Local volatility is an important quantity in option pricing, portfolio hedging, and risk management. It is not directly observable from the market; hence calibrations of local volatility models are necessary using observable market data. Unlike most existing point-estimate methods, we cast the large-scale nonlinear inverse problem into the Bayesian framework, yielding a posterior distribution of the local volatility, which naturally quantifies its uncertainty. This extra uncertainty information enables traders and risk managers to make better decisions. To alleviate the computational cost, we apply Karhunen--L\`oeve expansion to reduce the dimensionality of the Gaussian Process prior for local volatility. A modified two-stage adaptive Metropolis algorithm is applied to sample the posterior probability distribution, which further reduces computational burdens caused by repetitive numerical forward option pricing model solver and time of heuristic tuning. We demonstrate our methodology with both synthetic and market data.

翻译：本地波动性是选择定价、投资组合套期保值和风险管理方面的一个重要数量。 它无法直接从市场中观测到。 因此,使用可观测的市场数据对本地波动性模型进行校准是必要的。 与大多数现有的点估计方法不同,我们将大规模非线性反向问题投放到巴伊西亚框架,产生本地波动性的后方分布,这自然会量化其不确定性。 额外的不确定性信息使贸易商和风险管理人员能够做出更好的决定。 为了降低计算成本,我们应用Karhunen-L ⁇ oeve扩展来降低高斯进程在当地波动之前的维度。 修改后的两阶段适应性大都会算法被应用于外缘概率分布样本,这进一步减少了由重复的远期数字定价模型求价解算和超常调时间造成的计算负担。 我们用合成数据和市场数据来展示我们的方法。