Risk Measures and Financial Innovation with Backward Stochastic Difference Equations

44 Pages Posted: 1 Apr 2016 Last revised: 22 Apr 2019

See all articles by Ezequiel Antar

Ezequiel Antar

University of Cambridge - Centre for Financial Research

M. A. H. Dempster

University of Cambridge - Centre for Financial Research; Cambridge Systems Associates Limited; University of Cambridge - Judge Business School

Date Written: April 19, 2019

Abstract

Economic agents are exposed to the uncertain outcomes of future events. By enabling the exchange of securities, financial markets allow agents to reallocate their exposures in more efficient and mutually convenient arrangements to reduce perceived risks. The complexity and changing nature of the world in which agents operate make all markets incomplete, which means that there is scope for further risk reduction by the introduction of new market securities -- financial innovation -- to cover previously un-priced risks. This paper provides a convenient mathematical framework to solve the problem of dynamic equilibrium pricing and optimal design of new financial securities. Our mathematical tool for studying trading market equilibria is a novel theory of backward stochastic difference equations (BSΔEs) in discrete time, which we develop in analogy to the currently incomplete theory of backward stochastic differential equations (BSDEs) in continuous time. The new tool is used to define a family of dynamic risk measures which is used to solve the problems of optimal trading for a single economic agent, equilibrium pricing of new securities amongst multiple agents, and the optimal design of new securities. Our approach allows the characterization of unique dynamic trading equilibria, optimal instrument design, inter-agent risk transfer and the implications for real-life financial market structures which elude the continuous time BSDE theory. A simple intuitive numerical example is presented to illustrate the concepts introduced.

Keywords: BSΔEs, risk measures, dynamic trading equilibria, equilibrium pricing, unpriced risks, optimal securities, risk transfer

JEL Classification: C61, C62, C63, D81, G12

Suggested Citation

Antar, Ezequiel and Dempster, M. A. H., Risk Measures and Financial Innovation with Backward Stochastic Difference Equations (April 19, 2019). Available at SSRN: https://ssrn.com/abstract=2721885 or http://dx.doi.org/10.2139/ssrn.2721885

Ezequiel Antar

University of Cambridge - Centre for Financial Research ( email )

Centre for Mathematical Sciences
Wilberforce Road
Cambridge, CB3 0WA
United Kingdom

M. A. H. Dempster (Contact Author)

University of Cambridge - Centre for Financial Research ( email )

Centre for Mathematical Sciences
Wilberforce Road
Cambridge, CB3 0WA
United Kingdom

Cambridge Systems Associates Limited ( email )

5-7 Portugal Place
Cambridge, CB5 8AF
United Kingdom

University of Cambridge - Judge Business School ( email )

Trumpington Street
Cambridge, CB2 1AG
United Kingdom

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