Hedging Strategies For European Contingent Claim Under Two Different Models

With the development of economic society, financial market progresses rapidly, which brings a boom in the study of derivatives. This paper considers the optimal hedging strategy for European contingent claim under two different risky asset models.Chapter One and Chapter Two introduce the significance and important development in the field, as well as some general notations and preliminaries.In Chapter Three and Chapter Four, before presenting the obtained result, we first analyze roughly historical data in order to clarify the consideration when setting the model and to show the logicality of selected model.In Chapter Three, we consider the hedging problem for a class of stochastic differential delay equations with Markov switching. When the price process of a risky asset follows the considered model, we derive a martingale representation for the price process of a contingent claim written on the risky asset with respect to an equivalent martingale measure obtained by the Esscher transform. Then, under some conditions for the coefficients of the model, we identify the continuous-time hedging strategy by minimizing the residual risk in the incomplete market due to the additional source of uncertainty introduced by the regime switching.Chapter Four focuses on the optimal hedging strategies for a class of stochastic differential equations with a generalized diffusion term. When the price process of a risky asset with dividend follows the considered model, we obtain a risk-neutral measure with Girsanov Theorem for an European option written on the risky asset. Then, we identify the continuous hedging strategy, as well we the applicable discrete variance-optimal hedging strategy by minimizing the total hedging risk.