Maximizing The Probability Of A Perfect Hedge Using An Imperfectly Correlated Instrument

Let Xϕ denote the trading wealth generated using a strategy ϕ, and let CT be a contingent claim which is not spanned by the traded assets. Consider the problem of finding the strategy which maximizes the probability of terminal wealth meeting or exceeding the claim value at some fixed time horizon, i.e., of finding $\sup_{\phi} {\mathbb P}^x (X^{\phi}_T \geq C_T)$. This problem is sometimes referred to as the quantile hedging problem.We consider the quantile hedging problem when the traded asset and the contingent claim are correlated geometric Brownian motions. This fits with several important examples. One of the benefits of working with such a concrete model is that although it is incomplete we can still do calculations. In particular, we can consider some detailed issues such as the impact of the timing at which information about CT is revealed.