Option Pricing And Hedging In Incomplete Market

This thesis develops a new Levy process based on which a series of asset pricing models considering stochastic volatilities and the leverage effect are proposed. The new distribution class, termed NTS, is constructed to model the asset returns by subordinating a drifted Brownian motion through a strictly increasing Tempered Stable process that gen-eralizes the Variance Gamma (VG) and the Normal Inverse Gaussian (NIG). The newly added parameter is to create subclasses for all the distributions discovered in financial mar-ket. The empirical test suggests that time series of Technology stock returns in US market reject both the VG and the NIG distribution and admit instead another subclass of the NTS distribution. In order to solve the option pricing and hedging problems, we derive explicit formula for option prices by means of the Fourier transform method and the convolu-tion theorem, and develop numerical solutions for the variance-optimal hedging strategies, respectively in continuous-time trading and in discrete-time trading. The approaches are novel and are more efficient than those previously suggested in the literature. Based on data on Apple stock prices and the corresponding option contracts, numerical studies for different models also suggest that the algorithm has better accuracy and stability for the pricing and hedging problems. The main work of this thesis is:1. we set up a new pricing framework and derive the formula for option prices. A pure-jump Levy process with infinite activity, termed NTS, is proposed and applied to models considering stochastic volatilities (termed NTSSV) and the leverage effect (termed NTSSVR). The closed form representation of the option price can be easily obtained, by treating the option price as the convolution of the modified terminal payoff and the characteristic function of the asset return, and is computed via the efficient fast Fourier transform (FFT). It should be stressed that the pricing method developed in this thesis are for the general case and not limited to the NTS-related models. As a matter of fact, the option price is explicitly obtained as long as the characteristic function of the underlying asset return is available.2. We develop numerical solutions for the variance-optimal hedging strategies, respec-tively in continuous-time trading and in discrete-time trading. First, we make the continuous-time variance-optimal hedging strategies explicitly computable in mod-els without the leverage effect, which is relied on the stochastic integral represen-tation of the martingale version of stock and option prices under the risk-neutral measure. This hedging method can be applied to many models, including the BS, the NTS, the NTSSV and the Heston model without the leverage effect. Second, we derive a numerical solution by means of a two-dimensional Fourier cosine method for the discrete-time variance-optimal hedging of the European-style derivatives in a more general mathematical setting that only requires the underlying process to be additive.3. We conduct an extensive empirical investigation to test both the fitting capacity and the predictive quality of various models. The numerical experiment consists of the equity price fitting test and the option’s numerical analysis. The former adopted data of 15 indices and stocks listed in US market to calibrate the asset return by assuming it follows an NTS distribution, in order to show if there’s necessity to add the fourth parameter into the model. The models involved in the option’s numerical analysis include the NTS, the NTSSVR, the BSJSV and the BS model as a benchmark. The experimental designs contains the in-sample pricing test, the out-of-sample pricing test, the hedging test, and the implied-volatility patten to give additional insight into model performance.