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 less
risk-averse for very low values of wealth. This means that SAHARA utility can be used
to characterize risk gambling behavior of an economic agent in a financial crisis. The
class 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 clear
economic meaning such as a target default probability and a target relative risk aversion
coefficient. We investigate the optimal investment problem under SAHARA utility and
derive the optimal strategies in an explicit form using dual optimization methods. We
also show how SAHARA utility functions can be used for indifference pricing in incom-
plete markets. Throughout the paper, we compare SAHARA with exponential and power
utility functions to highlight their qualitative differences.
Keywords: SAHARA utility, optimal investment problem, primal/dual approach, utility
indifference pricing
JEL-Codes: G11, G13, G22, D52, C61