Risk analysis, valuation and hedging of financial derivatives

Abstract

When it comes to dealing with financial products, pricing and hedging is indispensable activities for market participants. The principle of dynamic replication provides a framework to achieve both valuation and management of derivatives. However, over the past decades, many empirical studies show that dynamic hedging strategies tend to fail for a variety of reasons in reality. As a consequence, an alternative approach, static hedging, has been suggested. This thesis discusses an approach to analyzing, evaluating and managing financial derivatives via static replications. In particular, I focus on several improvements from two perspectives: (1) theoretical foundation of static hedge with the theory of integral equations

and (2) extension of applicability to meet the growth in complex financial products.

From the first perspective, I propose a new systematic approach to hedging a wide class of financial products under mild assumptions on implied volatilities, and under a general Markovian diffusion with killing. I call it boundary matching approach. While dynamic hedging is elaborately derived from a continuous time finance model, there is no such a fundamental theory to static hedging. I establish a continuous time version of static replications with integral equations whose rich theory is essential for characterizing and quantifying static hedging portfolios. Then, the existence and uniqueness of static hedging portfolio for target derivatives are studied. To do this, I study the associated Volterra integral equation and generalized Abel integral equations. This framework allows us to obtain analytic expressions of hedge weights and values of complex derivatives. Furthermore, improved numerical schemes are designed to make the proposed method practically feasible. Lastly, their convergence of discrete portfolios is rigorously proven, which has been confirmed only by numerical experiments in the literature.

In response to the criticism of static hedging about the restricted applicability, I broaden the area in which static hedging can be applied. In this thesis, I explore static replications of Parisian options, American options, barrier options, sequential barrier options, knock-in knock-out options and Autocallable barrier reverse convertible products(also known as ELS in Korea). To handle a wide spectrum of trigger features and payoff functions, I suggest a decomposition technique for Parisian options, and recursive method for Autocallable barrier reverse convertible products. A Parisian option is also a barrier-type option in which its payment is activated only if the underlying asset consecutively remains below a given barrier over a certain amount of time, the option window. Unlike most type of barrier options, the knock-in(out) boundaries of Parisian options are not depicted on $(t,S_t)$-plane. Thus, I investigate the decomposition of a Parisian option into other contingent claims for each of which a static hedging strategy can be devised. For structured products with autocallable, reverse convertible and barrier features, I propose a recursive method that utilizes strike-spread approach and calendar-spread approach in the literature.