Risk Measures and Financial Innovation with Backward Stochastic Difference Equations

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