The performance of optimal strategies for hedging a claim on a nontraded asset is analyzed. The claim is valued and hedged in a utility m a ximization framework, using exponential utility. A traded asset, correlated with that underlying the claim, is used for hedging, with the correlation typically close to 1. Using a distortion method 30, 31] we derive a nonlinear expectation representation for the claim's ask price and a formula for the optimal hedging strategy. W e generate a perturbation expansion for the price and hedging strategy in powers of 2 = 1 ; 2. The terms in the price expansion are found to be proportional to the central moments of the claim payo under a measure equivalent t o t h e p h ysical measure. The resulting fast computation capability is used to carry out a simulation based test of the optimal hedging program, computing the terminal hedging error over many asset price paths. These errors are compared with those from a naive strategy which uses the traded asset as a proxy for the non-traded one. The distribution of the hedging error acts as a suitable metric to analyze hedging performance. We nd that the the optimal policy improves hedging performance, in that the hedging error distribution is more sharply peaked around a non-negative pro t. The frequency of pro ts over losses is increased, and this is measured by the median of the distribution, which i s a l w ays increased by the optimal strategies.

We explore the impact of drift parameter uncertainty in a basis risk model, an incomplete market in which a claim on a nontraded asset is optimally hedged using a correlated traded stock. Using analytic expansions for indifference prices and hedging strategies, we develop an efficient procedure to generate terminal hedging error distributions when the hedger has erroneous estimates of the drift parameters. These show that the effect of parameter uncertainty is occasionally benign, but often very destructive. In light of this, we develop a filtering approach in which the hedger updates her parameter estimates from observations of the asset prices, and we find an analytic solution to the hedger's combined filtering and control problem in the case that the drift of the traded asset is known with certainty.

Several stylized theoretical models of futures basis behavior under nonzero transactions costs predict nonlinear mean reversion of the futures basis towards its equilibrium value. Nonlinearly mean-reverting models are employed to characterize the basis of the S&P 500 and the FTSE 100 indices over the post-1987 crash period, capturing empirically these theoretical predictions and examining the view that the degree of mean reversion in the basis is a function of the size of the deviation from equilibrium. The estimated half lives of basis shocks, obtained using Monte Carlo integration methods, suggest that for smaller shocks to the basis level the basis displays substantial persistence, while for larger shocks the basis exhibits highly nonlinear mean reversion towards its equilibrium value.

An e cient algorithm is developed to price European options in the presence of proportional transaction costs, using the optimal portfolio framework of Davis (in: Dempster, M.A.H., Pliska, S.R. (Eds.), Mathematics of Derivative Securities. Cambridge University Press, Cambridge, UK). A fair option price is determined by requiring that an inÿnitesimal diversion of funds into the purchase or sale of options has a neutral e ect on achievable utility. This results in a general option pricing formula, in which option prices are computed from the solution of the investor's basic portfolio selection problem, without the need to solve a more complex optimisation problem involving the insertion of the option payo into the terminal value function. Option prices are computed numerically using a Markov chain approximation to the continuous time singular stochastic optimal control problem, for the case of exponential utility. Comparisons with approximately replicating strategies are made. The method results in a uniquely speciÿed option price for every initial holding of stock, and the price lies within bounds which are tight even as transaction costs become large. A general deÿnition of an option hedging strategy for a utility maximising investor is developed. This involves calculating the perturbation to the optimal portfolio strategy when an option trade is executed. ?