A Probabilistic Approach to Nonparametric Local Volatility

TL;DR

This paper introduces a probabilistic, nonparametric Gaussian process approach for calibrating local volatility models, enabling uncertainty quantification and dynamic surface prediction in financial derivatives pricing.

Contribution

It proposes a novel Gaussian process-based calibration method that encodes prior beliefs, avoids over-fitting, and provides uncertainty estimates and dynamic surface predictions.

Findings

Effective calibration of local volatility with uncertainty quantification.

Ability to infer dynamic volatility surfaces over time.

Application to S&P 500 data demonstrates practical utility.

Abstract

The local volatility model is a widely used for pricing and hedging financial derivatives. While its main appeal is its capability of reproducing any given surface of observed option prices---it provides a perfect fit---the essential component is a latent function which can be uniquely determined only in the limit of infinite data. To (re)construct this function, numerous calibration methods have been suggested involving steps of interpolation and extrapolation, most often of parametric form and with point-estimate representations. We look at the calibration problem in a probabilistic framework with a nonparametric approach based on a Gaussian process prior. This immediately gives a way of encoding prior beliefs about the local volatility function and a hypothesis model which is highly flexible yet not prone to over-fitting. Besides providing a method for calibrating a (range of) point-estimate(s), we draw posterior inference from the distribution over local volatility. This leads to a better understanding of uncertainty associated with the calibration in particular, and with the model in general. Further, we infer dynamical properties of local volatility by augmenting the hypothesis space with a time dimension. Ideally, this provides predictive distributions not only locally, but also for entire surfaces forward in time. We apply our approach to S&P 500 market data.