Optimal investment and indifference pricing when risk aversion is not monotone: SAHARA utility functions

Abstract. We develop a new class of utility functions, SAHARA utility, with the dis-tinguishing feature that they implement the assumption that agents may become lessrisk-averse for very low values of wealth. This means that SAHARA utility can be usedto characterize risk gambling behavior of an economic agent in a financial crisis. Theclass contains the most frequently used exponential and power utility functions as limit-ing cases and its two parameters can be easily calibrated in terms of quantities with a cleareconomic meaning such as a target default probability and a target relative risk aversioncoefficient. We investigate the optimal investment problem under SAHARA utility andderive the optimal strategies in an explicit form using dual optimization methods. Wealso show how SAHARA utility functions can be used for indifference pricing in incom-plete markets. Throughout the paper, we compare SAHARA with exponential and powerutility functions to highlight their qualitative differences.Keywords: SAHARA utility, optimal investment problem, primal/dual approach, utilityindifference pricingJEL-Codes: G11, G13, G22, D52, C61