Theoretical and Applied Probability on Stochastic Calculus, Numerical Methods, and Mathematical Finance

Cornell University, Ithaca NY

Investigators

Abstract

0202958 Protter The principal investigator and co-principal investigator will investigate several topics in probability within the subfields of stochastic calculus and Monte Carlo simulation. The problems selected are motivated primarily by applications in mathematical finance, but the results will be of more general theoretical significance. In stochastic calculus, new results about stopping times will lead to better models of credit risk and a deeper understanding of what makes possible successful hedging of financial risks. New progress in stochastic calculus will help extend theories of incomplete markets, i.e. markets containing risks that can not be perfectly hedged, as is true in practice. One Monte Carlo issue to be addressed is the optimal use of an algorithm that has recently become popular in option pricing. Another line of research will involve improving numerical techniques for solutions of stochastic differential equations in connection with simulation. A third Monte Carlo topic is the optimal application of some variance reduction techniques that are well suited to problems in nuclear physics, estimation of rare events probabilities, and some option pricing problems. The recent development of a new model for stock prices that incorporates sizes of trade will lead to new pricing technology for financial derivatives. The application of probability to finance has revolutionized an industry. In the past 20 years the creation of multi-trillion dollar derivative security markets has facilitated the world-wide flow of capital and thereby enhanced international commerce and productivity. Without the mathematical models which provide reliable pricing of derivative securities (e.g., stock options) and guide the management of their associated risk, these markets could not exist. The underlying theme of the mathematical success has been to compute precisely the price of financial derivatives which enable companies to lay off risk by buying financial instruments that protect them from unlikely but possibly disastrous events. Equally if not more important has been the description of a recipe for the seller of the instrument to follow in order to protect himself from the risk he accepts through the sale. A complete market is one in which the theory explains how to do this in principle, and in such a market the theory often provides an explicit guide to implementation of this recipe. In other words, in complete markets a new type of insurance has been created, and this has been made possible by existing probability theory. This type of "risk insurance" generated the revolution mentioned above. A real problem, however, is that in reality markets are not complete, and thus new mathematical techniques are needed to extend the theory and to make it more truly applicable. This has already begun, but it is in its infancy, and this extension of the theory will be a large focus of the proposed project. In addition, recently new models have been proposed to better incorporate liquidity issues and market frictions (such as transaction costs when implementing stock trades), in part by the PI himself. These models will continue to be developed, calibrated, and statistically verified.