Pricing and hedging in incomplete markets with model uncertainty

We search for a trading strategy and the associated robust price of unhedgeable assets in incomplete markets under the acknowledgement of model uncertainty. Our set-up is that we postulate the management of a firm that wants to maximise the expected surplus by choosing an optimal investment strategy. Furthermore, we assume that the firm is concerned about model misspecification. This robust optimal control problem under model uncertainty leads to (i) risk-neutral pricing for the traded risky assets, and (ii) adjusting the drift of the nontraded risk drivers in a conservative direction. The direction depends on the firm's long or short position, and the adjustment that ensures a robust strategy leads to what is known as “actuarial” or “prudential” pricing. Our results extend to a multivariate setting. We prove existence and uniqueness of the robust price in an incomplete market via the link between the semilinear partial differential equation and backward stochastic differential equations for viscosity and classical solutions.