关键词: 期权定价, 状态转移, 随机波动率, 上证50ETF期权, 粒子滤波

Abstract: Accurate pricing is one of the prerequisites to ensure options functioning well in financial markets. Stochastic volatility models are widely used in option pricing as they can generate volatility smiles as well as term structures. However, the shape of volatility curves generated by one-factor stochastic volatility models has weak correlation with real fluctuating level, and cannot accurately reflect the regime switching characteristics of the volatility process as well. Adding state variables which describe regime switching characteristics of the volatility processes to stochastic volatility models can hopefully better describe the shapes of volatility surfaces as well as their dynamic processes, thus improving pricing accuracy. The application of stochastic volatility models with regime switching features in European option pricing has received growing attention for the past few years. However, in terms of model construction, existing research typically limits the relationship between volatility processes and the switching regime, without involving the adaptation of models to market features, and also without further discussion of their performance in certain option markets. In terms of pricing methods, when only the long-term mean is assumed to rely on a switching regime, a closed solution for European option pricing can be obtained. However, for other stochastic volatility models with regime switching features, closed solutions for option pricing are difficult to obtain.Existing research generally uses perturbation analysis or numerical methods to obtain option prices, failing to achieve balance between pricing accuracy and computational efficiency.In terms of the pricing of SSE 50ETF options, models and methods in existing research contribute to the improvement of SSE 50ETF option pricing accuracy, but have not taken regime switching features of volatility into account. However, according to iVIX data, the volatility of SSE 50ETF has regime switching features. Therefore, we aim to construct a stochastic volatility model with regime switching features to better reflect the volatility characteristics of SSE 50ETF, and further conduct study on option pricing.
This paper constructs a series of stochastic volatility models with regime switching features on the basis of Heston model, which describe the regime of variance process with continuous Markov chain, and assumes all parameters in the variance process of Heston model to be any function of the Markovian regime. By specifying the concrete relationship between parameters and switching regimes, we can obtain models fitting different market features. Furthermore, we construct a semi-analytical pricing method suitable for European option pricing under the above model: By applying the properties of affine model, and obtaining the analytical formula of conditional characteristic function of the log-price distribution through Fourier transform under each path of the switching regime, option pricing is then achieved through Monte Carlo simulation of the state paths. In terms of calibration of parameters, we construct an algorithm to realize maximum likelihood estimation for model parameters based on particle filtering, by synthesizing stratified sampling and continuous importance resampling methods, based on iVIX and SSE 50ETF price data.
By analyzing the characteristics of SSE 50ETF price and volatility, we find that there are significant differences in fluctuating levels across different time periods. Therefore, using stochastic volatility models with regime switching features can help better describe the dynamic process of the logarithmic price and variance of SSE50 ETF. Furthermore, based on market features, we select various forms of the model constructed above to describe different regime-switching features, and conduct empirical research on the performance of the stochastic volatility models with regime switching features in the pricing of SSE 50ETF options. The results show that, in the pricing of SSE 50ETF options, the stochastic volatility models with regime switching features in which the long-term mean and volatility of instantaneous variance are dependent on the volatility state can better describe the volatile characteristics of SSE 50ETF prices and significantly improve the accuracy and robustness of option pricing, and in particular, it is of great necessity to consider the regime-switching features of volatility in pricing when the fluctuating level fluctuates violently, especially when it rapidly increases. Among these models, the RS-SV-123 model, whose mean reversion speed also depends on the switching regime, is prominent among all models in terms of pricing accuracy as well as robustness during periods of intense fluctuation, while the RS-SV-23 model, whose mean reversion speed does not depend on the switching regime, is more prominent in terms of pricing robustness during the whole period.

Key words: option pricing, regime switching, stochastic volatility, SSE 50 ETF options, particle filter