Higher order approximation of option prices in Barndorff-Nielsen and Shephard models
提出一种基于混合公式和泰勒展开的递归算法，用于Barndorff-Nielsen和Shephard随机波动率模型中的欧式期权定价，可得到任意阶的闭式近似，并在逆高斯或伽马背景下与特征函数方法比较，误差分析部分基于已有结果。
We present an approximation method based on the mixing formula [Hull, J. and White, A., The pricing of options on assets with stochastic volatilities. J. Finance, 1987, 42, 281–300; Romano, M. and Touzi, N., Contingent claims and market completeness in a stochastic volatility model. Math. Finance, 1997, 7, 399–412] for pricing European options in Barndorff-Nielsen and Shephard models. This approximation is based on a Taylor expansion of the option price. It is implemented using a recursive algorithm that allows us to obtain closed form approximations of the option price of any order (subject to technical conditions on the background driving Lévy process). This method can be used for any type of Barndorff-Nielsen and Shephard stochastic volatility model. Explicit results are presented in the case where the stationary distribution of the background driving Lévy process is inverse Gaussian or gamma. In both of these cases, the approximation compares favorably to option prices produced by the characteristic function. In particular, we also perform an error analysis of the approximation, which is partially based on the results of Das and Langrené [Closed-form approximations with respect to the mixing solution for option pricing under stochastic volatility. Stochastics, 2022, 94, 745–788]. We obtain asymptotic results for the error of the Nth order approximation and error bounds when the variance process satisfies an inverse Gaussian Ornstein–Uhlenbeck process or a gamma Ornstein–Uhlenbeck process.