Measuring the Risk-neutral Probability Distribution of Equity Index Options

Abstract: The focus of this master thesis is to develop a model that measures the risk-neutral probability distributionof the future value of a portfolio consisting of options on the S&P 500 index. The cornerstone of the model is an explicit and thorough construction of the local volatility surface. The parametric model of Coleman etal. (1998), with some modifications, is used, representing the local volatility surface through a bicubic spline. The local volatility surface is optimized to be consistent with market data on option prices, futures contracts and Overnight Index Swap (OIS) interest rates. Repricing of options is done through a finite difference method (FDM) approach presented by Andersen and Brotherton-Ratcliffe (1998), using the Crank-Nicholson scheme. An interior point solver is used to minimize the squared pricing error weighted by liquidity in each option. Fast and accurate gradients of the objective function are obtained using the Automatic Differentiation library Autograd.The local volatility surface is constructed for multiple dates and the systematic changes are analyzed through Principal Component Analysis (PCA) of the logarithmic local variance. A stochastic process is assigned to the local volatility surface using the sensitivities towards systematic changes identified through the PCA. Using a Gaussian Kernel Density Estimator, the probability density function (PDF) of the future value of the portfolio is measured. The method requires simulated portfolio values, which are achieved through FDM pricing using simulations of the local volatility surface and the underlying index. The cumulative distribution function (CDF) is finally computed by integration of the PDF. To evaluate the measured probability distribution, a normal CDF inversion of 106 measured out-of-sample CDF values are compared to theoretical normal distribution quantiles with Q-Q plots. The constructed local volatility surface is consistent with market prices to an extent where it is more accurate for more liquid options. It is in most cases realistic with respect to smoothness, but have an unexpectedly large offset from the at-the-money strike level in the skew structure. It is unstable from date to date and also significantly dependent on choice of parameters, limited by computational power, and input data. The unstable construction of the local volatility surface results in measurement noise that cause negative auto correlation in the principal components, which impairs their explanatory ability. The main result show that the shape of the probability distribution is measured accurately, but the standard deviation (or volatility) is overestimated.