Pricing and hedging in incomplete markets with model ambiguity

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 an agent who wants to maximise the expected surplus by choosing an optimal investment strategy. Furthermore, we assume that the agent 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 non-traded risk drivers in a conservative direction.

The direction depends on the agent’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 semi-linear partial differential equation and backward stochastic differential equations.

Keywords: model uncertainty; indifference pricing; hedging; incomplete markets; robustness; optimisation
MSC2000 subject classification: Primary: 91B06; 91B30; 49L99; 49J35; secondary: 62P05; 35A01; 35A02

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