Pricing and Empirical Research on European Option under 4/2-CIR Model

doi:10.12005/orms.2024.0093

Abstract

Abstract: With the increasing number of endogenous and exogenous shocks, the financial market has entered an unprecedented period of instability, which increases the financial market risk. As an important financial derivative product, option can effectively hedge the market risk, and the key to using option to hedge the market risk is to price it reasonably. The core of the research on option pricing is to construct a model that fits in with the dynamic change characteristics of the underlying asset. The uncertainty of the future of the financial market keeps the price of financial assets appreciated or depreciated, that is, the volatility of the underlying asset price of the option is not a definite constant but has a certain degree of volatility. Therefore, in order to ensure the reliability of the option pricing results, in the process of constructing the pricing model we must take into account the stochastic volatility characteristics of the underlying asset price. In addition, with the continuous change in national economic policies and financial market conditions, the market interest rate is no longer a constant. An accurate description of the stochastic characteristics of market interest rates is conducive to improving the pricing accuracy of financial derivatives. Therefore, how to accurately describe the dynamic process of financial asset prices and the stochastic fluctuation characteristics of interest rates is an urgent problem to be solved.
In this paper, we fully consider the dynamic change characteristics of financial asset prices and the impact of interest rate stochasticity on the pricing results of the option pricing model, and explore the option pricing problem under the 4/2 stochastic volatility model with stochastic interest rate under the assumption that the volatility obeys the mean-reverting process. Firstly, based on the dynamic characteristics of financial asset prices and the randomness of interest rates, the 4/2-CIR stochastic hybrid model is constructed, and the characteristic function of the underlying log asset price and the European option pricing formula based on the 4/2-CIR model are obtained using It’s formula and fast Fourier transform methods. Secondly, a numerical analysis is conducted based on the 4/2-CIR stochastic hybrid model to explore the impact of interest rate stochasticity on the model pricing results, and a sensitivity analysis is conducted on the main parameters in the 4/2-CIR stochastic hybrid model to study the impact of model parameters on the option pricing results. Finally, based on the SSE 50 ETF options market data, the particle swarm optimization algorithm is used to estimate the unknown parameters in the model, and the pricing accuracy and error of the model are analyzed based on the model parameter estimation.
The results show that: the larger the maturity period, the more obvious the impact of interest rate stochastic characteristics on the pricing results of the model, that is, under the stochastic volatility model, to consider the impact of the interest rate factor on the pricing of options is of great practical significance. The speed of mean reversion in volatility and interest rates, as well as changes in the level of mean reversion, moves in the opposite direction of option price movements. The volatility of volatility has a positive effect on option prices. The volatility of interest rates has a non-significant effect on option prices. In conclusion, the stochastic interest rate has a significant effect on the model pricing results. The option price is insensitive to the volatility parameter of interest rate, while it is more sensitive to all other parameters. In addition, the 4/2-CIR stochastic hybrid model has smaller absolute error, mean square error and average absolute percentage error and more accurate pricing results than the classical B-S model and the 4/2 stochastic volatility model.

Key words: 4/2-CIR stochastic hybrid model, option pricing, fast Fourier transform, sensitivity analysis

摘要： 在假设波动率服从均值回复过程的条件下,探讨了具有随机利率的4/2随机波动率模型下的期权定价问题。首先,基于4/2随机波动率模型提出4/2-CIR随机混合模型,并利用快速傅里叶变换方法推出4/2-CIR随机混合模型下的欧式期权定价公式。其次,通过数值分析的方法,对比 4/2随机波动率模型与4/2-CIR随机混合模型的定价结果,分析新模型的定价性能,并运用交叉验证法,对模型中参数进行敏感性分析。最后,选取上证50ETF期权数据进行实证分析。研究发现:随机利率对模型定价结果具有显著影响;期权价格对利率的波动率参数不敏感,而对其它参数都较敏感;与经典B-S模型及4/2随机波动率模型相比,4/2-CIR随机混合模型的定价误差更小,定价结果更接近真实值。

关键词: 4/2-CIR随机混合模型, 期权定价, 快速傅里叶变换, 敏感性分析

CLC Number:

O212

GUO Jingjun, MA Aiqin, ZHANG Cuiyun. Pricing and Empirical Research on European Option under 4/2-CIR Model[J]. Operations Research and Management Science, 2024, 33(3): 162-168.

郭精军, 马爱琴, 张翠芸. 基于4/2-CIR模型的欧式期权定价及实证研究[J]. 运筹与管理, 2024, 33(3): 162-168.

References

[1]BLACK F, SCHOLES M S. The pricing of options and corporate liabilities[J]. Journal of Political Economy, 1973, 81(3): 637-654.
[2]HULL J, WHITE A. The pricing of options on assets with stochastic volatilities[J]. Finance, 1987, 42(2): 281-300.
[3]STEIN E M, STEIN J C. Stock price distributions with stochastic volatility: An analytic approach[J]. Review of Financial Studies, 1991, 4(4): 727-752.
[4]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): 327-343.
[5]吴鑫育,李心丹,马超群.基于随机波动率模型的上证50ETF期权定价研究[J].数理统计与管理,2019,38(1):115-131.
[6]HESTON S L. A simple new formula for options with stochastic volatility[J]. Social Science Electronic Publishing, 1997, 15(4): 23-44.
[7]GRASSELLI M. The 4/2 stochastic volatility model: A unified approach for the Heston and the 3/2 model[J]. Mathematical Finance, 2017, 27(4): 1013-1034.
[8]LIN W, LI S H, LUO X G, et al. Consistent pricing of VIX and equity derivatives with the 4/2 stochastic volatility plus jumps model[J]. Journal of Mathematical Analysis and Applications, 2017, 447(2): 778-797.
[9]王波,朱顺伟,邓亚东,等. 4/2随机波动率模型下的期权定价[J].系统管理学报,2020,29(1):192-198.
[10]李蓬实,杨建辉,林焰.快速均值回归随机波动率模型参数估计及应用[J].运筹与管理,2020,29(2):137-143.
[11]VASICEK O. An equilibrium characterization of the term structure[J]. Journal of Financial Economics, 1977, 5(4): 627-627.
[12]COX J C, INGERSOLL J E, ROSS S A. A theory of the term structure of interest rates[J]. Econometrica, 1985, 53(2): 385-408.
[13]郭精军,张亚芳. 次分数Vasicek随机利率模型下的欧式期权定价[J].应用数学,2017,30(3):503-511.
[14]ZHANG X, SHU H, KAN X, et al. The call option pricing based on investment strategy with stochastic interest rate[J]. Journal of Mathematical Finance, 2018, 8(1): 43-57.
[15]FRUTOS J D, GATCĆN V. An extension of Heston’s SV model to stochastic interest rates[J]. Journal of Computational and Applied Mathematics, 2019, 354: 174-182.
[16]HE X J, ZHU S P. 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.
[17]RECCHIONI M C, SUN Y. An explicitly solvable Heston model with stochastic interest rate[J]. European Journal of Operational Research, 2016, 249(1): 359-377.