您的位置：山东大学 -> 科技期刊社 -> 《山东大学学报（理学版）》

《山东大学学报(理学版)》 ›› 2025, Vol. 60 ›› Issue (3): 22-32.doi: 10.6040/j.issn.1671-9352.0.2024.105

• 金融数学 • 上一篇    下一篇

区制转换与Hawkes跳扩散模型下的脆弱欧式期权定价

杜慧源,范小明*   

  1. 西南交通大学数学学院, 四川 成都 611756
  • 发布日期:2025-03-10
  • 通讯作者: 范小明(1972— ),男,教授,博士,研究方向为动力系统与随机微分方程、金融统计. E-mail:fanxm@swjtu.edu.cn
  • 作者简介:杜慧源(2000— ),女,硕士研究生,研究方向为金融统计. E-mail:duhuiyuan1127@163.com*通信作者:范小明(1972— ),男,教授,博士,研究方向为动力系统与随机微分方程、金融统计. E-mail:fanxm@swjtu.edu.cn
  • 基金资助:
    国家自然科学基金资助项目(12371178);西南交大创新项目(P113123G02004)

Vulnerable European option pricing in a regime-switching and Hawkes jump diffusion model

DU Huiyuan, FAN Xiaoming*   

  1. School of Mathematics, Southwest Jiaotong University, Chengdu 611756, Sichuan, China
  • Published:2025-03-10

PDF (PC)

40

摘要： 研究随机波动率和随机利率模型下含交易对手违约风险的期权定价问题,该模型中波动率和利率过程的均值回复水平均由有限状态空间的连续马尔可夫过程控制,并假设标的资产价格过程和交易对手的资产价格过程均含有跳,且它们的跳均服从具有自刺激性的Hawkes过程,以及假设波动率过程中也含有跳。利用测度变换、求解贴现特征函数、多元傅里叶变换等方法,推导欧式脆弱期权的解析定价公式;然后利用快速傅里叶变换(fast Fourier transform, FFT)方法计算期权解析定价公式的有效逼近,并通过蒙特卡罗仿真检验逼近的准确性;最后,对所提模型中不同参数对脆弱看涨期权价格的敏感性进行分析,并通过数值实验对比所提出模型与不具有马尔可夫区制转换(Markov regime-switching, MRS)的随机利率模型的差异,说明在模型中引入区制转换对期权定价结果的影响。

关键词: 脆弱欧式期权, 跳跃聚集, 区制转换, 快速傅里叶变换

Abstract: Option pricing with counterparty default risk under stochastic volatility and stochastic interest rate models is studied. In this model, the mean reversion levels of volatility and interest rate process are controlled by a continuous Markov process in a finite state space, and it is assumed that both the underlying asset price process and the counterparty asset price process contain jumps, and their jumps obey the Hawkes process with self-stimulation, and it is assumed that the volatility process also contains jumps. The analytical pricing formula of European vulnerable option is derived by means of measure transform, solution of discount characteristic function and multivariate Fourier transform. Then the fast Fourier transform method is used to calculate the effective approximation of the option analytic pricing formula, and the accuracy of the approximation is verified by Monte Carlo simulation. Finally, the sensitivity of different parameters in the proposed model to the price of vulnerable call options is analyzed, and the difference between the proposed model and the stochastic interest rate model without Markov regime-switching(MRS)is compared by numerical experiments, and the impact of the introduction of regime-switching in the model on the option pricing results is illustrated.

Key words: vulnerable European option, jump clustering, regime-switching, fast Fourier transform

中图分类号: 

