Abstract:

The optimal pricing and hedging of claims written on a non-traded
asset is studied using both traditional dynamic programming and modern
duality techniques, and comparisons between the two methods are
drawn. A traded asset, imperfectly correlated with the non-traded
asset, is available for hedging, and we focus primarily on the case
where the agent's preferences are described by an exponential utility
function. Representations for utility-based prices of the claim are
obtained using the two methods, and the link between them is
elucidated. The dynamic programming method yields a nonlinear partial
differential equation for the value function representing the agent's
maximum utility. The PDE is linearised by applying a power
transformation to the value function. This distortion technique
results in a probabilistic representation for the claim's value,
involving a measure under which the drift coefficient of the
non-traded asset is modified, but which is not necessarily a local
martingale measure. The dual approach yields a different
representation for the utility-based price, involving an optimization
(over the class of local martingale measures) of a functional
involving the claim payoff and an entropic penalty term. The
equivalence of the two representations is demonstrated by showing that
they yield the same hedging strategies for the derivative.