Stochastic Analysis with Applications to Finance

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This is a three-part proposal. In the first part, the limit order book of an electronic exchange will be modeled as a queueing system, evolving as a measure-valued process, where the measure in question is the set of order sizes located on the price axis. Like many queueing systems, the evolution of the order book at the level of discrete orders is prohibitively complex. Therefore, the theory of heavy traffic limits for queueing systems will be used to obtain continuous approximations to the discrete systems. In the second part of the proposal, models for asset price volatility will be developed. Local volatility models currently in use describe volatility as a function of time and the underlying asset price. We propose to extend this concept to models in which volatility is a function of time, the underlying asset price, and one or more factors that determine the payoff of derivative securities written on the underlying asset. The third part of the proposal continues work on optimal investment and consumption in the presence of transaction costs. These problems are well understood and there are closed-form solutions in cases where transaction costs are zero, but there are no closed-form solutions in the more realistic case when transaction costs are positive.It is proposed to develop asymptotic expansions in the small transaction cost parameter of the unknown solution around the zero-transaction-cost known solution.In overview, it is proposed to build mathematical models designed to reduce risk and increase efficiency in financial markets. Innovations in the financial markets have changed their structure, and analysis needs to keep pace. In particular, the advent of order-driven electronic exchanges has spawned the growth of algorithmic trading, whereby firms trade large quantities of assets quickly. Does this make markets liquid? Does it dampen or amplify volatility? It is difficult to answer these questions because we do not have good mathematical models for these exchanges. Neither can we predict with any accuracy how various order-matching protocols on these exchanges will affect volatility, bid-ask spreads, and general market efficiency. A major part of this proposal is the development of mathematical models for electronic exchanges. A second aspect of this proposal is to improve mathematical models for asset price volatility. The celebrated Black-Scholes model assumes that volatility is constant, but departures from this assumption in financial markets are too significant to ignore. The third part of this proposal is to better understand the effect of transaction costs on problems of optimal investment and consumption. This problem is at the heart of equilibrium analysis of financial markets, but most studies to date make the unrealistic assumption that transaction costs are zero.