Probabilistic aspects of finance
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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- Fabio Maccheroni & Massimo Marinacci & Aldo Rustichini, 2006.
"Ambiguity Aversion, Robustness, and the Variational Representation of Preferences,"
Econometric Society, vol. 74(6), pages 1447-1498, November.
- Fabio Maccheroni & Massimo Marinacci & Aldo Rustichini, 2004. "Ambiguity Aversion, Robustness, and the Variational Representation of Preferences," Carlo Alberto Notebooks 12, Collegio Carlo Alberto, revised 2006.
- David G. Hobson, 1998. "Robust hedging of the lookback option," Finance and Stochastics, Springer, vol. 2(4), pages 329-347.
- M. Musiela & T. Zariphopoulou, 2009. "Portfolio choice under dynamic investment performance criteria," Quantitative Finance, Taylor & Francis Journals, vol. 9(2), pages 161-170.
- Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
- Harrison, J. Michael & Kreps, David M., 1979. "Martingales and arbitrage in multiperiod securities markets," Journal of Economic Theory, Elsevier, vol. 20(3), pages 381-408, June.
- Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447.
- Aurelien Alfonsi & Antje Fruth & Alexander Schied, 2010. "Optimal execution strategies in limit order books with general shape functions," Quantitative Finance, Taylor & Francis Journals, vol. 10(2), pages 143-157.
- Alexander Schied, 2007. "Optimal investments for risk- and ambiguity-averse preferences: a duality approach," Finance and Stochastics, Springer, vol. 11(1), pages 107-129, January.
- P. Weber & B. Rosenow, 2005. "Order book approach to price impact," Quantitative Finance, Taylor & Francis Journals, vol. 5(4), pages 357-364.
- Föllmer, Hans & Kabanov, Jurij M., 1997. "Optional decomposition and lagrange multipliers," SFB 373 Discussion Papers 1997,54, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
- Jim Gatheral, 2010. "No-dynamic-arbitrage and market impact," Quantitative Finance, Taylor & Francis Journals, vol. 10(7), pages 749-759.
- Markus K. Brunnermeier & Lasse Heje Pederson, 2003.
LSE Research Online Documents on Economics
24829, London School of Economics and Political Science, LSE Library.
- Brunnermeier, Markus K & Pedersen, Lasse Heje, 2004. "Predatory Trading," CEPR Discussion Papers 4639, C.E.P.R. Discussion Papers.
- Markus K Brunnermeier & Lasse Heje Pederson, 2003. "Predatory Trading," FMG Discussion Papers dp441, Financial Markets Group.
- Markus K. Brunnermeier & Lasse Heje Pedersen, 2004. "Predatory Trading," NBER Working Papers 10755, National Bureau of Economic Research, Inc.
- Lasse H. Pedersen & Markus Brunnermeier, 2004. "Predatory Trading," Econometric Society 2004 North American Winter Meetings 425, Econometric Society.
- Bruce Ian Carlin & Miguel Sousa Lobo & S. Viswanathan, 2007. "Episodic Liquidity Crises: Cooperative and Predatory Trading," Journal of Finance, American Finance Association, vol. 62(5), pages 2235-2274, October.
- Frittelli, Marco & Rosazza Gianin, Emanuela, 2002. "Putting order in risk measures," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1473-1486, July.
- Sasha F. Stoikov & Thaleia Zariphopoulou, 2005. "Dynamic Asset Allocation And Consumption Choice In Incomplete Markets ," Australian Economic Papers, Wiley Blackwell, vol. 44(4), pages 414-454, December.
- Deprez, Olivier & Gerber, Hans U., 1985. "On convex principles of premium calculation," Insurance: Mathematics and Economics, Elsevier, vol. 4(3), pages 179-189, July.
- Alfonsi Aurélien & Alexander Schied & Alla Slynko, 2012. "Order Book Resilience, Price Manipulation, and the Positive Portfolio Problem," Post-Print hal-00941333, HAL.
- Obizhaeva, Anna A. & Wang, Jiang, 2013.
"Optimal trading strategy and supply/demand dynamics,"
Journal of Financial Markets,
Elsevier, vol. 16(1), pages 1-32.
- Anna Obizhaeva & Jiang Wang, 2005. "Optimal Trading Strategy and Supply/Demand Dynamics," NBER Working Papers 11444, National Bureau of Economic Research, Inc.
- Gilboa, Itzhak & Schmeidler, David, 1989.
"Maxmin expected utility with non-unique prior,"
Journal of Mathematical Economics,
Elsevier, vol. 18(2), pages 141-153, April.
- Rama Cont & Adrien De Larrard, 2013. "Price Dynamics in a Markovian Limit Order Market," Post-Print hal-00552252, HAL.
- Marco Avellaneda & Sasha Stoikov, 2008. "High-frequency trading in a limit order book," Quantitative Finance, Taylor & Francis Journals, vol. 8(3), pages 217-224.
- Charles-Albert Lehalle, 2013. "Market Microstructure Knowledge Needed for Controlling an Intra-Day Trading Process," Papers 1302.4592, arXiv.org.
- Robert C. Merton, 1973.
"Theory of Rational Option Pricing,"
Bell Journal of Economics,
The RAND Corporation, vol. 4(1), pages 141-183, Spring.
- Haydyn Brown & David Hobson & L. C. G. Rogers, 2001. "Robust Hedging of Barrier Options," Mathematical Finance, Wiley Blackwell, vol. 11(3), pages 285-314.
- Borkar, V. S. & Konda, V. R. & Mitter, S. K., 2004. "On De Finetti coherence and Kolmogorov probability," Statistics & Probability Letters, Elsevier, vol. 66(4), pages 417-421, March.
- Anton Bovier & Jiří Černý & Ostap Hryniv, 2006. "The Opinion Game: Stock Price Evolution From Microscopic Market Modeling," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 9(01), pages 91-111.
- Kramkov, D.O., 1994. "Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets," Discussion Paper Serie B 294, University of Bonn, Germany.
- Alexander Schied, 2012. "A control problem with fuel constraint and Dawson-Watanabe superprocesses," Papers 1207.5809, arXiv.org, revised Dec 2013.
- Bick, Avi & Willinger, Walter, 1994. "Dynamic spanning without probabilities," Stochastic Processes and their Applications, Elsevier, vol. 50(2), pages 349-374, April.
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