  In this paper we consider a class of BSDEs with drivers of quadratic growth,
on a stochastic basis generated by continuous local martingales. We first
derive the Markov property of a forward--backward system (FBSDE) if the
generating martingale is a strong Markov process. Then we establish the
differentiability of a FBSDE with respect to the initial value of its forward
component. This enables us to obtain the main result of this article, namely a
representation formula for the control component of its solution. The latter is
relevant in the context of securitization of random liabilities arising from
exogenous risk, which are optimally hedged by investment in a given financial
market with respect to exponential preferences. In a purely stochastic
formulation, the control process of the backward component of the FBSDE steers
the system into the random liability and describes its optimal derivative hedge
by investment in the capital market, the dynamics of which is given by the
forward component.
