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随机利率随机波动率混合指数跳扩散模型下的期权定价

吴胤昊1, 陈荣达1,2, 汪圣楠1, 俞静婧1   

  1. 1. 浙江财经大学金融学院, 杭州 310018;
    2. 嘉兴学院, 嘉兴 314001
  • 收稿日期:2021-10-25 修回日期:2022-02-27 发布日期:2022-10-18
  • 基金资助:
    国家自然科学基金重点项目(71631005)资助课题.

吴胤昊, 陈荣达, 汪圣楠, 俞静婧. 随机利率随机波动率混合指数跳扩散模型下的期权定价[J]. 系统科学与数学, 2022, 42(8): 2207-2234.

WU Yinhao, CHEN Rongda, WANG Shengnan, YU Jingjing. Option Pricing Under the Mixed-Exponential Jump Diffusion Model with Stochastic Interest Rate and Stochastic Volatility[J]. Journal of Systems Science and Mathematical Sciences, 2022, 42(8): 2207-2234.

Option Pricing Under the Mixed-Exponential Jump Diffusion Model with Stochastic Interest Rate and Stochastic Volatility

WU Yinhao1, CHEN Rongda1,2, WANG Shengnan1, YU Jingjing1   

  1. 1. School of Finance, Zhejiang University of Finance and Economics, Hangzhou 310018;
    2. Jiaxing University, Jiaxing 314001
  • Received:2021-10-25 Revised:2022-02-27 Published:2022-10-18
文章在Heston模型的基础上同时将随机利率与混合指数跳引入模型,并且考虑到金融市场中已存在的负利率事实,因此不同于经典的CIR假设,文章放松了利率不能为负的限制,假定其满足Hull-White过程,提出了Heston-HW-MEJ混合模型.在模型的求解方面,文章推导了含对数价格特征函数的定价半闭解,并采用快速Fourier变换方法进行计算.数值分析的结果表明,文章提出的模型较好地捕捉了尖峰厚尾、盈亏不对称等市场特征与隐含波动率曲面形态,并且快速Fourier变换方法求解相比于Monte Carlo模拟快速、高效.最后通过香港恒生指数期权市场与上证50ETF期权市场数据对多个模型进行了实证比较,结果表明,文章的模型在低利率市场具有更好的定价效果,但在利率相对较高的市场略差于Heston-CIR-MEJ模型.
Based on Heston's model, this paper introduces stochastic interest rate and mixed exponential jumps into the model. And taking into account the fact that there is already negative interest rate in the financial market, this paper tries to relax the restriction that interest rate cannot be negative, assuming that it satisfies the Hull-White process which is different from the classical CIR hypothesis. Finally, the hybrid Heston-HW-MEJ model is proposed. However, the complexity of the multi-stochastic element system makes the closed-form solution usually does not exist, so in terms of model solving, this paper adopts the fast Fourier transform method to obtain the semi-closed solution of pricing containing the characteristic function of logarithmic price, which is the main work of this paper to derive this characteristic function. The results of numerical analysis show that the model proposed in this paper captures the market characteristics well, such as heavy tail, leverage effect, profit and loss asymmetry, etc, and the fast Fourier transform method is fast and efficient compared with Monte Carlo simulation in solving. And in comparison with the Black Scholes model, the call option prices obtained from the Heston-HW-MEJ model are higher, indicating that the Black Scholes model underestimates the volatility of the underlying, and the Heston-HW-MEJ model implies greater market uncertainty. Finally, through the data of Hong Kong Hang Seng Index option market and Shanghai 50ETF option market, the empirical comparison of several models shows that the model in this paper has better pricing effect in low interest rate market, but it is slightly worse than the Heston-CIR-MEJ model in relatively high interest rate market.

MR(2010)主题分类: 

