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September 2013 Probabilistic aspects of finance
Hans Föllmer, Alexander Schied
Bernoulli 19(4): 1306-1326 (September 2013). DOI: 10.3150/12-BEJSP05

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.

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Hans Föllmer. Alexander Schied. "Probabilistic aspects of finance." Bernoulli 19 (4) 1306 - 1326, September 2013. https://doi.org/10.3150/12-BEJSP05

Information

Published: September 2013
First available in Project Euclid: 27 August 2013

zbMATH: 1279.91053
MathSciNet: MR3102553
Digital Object Identifier: 10.3150/12-BEJSP05

Keywords: algorithmic trading , arbitrage pricing theory , coherent risk measure , convex risk measure , hedging , incomplete market , Knightian uncertainty , market impact model , model uncertainty , monetary measure of risk , pathwise Itô calculus , price impact , superhedging , variance swap

Rights: Copyright © 2013 Bernoulli Society for Mathematical Statistics and Probability

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Vol.19 • No. 4 • September 2013
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