论文标题
线性粗糙波动下期权定价的错误率较弱
Weak error rates for option pricing under linear rough volatility
论文作者
论文摘要
在定量融资中，对基础资产的波动率结构进行建模对于定价选项至关重要。粗糙的随机波动率模型，例如粗糙的Bergomi模型[Bayer，Friz，Gainteral，定量金融16（6），887-904，2016]，试图基于观察到的观察到的观察，即具有小数的Brownian Motion具有小型的Hurst参数，$ h <1/2 $ h <1/2 $ $ h <1/2 $ $ h <1 $ $ hecals tesscals loce of tays compartional cartistions complocy oby obs coss的数据。资产价格的时间序列和期权衍生的价格数据都表明，$ h $通常的价值接近$ 0.1 $或以下，即比布朗尼运动更粗糙。这种变化改善了基础资产价格的期权价格和时间序列的拟合度，同时保持了简约的价格。但是，在粗糙的波动模型中，驱动部分布朗运动的非马克维亚性质对理论和数值分析以及计算实践构成了严重的挑战。虽然已知明确的Euler方法会收敛到粗糙的Bergomi和类似模型的解决方案，但其强大的收敛速度仅为$ H $。对于粗糙的Stein-Stein模型，对于Euler方法的弱收敛性，我们证明了$ h + 1/2 $，这将波动视为驱动分数布朗尼运动的线性函数，而且令人惊讶的是，对于二次支付功能，我们证明了一个率。我们的证明使用Talay-tubaro扩展和基础的仿射马尔可夫表示，并通过数值实验进一步支持。这些收敛的结果为推导粗糙Bergomi模型的较弱速率提供了第一步，该模型将波动率视为驱动部分布朗尼运动的非线性功能。
In quantitative finance, modeling the volatility structure of underlying assets is vital to pricing options. Rough stochastic volatility models, such as the rough Bergomi model [Bayer, Friz, Gatheral, Quantitative Finance 16(6), 887-904, 2016], seek to fit observed market data based on the observation that the log-realized variance behaves like a fractional Brownian motion with small Hurst parameter, $H < 1/2$, over reasonable timescales. Both time series of asset prices and option-derived price data indicate that $H$ often takes values close to $0.1$ or less, i.e., rougher than Brownian motion. This change improves the fit to both option prices and time series of underlying asset prices while maintaining parsimoniousness. However, the non-Markovian nature of the driving fractional Brownian motion in rough volatility models poses severe challenges for theoretical and numerical analyses and for computational practice. While the explicit Euler method is known to converge to the solution of the rough Bergomi and similar models, its strong rate of convergence is only $H$. We prove rate $H + 1/2$ for the weak convergence of the Euler method for the rough Stein-Stein model, which treats the volatility as a linear function of the driving fractional Brownian motion, and, surprisingly, we prove rate one for the case of quadratic payoff functions. Our proof uses Talay-Tubaro expansions and an affine Markovian representation of the underlying and is further supported by numerical experiments. These convergence results provide a first step toward deriving weak rates for the rough Bergomi model, which treats the volatility as a nonlinear function of the driving fractional Brownian motion.
