Parametric and Nonparametric Approaches for Estimating Risk-Neutral Density

In this dissertation, we propose one parametric method and one nonparametric method for estimating the risk-neutral density, which is a fundamental concept in pricing financial derivatives, risk management, and assessing financial markets' perceptions over significant political or economic events. In Chapter 2, we propose an innovative parametric estimate of risk-neutral density using the Normal Inverse Gaussian distribution (NIG). It has been known that the estimation bias comes from two sources, the discontinuity of available option strikes and the asymmetry of available put and call option strikes. To reduce the bias, we propose two new methods as opposed to the existing ones in the literature for each of the two bias sources, respectively. We thus have four combinations of the bias-reduction approaches. We evaluate the performance of all the four combinations by running a comprehensive empirical study using 20 years' S&P 500 index option data. The two new methods proposed by us significantly outperform the classical ones in the literature regarding feasible domain coverage and option price estimation. In Chapter 3, we propose a new nonparametric method which estimates the risk-neutral density by natural cubic splines (NCS hereafter). Our method targets the logarithm of the underlying asset price so that the restriction to the positive domain is released. We also deliberatively put the knots on all the unique option strikes to make our NCS estimate flexible enough to capture the market information contained in option prices. We run a comprehensive empirical study on the proposed NCS method, as well as other relevant approaches in the literature on 20 years of S&P 500 index option data. The empirical study shows that our proposed NCS method is more robust than the historical piecewise constant method which can only produce a discontinuous density, especially for the cases where the options have longer than half a year to maturity. Moreover, our NCS method outperforms other historical continuous methods in terms of optimization feasibility and option price estimation. In Chapter 4, we theoretically prove the consistency property of our NCS method. We prove that under reasonable assumptions, the fair prices of options based on the estimated risk-neutral density converge to their market values on average. We then verify the theoretical property numerically.