The analysis of PDES arising in non-linear and non-standard option pricing

This thesis examines two distinct classes of problem in which nonlinearities arise in option pricing theory. In the first class, we consider the effects of the inclusion of finite liquidity into the Black-Scholes-Merton option pricing model, which for the most part result in highly nonlinear partial differential equations (PDEs). In particular, we investigate a model studied by Schonbucher and Wilmott (2000) and furthermore, show how many of the proposed existing models in the literature can be placed into a unified analytical framework. Detailed analysis reveals that the form of the nonlinearities introduced can lead to serious solution difficulties for standard (put and call) payoff conditions. One is associated with the infinite gamma and in such regimes it is necessary to admit solutions with discontinuous deltas, and perhaps even more disturbingly, negative option values. A second failure (applicable to smoothed payoff functions) is caused by a singularity in the coefficient of the diffusion term in the option-pricing equation. It is concluded in this case is that the model irretrievably breaks down and there is insufficient 'financial modelling' in the pricing 'equation. The repercussions for American options are also considered. In the second class of problem, we investigate the properties of the recently introduced British option (Peskir and Samee, 2008a,b), a new non-standard class of early exercise option, which can help to mediate the effects of a finitely liquid market, since the contract does not require the holder to enter the market and hence incur liquidation costs. Here we choose to focus on the interesting nonlinear behaviour of the early-exercise boundary, specifically for times close to, and far from, expiry. In both classes, detailed asymptotic analysis, coupled with advanced numerical techniques (informed by the asymptotics) are exploited to extract the relevent dynamics.EThOS - Electronic Theses Online ServiceGBUnited Kingdo