Real options with constant relative risk aversion

Real options problems have recently attracted much attention worldwide. One such problem is how to deal with claims on ‘non-traded’ assets. Often there is another traded asset which is correlated to the non-traded asset, and this traded asset is used as a proxy for hedging purposes. We introduce a second (non-traded) log Brownian asset into the well known Merton investment model with power-law utility. The investor has a claim on units of the non-traded asset and the question is how to price and hedge this random payoff. The presence of the second Brownian motion means that we are in the situation of incomplete markets. We propose an approximation to the solution for the ‘optimal’ reservation price and hedge which is accurate when the position is small in comparison to wealth. The resulting loss when a suboptimal proxy strategy is followed is shown to be approximately quadratic in 1−ρ.