Title: Probabilistic Aspects of Finance I
Speaker: Professor Ioannis KARATZAS
Speaker Info: Columbia
Date: Friday, October 23, 1998
In the first of these talks, we shall present the standard model of Samuelson-Merton and Black-Scholes for a financial market consisting of one risk-free asset (bond) and several risky assets (stocks). Within its context, we shall introduce and discuss notions of portfolio and consumption strategies, arbitrage and its absence, equivalent martingale measures, contingent claims, complete and incomplete markets. We shall broach the problem of hedging contingent claims, such as options, and show how it can be solved in the context of complete markets using the methodologies of Stochastic Analysis and of linear parabolic Partial Differential Equations. Along with other examples of this methodology, we shall present the famous Black-Scholes (1973) formula for the price of a European call-option. Finally, we shall indicate briefly how this methodology has to be modified when dealing with American options, as well with incomplete/constrained markets, different interest rates for borrowing and lending, transaction costs, etc. All these developments are now well documented in the literature; see, for instance, the recent monograph by Karatzas & Shreve (1998).
In the second lecture we shall try to examine what happens when some of the assumptions of the standard model are not satisfied. In particular, we shall assume that we start with funds insufficient for hedging without risk, and that there is uncertainty about the stock appreciation rates. This uncertainty will be modelled in a Bayesian fashion, i.e., by positing that the appreciation rates are unobservable random variables, independent of the primary sources of randomness in the market and with known probability distributions. Based on an observation-flow of past-and-present stock-prices, what is then the portfolio that maximizes the probability of perfect hedge? We shall provide an explicit solution to this adaptive stochastic control problem, using techniques of filtering, the Neyman-Pearson lemma, and dynamic programming methodologies leading to a parabolic version of the famous Monge-Ampere equation. These results are taken from Karatzas (1997) and extend those of Kulldorff (1993) and Heath. Similar techniques can be used to solve more general utility maximization problems in a similar context, as in Karatzas & Zhao (1998).