Characterizations of and closed-form solutions for plain vanilla and exotic derivatives

Abstract

This dissertation is composed of three stand-alone research projects on the valuation of contingent claims.

The first essay proposes an extension of the Kou (2002) double exponential jump-diffusion model. Displacing the two exponential tails introduces additional degrees of asymmetry in the jump size distribution. The model dynamics are supported by a general equilibrium framework. Our main contribution is to derive closed-form solutions for European plain vanilla options. A further extension to displaced gamma tails is possible while retaining full analytical tractability. We propose an efficient routine to estimate the physical model parameters through maximum likelihood. Our empirical analysis covers a diverse sample of assets across equities, commodities and foreign exchange. We find that for the vast majority of assets, the original Kou (2002) model can be rejected in favour of our newly introduced displaced double exponential dynamics.

The second essay proposes an approach to valuation and risk management of deferred start barrier options within the Black and Scholes (1973) framework. We provide closed-form solutions which are functions of the implied volatility smile. Our barrier options are contingent claims on two perfectly correlated assets that diffuse with different volatilities. While the terminal payoff is a function of one of the assets, the barrier trigger is determined by the path of the other. To mitigate the dynamic hedging problems associated with large discontinuous sensitivities, we suggest the application of an additional exponential bending of the barrier close to maturity. By generalizing the method of images, we obtain closed-form solutions for both deferred start piecewise exponential barrier options and associated rebates.

The third essay models logarithmic asset prices under the physical probability measure as additive jump-diffusion processes. The corresponding risk-neutral probability measure is defined through an Esscher transform. We are interested in the conditions under which the jump size distributions under the two probability measures fall into the same parametric class. We show that it is both necessary and sufficient for the jump size distribution to follow a natural exponential mixture family at all points of time. Examples for applications of this result in financial engineering are provided.