A Probabilistic Approach to Nonparametric Local Volatility

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 overfitting. 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.

Bayesian inference of PolII dynamics over the exclusion process

“…We developed a general Bayesian framework to study the dynamics of a one-dimensional transport model given time-resolved density profiles. The general problem addressed here is the identification of the PDE parameters that best describe data as a subset of the true PDE solution (see, e.g., [41,[44][45][46] and references therein). We focused on the hydrodynamic TASEP with smoothlyvarying jump rates (which are the parameters to be inferred) as a paradigmatic and well-characterised model of transport.…”

“…where θ min := (m min , σ min , κ min , σ f min , l min ), θ max := (m max , σ max , κ max , σ f max , l max ), and (µ ξ , σ ξ ) are referred to as hyperparameters. By virtue of the Bayes theorem the joint posterior probability for θ and f P (f , m, σ , κ, σ f , l|y) ∝ P (y|f , σ , κ) P (f |m, σ f , l)P (m)P (σ )P (κ)P (σ f )P (l), ( 15) which we draw random samples from by Markov chain Monte Carlo (MCMC) sampling scheme, more specifically block Gibbs sampling with elliptical slice sampling at each block [40,41] (Appendix C). Evaluating the likelihood also requires computing from equation ( 11) with initial condition (x, 0) = κ y * (x), ∀x ∈ [0, L].…”

“…In the third step, given the values for f , l, and σ f , the likelihood hyperparameters m, σ , and κ are updated by sampling from the unnormalised conditional P (y|f , m, σ )P (m)P (σ )P (κ). We refer the reader to [41] for more details.…”