A BSDE Approach to Convex Risk Measures for Derivative Securities

Abstract

A backward stochastic differential equation (BSDE) approach is used to evaluate convex risk measures for unhedged positions of derivative securities in a continuoustime economy. The convex risk measure is represented as the solution of a BSDE. We use the Clark-Ocone representation result together with Malliavin calculus to identify the integrand in the martingale representation associated with the BSDE. In the Markov case, we relate the BSDE solution to a partial differential equation solution for convex risk measure evaluation.

Backward stochastic difference equations for dynamic convex risk measures on a binomial tree

“…Some versions of dynamic coherent risk measures appeared in [2], [5], [12], and [34]- [36]. Frittelli and Rosazza-Gianin [26], Detlefsen and Scandolo [15], Klöppel and Schweizer [28], Jobert and Rogers [27], and others studied dynamic convex risk measures from a theoretical perspective; see also [19], [20], and [33]. Time-consistency is an important property of dynamic risk measures.…”

Backward stochastic difference equations for dynamic convex risk measures on a binomial tree

“…Some versions of dynamic coherent risk measures appeared in [2], [5], [12], and [34]- [36]. Frittelli and Rosazza-Gianin [26], Detlefsen and Scandolo [15], Klöppel and Schweizer [28], Jobert and Rogers [27], and others studied dynamic convex risk measures from a theoretical perspective; see also [19], [20], and [33]. Time-consistency is an important property of dynamic risk measures.…”