  • O211.6
[1] BLACK F, SCHOLES M. The pricing of options and corporate liabilities[J]. Journal of Political Economy, 1973, 81(3):637-654.
[2] HE Xinjiang. A closed-form pricing formula for European options under the Heston model with stochastic interest rate[J]. Journal of Computational and Applied Mathematics, 2018, 335:323-333.
[3] PILLAY E, OHARA G J. FFT based option pricing under a mean reverting process with stochastic volatility and jumps[J]. Computational Applied Mathematics, 2011, 235(12):3378-3384.
[4] JOHNSON H, STULZ R. The pricing of options with default risk[J]. Journal of Finance, 1987, 42(2):267-280.
[5] TIAN Lihui, WANG Guanying, WANG Xingchun, et al. Pricing vulnerable options with correlated credit risk under jump-diffusion processes[J]. Journal of Futures Markets, 2014, 34(10):957-979.
[6] JING Bo, LI Shenghong, MA Yong. Pricing VIX options with volatility clustering[J]. Journal of Futures Markets, 2020, 40(6):928-944.
[7] CHEN Li, MA Yong, XIAO Weilin. Pricing defaultable bonds under Hawkes jump-diffusion processes[J]. Finance Research Letters, 2022, 47:102738.
[8] MA Y, PAN D, SHRESTHA K, et al. Pricing and hedging foreign equity options under Hawkes jump-diffusion processes[J]. Physica A:Statistical Mechanics and its Applications, 2020, 537:122645.
[9] HAMILTON J D. A new approach to the economic analysis of nonstationary time series and the business cycle[J]. Econometrica, 1989, 57(2):357.
[10] XIE Yurong, DENG Guohe. Vulnerable European option pricing in a Markov regime-switching Heston model with stochastic interest rate[J]. Chaos, Solitons & Fractals, 2022, 156:111896.
[11] HAN Miao, SONG Xuefeng, NIU Huawei, et al. Pricing vulnerable options with market prices of common jump risks under regime-switching models[J]. Discrete Dynamics in Nature and Society, 2018, 2018:8545841.
[12] FAN K, SHEN Y, SIU K T, et al. An FFT approach for option pricing under a regime-switching stochastic interest rate model[J]. Communications in Statistics: Theory and Methods, 2016, 46(11):5292-5310.
[13] ELLIOTT R J, AGGOUN L, MOORE J B. Hidden Markov models: estimation and control[M]. New York: Springer, 1994.
[14] 马勇,吕建平. Hawkes跳扩散模型下的脆弱期权定价[J]. 系统工程学报, 2022, 37(5):605-616. MA Yong, LYU Jianping. Vulnerable option pricing under Hawkes jump diffusion model[J]. Journal of Systems Engineering, 2022, 37(5):605-616.
[15] DUFFIE D, PAN J, SINGLETON K. Transform analysis and asset pricing for affine jump-diffusions[J]. Econometrica, 2000, 68(6):1343-1376.
[16] ELLIOTT J R, NISHIDE K, OSAKWE U C. Heston-type stochastic volatility with a Markov switching regime[J]. Journal of Futures Markets, 2016, 36(9):902-919.
[17] TENG L, EHRHARDT M, GÜNTHER M. On the Heston model with stochastic correlation[J]. International Journal of Theoretical and Applied Finance, 2016, 19(6):1-30.
[18] BUFFINGTON J, ELLIOTT RJ. American options with regime switching[J]. International Journal of Theoretical and Applied Finance, 2002, 5(5):497-514.
[1] 陈祥利. 脆弱期权的公司价值分形定价模型[J]. J4, 2010, 45(11): 109-114.
Viewed
Full text
40
HTML PDF
Just accepted Online first Issue Just accepted Online first Issue
0 0 0 0 0 40

  From Others local
  Times 11 29
  Rate 28% 72%

Abstract
63
Just accepted Online first Issue
0 0 63
  From Others local
  Times 61 2
  Rate 97% 3%

Cited
Web of Science  Crossref   ScienceDirect  Search for Citations in Google Scholar >>
 
This page requires you have already subscribed to WoS.
  Shared   
  Discussed   
[1] 孙守斌,孟广武,赵 峰 . 序同态的Dα-连续性[J]. J4, 2007, 42(7): 49 -53 .
[2] 郭亭,鲍晓明 . P137G点突变对嗜热细菌木糖异构酶酶活性及热稳定性的影响[J]. J4, 2006, 41(6): 145 -148 .
[3] 刁科凤,赵　平 . 具有最小连通点对图的C-超图的染色讨论[J]. J4, 2007, 42(2): 56 -58 .
[4] 薛岩波 杨波 陈贞翔. 小波分析在土木工程结构健康监测系统中的应用研究[J]. J4, 2009, 44(9): 28 -31 .
[5] 廖明哲. 哥德巴赫的两个猜想[J]. J4, 2013, 48(2): 1 -14 .
[6] 王开荣,高佩婷. 建立在DY法上的两类混合共轭梯度法[J]. 山东大学学报（理学版）, 2016, 51(6): 16 -23 .
[7] 伍代勇, 张海. 具有Allee效应和时滞的单种群离散模型的稳定性和分支[J]. 山东大学学报（理学版）, 2014, 49(07): 88 -94 .
[8] 李曙光,杨振光,何志红 . 多纤波分复用链网与环网中的利润极大化问题[J]. J4, 2006, 41(5): 7 -11 .
[9] 刘修生 . C［a,b］上半范数的两个问题[J]. J4, 2006, 41(5): 84 -86 .
[10] 江雪莲,石洪波*. 产生式与判别式组合分类器学习算法[J]. J4, 2010, 45(7): 7 -12 .
版权所有 © 2018 《山东大学学报(理学版)》编辑部　
地址: 山东省济南市山大南路27号山东大学中心校区（250100）　电话: +86-0531-88366917　E-mail: xblxb@sdu.edu.cn
本系统由北京玛格泰克科技发展有限公司设计开发