Dr. Martin Tegner, University of Oxford

A probabilistic
approach to non-parametric local volatility

 The local volatility model is a celebrated model widely used for pricing and hedging financial derivatives. While the model’s main appeal is its capability of reproducing any given surface of observed option prices—it provides a perfect fit—the essential component of the model is a latent function which can only be unambiguously determined 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-estimates as result. We seek to look at the calibration problem in a probabilistic framework with a fully nonparametric approach based on Gaussian process priors. This immediately gives a way of encoding prior believes about the local volatility function and a hypothesis model which is highly flexible whilst being prone to overfitting. Besides providing a method for calibrating a (range of) point-estimate(s), we seek to draw posterior inference on the distribution over local volatility. This to better understand the uncertainty attached with the calibration in particular, and with the model in general. Further, we seek to understand dynamical properties of local volatility by augmenting the hypothesis space with a time dimension. Ideally, this gives us means of inferring predictive distributions not only locally, but also for entire surfaces forward in time.

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