PROCONFIN · Topics on probability and convexity in finance

Topics on probability and convexity in finance

Randomness is an important constituent in the fields of Economics and Finance, mainly due to the inability of humans to perfectly forecast future, even under the immense availability of data. The analytical theory of Probability naturally forms the most prominent quantitative research tool in problems related to Finance. Substantial is also the use of Convex Analysis, especially in problems dealing with optimal investment and/or consumption, hedging of complex financial instruments, risk management and equilibrium theory. While the field of Financial Mathematics has witnessed a plethora of major achievements, there is ever-present need for more in-depth resolution of important problems. This proposal aims at addressing a representative collection of three areas: (1) Financial equilibria with heterogeneous agents in incomplete markets; (2) Viability of financial models with investment constraints and infinite number of traded assets; and (3) Hedging under model uncertainty. All the aforementioned three directions have been the subject of recent studies, resulting in an improvement of the quality of financial modelling, allowing for market imperfections and seeking to comprehend them, and manipulating the risk involved with complicated financial positions by exploiting the structure of simpler traded assets. During the four years since the initiation of the project, there has been significant work in all three areas mentioned above, which resulted in submission of papers to outstanding journals, with some already published. I have presented these results in several international conferences, and have both travelled and invited collaborators in order to work closely on the main objectives and certain ramifications. A PhD student worked under my supervision in relation to item (1) of the objectives. Main results, in reference to the three research areas, include: (A) A study of incomplete financial equilibria when endowments of interacting agents are close to Pareto optimality (with Hao Xing, LSE, and Gordan Zitkovic, UT Austin); Nash equilibria and strategic behaviour of agents in risk-sharing transactions (with Michael Anthropelos, Univerisy of Pireaus and George Vichos, PhD student). (B) A study of convex structure of the space of nonnegative random variables that will enable applications in both areas (2) and (3); a study of the robust version of fundamental theorem of asset pricing and hedging (with Sara Biagini, LUISS Guido Carli, Rome, Bruno Bouchard, Universite Paris Dauphine, and Marcel Nutz, Columbia University); a robust representation of utilities and cost functions (joint with Samuel Drapeau, Shanghai Jiao Tong University); a unifying study of hedging and valuation in markets with potentially large number of assets. (C) A study of robust long-term growth optimisation in environments with stable distribution of capital (with Scott Robertson, Boston University); a study of drawdown-constrained robust wealth targeting (with Eckhard Platen, University Technology Sydney and Jan Obloj, Oxford University). Results as the above, apart from an obvious mathematical significance, have further practical implications. Equilibrium models help in restricting the class of reasonable dynamics for liquid assets, and provide a quantitative analysis of sensitivity with respect to fundamental inputs. Models with an infinite number of assets are idealised limits of large financial markets, and their study helps in understanding their complexity and structure. Providing hedging and trading strategies that work in a robust way helps eliminate the risk under adverse and unpredictable market movements.

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