Support Vector Regression Based Approaches to the Recovery of Implied Information on Option Prices

Abstract

As a tool of managing risks, options have continued blooming in past years. The option prices contain much implied information of the market participators, for example, the market expectations of the distribution of the underlying asset price, and the investors risk preferences, and so on. The risk-neutral density (RND), which can be estimated from cross sections of European option prices, expresses the distribution of the underlying asset price directly. The RND can be recovered from cross sections of traded European option prices as it is proportional to the second derivative of the option prices with respect to the strike. However, it is not easy to obtain a well-behaved RND due to the data limitations, the complex constraints and the ill-posedness of the problem. A number of research works have been published to solve the problem and the topic continues blooming right now. With the RND estimated from the option prices at hand, one can find various applications, for example, estimating the risk aversion function (RAF), which reflects the investors risk preferences. Other applications include pricing illiquid derivatives, assessing market beliefs, estimating the higher moments of the RND and probability weighting function, managing risks, market timing and so on.
In this thesis, we first propose a nonparametric method for estimating the RND from a set of European option bid-ask quotes based on a linear programming support vector regression (LPSVR). The RND is modeled as a linear combination of the kernel function. The coefficients of the kernel function are determined by minimizing the objective function, which includes both the error term and the regularization term. The shape constraints are added into the optimization problem naturally. Due to the continuous constraint on the variables, the problem is finally transformed into a semi-infinite linear programming problem. Unlike the parametric approach, this nonparametric approach makes no strong assumptions on data generating process, and thus is flexible enough for the real market data. The method allows us to fit beyond the range of data due to the design of the kernel function, leading to a density with full tails. To give an arbitrage-free density, all the shape constraints can be incorporated into the fitting problem. In addition, the method does not need preprocessing of the data and is robust to the bid-ask spreads. The Monte-Carlo simulations and the empirical tests have been carried out to demonstrate the excellent accuracy and stability of the method. The results show that the LPSVR method is a promising alternative for estimating the RND.
Although the LPSVR method has good performance, the estimated RND usually has a wavy shape, resulting in larger fluctuation in accuracy. To over-come this problem, in the next part of the thesis, we propose a method for estimating the RND based on multiple kernel learning (MKL), which fits the data through a combination of multiple kernels instead of a single one. The multiple kernels are constructed by a linear combination of single kernels with different shape parameters. The shape parameter set is determined by a nonlinear least squares method according to option prices. The hyper-parameter is determined by cross validation approach. The problem is finally transformed to a semi-infinite linear programming problem. The method not only inherits the advantages of the LPSVR-based method, but also has a smaller fluctuation in accuracy, demonstrated by the Monte-Carlo simulations and the empirical tests. The results show that the MKL-based method is a more effective method for estimating the RND compared with the existing versions.
The estimated RND can be further separated into two parts, the subjective probability and the risk aversion adjustment. If the investors are risk-neutral, the RND and the subjective probabilities will be the same. However, the investors may have their preferences, which indicated by the risk aversion adjustment. If we know the two probabilities, we can derive the preferences of the investors empirically. In the third part of the thesis, we empirically recover the risk aversion function from the market option prices, which link the RND and subjective density of the underlying assets. The risk aversion functions are investigated both in American and Hong Kong markets for different times to maturity. We also analyze the reason why the risk aversion function looks different from the economic theory.

Date of Award	20 Jun 2016
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Chuangyin DANG (Supervisor)