Using the functional Itô's calculus and forward-backward stochastic differential equations (FBSDEs), a new approach for evaluating dynamic convex risk measures for European-style derivative securities is proposed in a general, non-Markovian, continuous-time financial market. Firstly a dynamic convex risk measure for an unhedged position of derivative securities is represented as the conditional g-expectation which is given by the solution of the backward system in a FBSDE. Then we use the functional Itô's calculus, a martingale representation and the unique decomposition of special semimartingales to identify the solution of the backward system in the FBSDE. In particular, the control component in the backward system is identified using functional derivatives. Whereas the first component of the backward system satisfies a functional partial differential equation.
|Number of pages||20|
|Journal||Communications on stochastic analysis|
|Publication status||Published - 2012|