Bayesian nonparametric approaches for financial option pricing

The price of a financial option equals the discounted expected payoff of the option under the risk-neutral measure, and an option's Greeks are formulas that give the change in an option price with respect to parameters of interest (e.g. the price of the underlying asset). The density that reproduces the observed option price is called the risk-neutral or state price density and is used for a variety of important activities in finance, including providing an arbitrage-free tool for pricing complex and less liquid securities. The importance of understanding this density with respect to asset pricing and risk management has led to a competing number of approaches for making inference about the state price density. Unlike the option prices, Greeks can not be observed in the market and have to be calculated. As Greeks are important for measuring and managing risk as well as executing dynamic trading strategies, developing methods to calculate them efficiently and accurately is of critical importance both in theory and in practice (Broadie and Glasserman, 1996).;We start by proposing a finite-dimensional model for the state price density in a Bayesian framework. This modeling approach can be viewed as a Bayesian Quadrature model, where the locations and weights of support points in the finite-dimensional representation of the risk-neutral density are random variables. This modeling approach allows a 'prior' reference distribution which can be a parametric distribution (e.g. the lognormal density) or which can be uniform and completely non-informative, and it also provides a posterior distribution of the state price density that is consistent with the observed option prices. We asses the performance of the proposed model using simulation studies based on synthetic data and then by contrasting the method with a number of competing methods using S&P; 500 index option data.;In contrast to European options, American options can be exercised anytime prior to maturity. We show how our Bayesian Quadrature approach can be extended to make inference for American options. To tackle this problem, we propose a Bayesian implied random tree model as an extension of the Bayesian Quadrature approach by building a unique binomial tree similar to Rubinstein (1994). The benefits of our approach are demonstrated via simulation study and empirical studies using S&P; 100 index option data.;Although finite-difference methods are commonly used to calculate Greeks, these estimates can often be biased and suffer from erratic behavior when the payoff function is discontinuous. We provide new and simple mathematical formulas that overcome these problems and that are applicable to a wide range of complicated options and underlying processes. Moreover, we provide an innovative Bayesian approach to calculate Greeks using observed option prices without any parametric assumptions on the underlying process, so that the proposed method avoids the model misspecification problem. We demonstrate the performance of our methods through simulation studies.