Probabilistic Aspects of Finance

Abstract

In the past decades, advanced probabilistic methods have had significant impact on the field of finance, both in academia and in the financial industry. Conversely, financial questions have stimulated new research directions in probability. In this survey paper, we review some of these developments and point to some areas that might deserve further investigation. We start by reviewing the basics of arbitrage pricing theory, with special emphasis on incomplete markets and on the different roles played by the "real-world" probability measure and its equivalent martingale measures. We then focus on the issue of model ambiguity, also called Knightian uncertainty. We present two case studies in which it is possible to deal with Knightian uncertainty in mathematical terms. The first case study concerns the hedging of derivatives, such as variance swaps, in a strictly pathwise sense. The second one deals with capital requirements and preferences specified by convex and coherent risk measures. In the final two sections we discuss mathematical issues arising from the dramatic increase of algorithmic trading in modern financial markets. Why does Brownian motion appear in the financial context? Here is a first rough argument. At each fixed time, the price of a stock could be seen as a temporary equilibrium resulting from a large number of decisions to buy or sell, made in a random and more or less independent manner: Many coins are thrown successively, and so Brownian motion should arise as a manifestation of the central limit theorem. This is the "Coin-Tossing View of Finance", as it is called by J. Cassidy in How Markets Fail [17]. This rough argument can be refined by using microeconomic assumptions on the behavior of agents and on the ways they generate a random demand, and then the