Numerical Method for KoBol Fractional Option Pricing Model

doi:10.11908/j.issn.0253-374x.20082

Home > Archive>Volume 48, Issue 10, 2020 >1495-1505. DOI:10.11908/j.issn.0253-374x.20082

Numerical Method for KoBol Fractional Option Pricing Model
DOI:
                        10.11908/j.issn.0253-374x.20082
                    
CSTR:
                        [cstr]
                    
Author:
                        ZHANG LingxiZHANG Lingxi
School of Mathematical Sciences， Tongji University， Shanghai 200092， China
Find this author on All Journals
Find this author on BaiDu
Search for this author on this site
YIN JunfengYIN Junfeng
School of Mathematical Sciences， Tongji University， Shanghai 200092， China
Find this author on All Journals
Find this author on BaiDu
Search for this author on this site

                    
Affiliation:School of Mathematical Sciences， Tongji University， Shanghai 200092， China
Clc Number:O241
Fund Project:

Article

Figures

Metrics

Reference [29]

Related [20]

Cited by

Materials

Comments

Abstract:

Since the Black-Scholes model was proposed， many option pricing models have been proposed， which has become a hotspot in financial engineering. For the past few years， option pricing models based on Lévy process such as FMLS， CGMY and KoBol have drawn great attention because of their capability to represent the dynamic characteristics of underlying asset. Solving these models would eventually come down to solving a class of fractional partial differential equations. In this paper， a numerical scheme is proposed for the class of FPDE and the stable condition of the scheme is given. The numerical experiments prove the feasibility and effectiveness of the proposed numerical scheme. Based on the practical data of SSE 50ETF and CSI 300ETF index option， the option price and volatility curve are calculated to verify the effectiveness of KoBol model in real market.

Key words:stock index option;option pricing;Lévy process;fractional partial differential equation;finite difference method

Reference

[1] 姜礼尚. 期权定价的数学模型和方法[M]. 2版. 北京：高等教育出版社， 2008.

[2] JOHN H. Options， futures， and other derivatives [M]. 10th ed. Upper Saddle River： Pearson Education， 2017.

[3] MERTON R C. Option pricing when underlying stock returns are discontinuous [J]. Journal of Financial Economics， 1975， 3（1）： 125.

[4] KOU S G. A Jump-diffusion model for option pricing [J]. Management Science， 2002， 48（8）： 1086.

[5] HESTON S L. A closed-form solution for options with stochastic volatility with applications to bond and currency options [J]. Review of Financial Studies， 1993， 6（2）： 327.

[6] HULL J， WHITE A. The pricing of options on assets with stochastic volatilities [J]. Journal of Finance， 1987， 42（2）： 281.

[7] DAVIS M H A， PANAS V G， ZARIPHOPOULOU T. European option pricing with transaction costs [J]. SIAM Journal on Control and Optimization， 1993， 31（2）： 470.

[8] GAN Xiaoting， YIN Junfeng， GUO Yunxiang. Finite volume method for pricing european and american options under jump-diffusion models [J]. East Asian Journal on Applied Mathematics， 2017， 7（2）： 227.

[9] GAN Xiaoting， YANG Ying， ZHANG Kun. A robust numerical method for pricing American options under Kou’s jump-diffusion models based on penalty method [J]. Journal of Applied Mathematics and Computing， 2020， 62（3）： 1.

[10] 甘小艇，殷俊锋，李蕊. 有限体积法定价跳扩散期权模型[J]. 同济大学学报（自然科学版）， 2016， 44（9）： 1458.

[11] ZHENG Ning， YIN Junfeng. Accelerated modulus-based matrix splitting iteration methods for linear complementarity problem [J]. Numerical Algorithms， 2013， 64（2）： 245.

[12] MANDELBROT B B. The variation of certain speculative prices [J]. The Journal of Business， 1963， 36（4）： 394.

[13] CARR P， WU Liuren. The finite moment log stable process and option pricing [J]. Journal of Finance， 2003， 58（2）： 753.

[14] KOPONEN I. Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process [J]. Physical Review E， 1995， 52（1）： 1197.

[15] BOYARCHENKO S I， LEVENDORSKII S Z. Non-gaussian merton-black-scholes theory [M]. Singapore： World Scientific， 2002.

[16] CARR P， HéLYETTE G， MADAN D B， et al. Stochastic volatility for Lévy processes [J]. Mathematical Finance， 2003， 13（3）：345.

[17] CARTEA A， DEL-CASTILLO-NEGRETE D. Fractional diffusion models of option prices in markets with jumps [J]. Physica A， 2006， 374（2）：749.

[18] MAROM O， MOMONIAT E. A comparison of numerical solutions of fractional diffusion models in finance [J]. Nonlinear Analysis Real World Applications， 2009， 10（6）：3435.

[19] WANG Wenfei， CHEN Xu， DING Deng， et al. Circulant preconditioning technique for barrier options pricing under fractional diffusion models [J]. International Journal of Computer Mathematics， 2015， 92（11/12）：2596.

[20] MENG Qingjiang， DING Deng， SHENG Qin. Preconditioned iterative methods for fractional diffusion models in finance [J]. Numerical Methods for Partial Differential Equations， 2015， 31（5）：1382.

[21] ZHANG H， LIU F， TURNER I， et al. Numerical simulation of a finite moment log stable model for a European call option [J]. Numerical Algorithms， 2017， 75（3）：569.

[22] 郭柏灵，蒲学科，黄凤辉. 分数阶偏微分方程及其数值解[M]. 北京：科学出版社， 2011.

[23] KANGRO R， NICOLAIDES R. Far field boundary conditions for black-scholes equations [J]. SIAM Journal on Numerical Analysis， 2000， 38（4）：1357.

[24] MEERSCHAERT M M， TADJERAN C. Finite difference approximations for two-sided space-fractional partial differential equations [J]. Applied Numerical Mathematics， 2006， 56（1）：80.

[25] TADJERAN C， MEERSCHAERT M M， SCHEFFLER H P. A second-order accurate numerical approximation for the fractional diffusion equation [J]. Journal of Computational Physics， 2006， 213（1）：205.

[26] SAAD Y. Iterative methods for sparse linear systems [M]. 2nd ed. Philadelphia： SIAM Publisher， 2003.

[27] CHAN R H F， JIN Xiaoqing. An introduction to iterative toeplitz solvers [M]. Philadelphia： SIAM Publisher， 2007.

[28] STRANG G. A proposal for toeplitz matrix calculations [J]. Studies in Applied Mathematics， 1986， 74（2）：171.

[29] CHAN T F. An optimal circulant preconditioner for toeplitz systems [J]. Siam Journal on Scientific & Statistical Computing， 1988， 9：766.

Get Citation

ZHANG Lingxi, YIN Junfeng. Numerical Method for KoBol Fractional Option Pricing Model[J].同济大学学报(自然科学版),2020,48(10):1495~1505

Copy

Article Metrics

Abstract:652
PDF: 156
HTML: 169
Cited by: 0

History

Received:March 17,2020
Revised:
Adopted:
Online: November 04,2020
Published:

Get Citation

Share

Article Metrics

History

Article QR Code