关键词: 期权定价, 随机波动率模型, 非仿射, 跳扩散, Fourier-Sinc方法, 特征函数微扰法

Abstract: The out-of-money (OTM) options with short expiration are quite popular in the market due to their cheap prices and lottery attributes. However, the accurate pricing of this type of options has always been a great challenge in financial industry due to the heavy affection from both market illiquidity and sentiment. The existing patching works based on the Heston stochastic volatility model, such as relaxing the constraint of the square root setting for the volatility diffusion, introducing the jump-diffusion process, and adding stochastic factors, have their own applicable scenarios, but generally fail to improve significantly the pricing accuracy of the short-term vanilla options. For this reason, this paper attempts to integrate the existing feature structures that have been proved to be effective and widely used, such as the non-affine volatility structure, the Poisson jump structure, and the two-factor volatility structure, into the Heston model, thus yielding a non-affine double Heston stochastic volatility model with jump (NDHJ).
Given the above comprehensive model, it is an urgent but quite challenging task to obtain the analytic formula of option pricing for practical use in real markets. In this paper, we apply the local perturbation method and the Fourier-Sinc approach to derive an approximate explicit formula for European option price. To be detailed, we decompose the conditional characteristic function into a deterministic component and an undetermined part which is further expanded into a series of terms by the perturbation method. Provided with the prerequisite of a tractable characteristic function, the Fourier-SINC method guarantees that the massive option prices can be numerically and efficiently computed in parallel. By doing this, our approach achieves a good tradeoff between the computational efficiency and accuracy.
Numerical experiments and empirical evidences show that:
(1)The price path simulated by the NDHJ model exhibits more volatility than that by a reduced version of NDHJ, i.e., the non-affine one-factor stochastic volatility model with jump (NHJ), even sharing the same parameter settings. This good performance of the NDHJ model is surely attributed to the double volatility components that drive very stable and quite steep paths, respectively. As a result, the NDHJ model exceeds the NHJ model in capturing the statistical characteristics, such as the “excess kurtosis”, “fat tails” and “skewness” empirically observed in the probability density function of asset return in real markets.
(2)Compared with the classical Heston model, the introduction of the non-affine structure and jump diffusion process contributes to producing steeper implied volatility (IV) curves. The NHJ model cannot fit well both the steep short-maturity IV curve and the smooth long-maturity IV curve simultaneously. As a result, the NHJ model is incompetent in characterizing the high-volatility markets. Therefore, it is necessary to use the NDHJ model to capture the overall characteristics of IV surfaces of options, especially for short-term OTM options, in different dimensions.
(3)Numerical experiments show that the pricing formula provided in this paper has high accuracy. The NDHJ model outperforms the alternatives in pricing the short-term OTM option.
(4)The empirical study validates the good performance of NDHJ model in forecasting the price of SSE 50ETF option. We select the close prices of SSE 50ETF option from January to June 2023 as the objective data, and exclude the ones with remaining trading days less than 5 days and the ones that do not satisfy no-arbitrage condition. The empirical evidences show that, compared with the Heston model and the NHJ model, the NDHJ model has the smallest in-sample fitting and out-of-sample prediction errors, and the overall pricing accuracy for options with different maturities and moneyness is improved, especially for those short-term OTM options.
The marginal contribution of this paper is two-folded: first, it extends the application of the local perturbation method from one-dimensional volatility models to two-dimensional volatility models with jump; second, this paper provides a universal framework of option valuation under a generalized stochastic volatility model.

Key words: option pricing, stochastic volatility model, non-affine, jump-diffusion, Fourier-Sinc, characteristic function perturbation method