Rough volatility models are a new class of stochastic volatility models that have been shown to provide a consistently good fit to implied volatility smiles of SPX options. They are continuous-time stochastic volatility models, whose volatility process is driven by a fractional Brownian motion with the corresponding Hurst parameter less than a half. Albeit the empirical success, the valuation of derivative securities under rough volatility models is challenging. The reason is that it is neither a semi-martingale nor a Markov process. This paper proposes a novel valuation framework for rough stochastic local volatility (RSLV) models. In particular, we introduce the perturbed stochastic local volatility (PSLV) model as the semi-martingale approximation for the RSLV model and establish its existence, uniqueness, Markovian representation and convergence. Then we propose a fast continuous-time Markov chain (CTMC) approximation algorithm to the PSLV model and establish its convergence. Numerical experiments demonstrate the convergence of our approximation method to the true prices, and also the remarkable accuracy and efficiency of the method in pricing European, barrier and American options. Comparing with existing literature, a significant reduction in the CPU time to arrive at the same level of accuracy is observed.

中文翻译：

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粗略随机局部波动率模型和应用的一般估值框架


粗略波动率模型是一类新的随机波动率模型，已被证明与 SPX 期权的隐含波动率微笑始终如一地提供良好的拟合。它们是连续时间随机波动率模型，其波动率过程由分数阶布朗运动驱动，相应的 Hurst 参数小于一半。尽管在实证上取得了成功，但在粗略波动率模型下对衍生证券的估值具有挑战性。原因是它既不是半马丁格尔也不是马尔可夫过程。本文提出了一种新的粗略随机局部波动率 （RSLV） 模型估值框架。特别是，我们引入了扰动随机局部波动率 （PSLV） 模型作为 RSLV 模型的半马丁格尔近似，并建立了它的存在、唯一性、马尔可夫表示和收敛性。然后，我们提出了一种针对 PSLV 模型的快速连续时间马尔可夫链 （CTMC） 近似算法并建立其收敛性。数值实验证明了我们的近似方法与真实价格的收敛性，以及该方法在为欧洲、壁垒和美式期权定价方面表现出的显着准确性和效率。与现有文献相比，观察到 CPU 时间显著缩短，以达到相同的准确度水平。