Bayesian Uncertainty Quantification of Local Volatility Model

TL;DR

This paper introduces a Bayesian framework for local volatility modeling in option pricing, providing uncertainty quantification and computational efficiency improvements through dimension reduction and adaptive sampling techniques.

Contribution

It presents a novel Bayesian approach to calibrate local volatility models, incorporating uncertainty quantification and efficient sampling methods.

Findings

Posterior distribution effectively captures local volatility uncertainty.

Karhunen--Loeve expansion reduces computational complexity.

Method performs well on synthetic and market data.

Abstract

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.