite Element Method for Integro-Differential Equation of Parabolic Type with Weakly Singular Kernel [J]. Acta mathematica scientia,Series A, 2010, 30(3): 764-775.
[8]	Li Changfeng ;Yuan Yirang. Domain Decomposition Upwind Difference Schemes for Multidimensional Parobolic Problems [J]. Acta mathematica scientia,Series A, 2007, 27(5): 879-889.
[9]	Wang Huan; Yuan Yirang. An Upwind-mixed Finite Element Method for Compressible Miscible Displacement [J]. Acta mathematica scientia,Series A, 2007, 27(1): 45-056.

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

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

Figures/Tables 3

References 26

Related Articles 9

Metrics

Comments

Recommended 0