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[1] Black F, Scholes M. The pricing of options and corporate liabilities. Journal of Political Economy, 1973, 81 (3): 637-654.
[2] 余湄, 程志勇, 邓军, 等. 一个新的期权定价方法:基于 混合次分数布朗运动的新视角. 系统工程理论与实践, 2021, 41 (11): 2761-2776. (Yu M, Cheng Z Y, Deng J, et al. A new option pricing method: Based on the perspective of sub-mixed fractional Brownian motion. Systems Engineering——Theory & Practice, 2021, 41 (11): 2761-2776.)
[3] Chen R D, Zhou H X, Yu L A, et al. An efficient method for pricing foreign currency options. Journal of International Financial Markets Institutions & Money, 2021, 74 (2): 101295.
[4] Heston S L. A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies, 1993, 6 (2): 327-343.
[5] Forde M, Jacquier A. robust approximations for pricing asian options and volatility swaps under stochastic volatility. Applied Mathematical Finance, 2010, 17 (3): 241-259.
[6] He X J, Zhu S P. An analytical approximation formula for European option pricing under a new stochastic volatility model with regime-switching. Journal of Economic Dynamics and Control, 2016, 71 (1): 77-85.
[7] 王宜峰, 孙雨尧, 蒋一琛. 基于傅里叶变换的银行触发性理财产品定价. 系统科学与 数学, 2018, 38 (2): 236-246. (Wang Y F, Sun Y Y, Jiang Y C. Pricing the bank's triggered financial products based on Fourier transform method. Journal of Systems Science and Mathematical Sciences, 2018, 38 (2): 236-246.)
[8] 林建伟,王志焕.随机利率背景下含有信用增进条款公司债券的定价. 系统科学与数学, 2021, 41 (7): 1938-1955. (Lin J W, Wang Z H. Pricing of company bonds with credit enhancement clause under stochastic interest rate. Journal of Systems Science and Mathematical Sciences, 2021, 41 (7): 1938-1955.)
[9] 李丹萍,林玉容,曾燕.随机利率与随机波动率模型下保险公司的均衡再保险-投资策略. 计 量经济学报, 2021, 1 (4): 904-920. (Li D P, Lin Y R, Zeng Y. Equilibrium reinsurance-investment strategy for insurers under stochastic interest rate and stochastic volatility models. China Journal of Econometrics, 2021, 1 (4): 904-920.)
[10] Chen R D, Li Z X, Zeng L Y, et al. Option pricing under the double exponential jump-diffusion model with stochastic volatility and interest rate. Journal of Management Science and Engineering, 2017, 2 (4): 252-289.
[11] He X J, Zhu S P. A closed-form pricing formula for European options under the Heston model with stochastic interest rate. Journal of Computational and Applied Mathematics, 2017, 335 (1): 323-333.
[12] Chang Y, Wang Y M. Option pricing under double stochastic volatility model with stochastic interest rates and double exponential jumps with stochastic intensity. Mathematical Problems in Engineering, 2020, 2020 (3): 1-13.
[13] Recchioni M C, Sun Y, Tedeschi G. Can negative interest rates really affect option pricing? Empirical evidence from an explicitly solvable stochastic volatility model. Quantitative Finance, 2017, 17 (8): 1-19.
[14] Burro G, Giribone P G, Ligato S, et al. Negative interest rates effects on option pricing: Back to basics? International Journal of Financial Engineering, 2017, 4 (2-3): 1-27.
[15] Merton R C. Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 1976, 3 (1): 125-144.
[16] 吴恒煜, 马俊伟, 朱福敏, 等. 基于Lé vy过程修正GJR-GARCH模型的权 证定价——对中国大陆和香港权证的实证研究. 系统工程理论与实践, 2014, 34 (12): 3009-3021. (Wu H Y, Ma J W, Zhu F M, et al. Warrants pricing based on GJR-GARCH model with Lé vy processes adjusting: An empirical analysis between Mainland and Hongkong. Systems Engineering——Theory & Practice, 2014, 34 (12): 3009-3021.)
[17] 王春发, 陈荣达. Markov调制Lé vy模型定价的Fourier-Cos方法. 高校 应用数学学报A辑, 2016, 31 (4): 390-404. (Wang C F, Chen R D. Option pricing in Markov regime switching Lé vy models using Fourier-Cosine expansions. Applied Mathematics A Journal of Chinese Universities $($Ser. A$)$, 2016, 31(4): 390-404.)
[18] Yang H X, Kanniainen J. Jump and volatility dynamics for the S&P 500: Evidence for infinite-activity jumps with non-affine volatility dynamics from stock and option markets. Review of Finance, 2017, 21 (2): 811-844.
[19] Yeap C, Kwok S S, Choy S. A flexible generalized hyperbolic option pricing model and its special cases. Journal of Financial Econometrics, 2018, 16 (3): 425-460.
[20] Cai N, Kou S G. Option pricing under a mixed-exponential jump diffusion model. Management Science, 2011, 57 (10): 2067-2081.
[21] Carr P, Stanley M, Madan D B. Option valuation using the fast Fourier transform. Journal of Computational Finance, 1999, 2 (4): 61-73.
[22] Kou S G. A jump-diffusion model for option pricing. Management Science, 2002, 48 (8): 1086-1101.
[23] Heyde C C, Kou S G. On the controversy over tailweight of distributions. Operations Research Letters, 2003, 32 (5): 399-408.
[24] Duffie D, Pan J, Singleton K. Transform analysis and asset pricing for affine jump-diffusions. Econometrica, 2000, 68 (6): 1343-1376.
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