25 Pages Posted: 11 Sep 2012 Last revised: 6 Feb 2013
Date Written: September 15, 2012
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 path wise 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.
Keywords: arbitrage pricing theory, hedging, incomplete market, superhedging, Knightian uncertainty, model uncertainty, pathwise Itô calculus, variance swap, monetary measure of risk, convex risk measure, coherent risk measure, algorithmic trading
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By Jim Gatheral