In this paper, we introduce and analyze the fractional Barndorff-Nielsen and Shephard (BN-S) stochastic volatility model. The proposed model is based upon two desirable properties of the long-term variance process suggested by the empirical data: long-term memory and jumps. The proposed model incorporates the long-term memory and positive autocorrelation properties of fractional Brownian motion with \(H>1/2\), and the jump properties of the BN-S model. We find arbitrage-free prices for variance and volatility swaps for this new model. Because fractional Brownian motion is still a Gaussian process, we derive some new expressions for the distributions of integrals of continuous Gaussian processes as we work towards an analytic expression for the prices of these swaps. The model is analyzed in connection to the quadratic hedging problem and some related analytical results are developed. The amount of derivatives required to minimize a quadratic hedging error is obtained. Finally, we provide some numerical analysis based on the VIX data. Numerical results show the efficiency of the proposed model compared to the Heston model and the classical BN-S model.

在本文中，我们介绍并分析了分数 Barndorff-Nielsen 和 Shephard (BN-S) 随机波动率模型。所提出的模型基于经验数据所建议的长期方差过程的两个理想特性：长期记忆和跳跃。所提出的模型结合了分数布朗运动的长期记忆和正自相关特性，\(H>1/2\)，以及 BN-S 模型的跳跃特性。我们为这个新模型找到了方差和波动率掉期的无套利价格。因为分数布朗运动仍然是一个高斯过程，我们推导出了一些新的连续高斯过程积分分布的表达式，因为我们致力于这些掉期价格的解析表达式。结合二次套期保值问题对该模型进行了分析，并得出了一些相关的分析结果。获得最小化二次对冲误差所需的导数数量。最后，我们提供了一些基于 VIX 数据的数值分析。数值结果显示了所提出模型与 Heston 模型和经典 BN-S 模型相比的效率。