An example of indifference prices under exponential preferences

Abstract

The aim herein is to analyze utility-based prices and hedging strategies. The analysis is based on an explicitly solved example of a European claim written on a nontraded asset, in a model where risk preferences are exponential, and the traded and nontraded asset are diffusion processes with, respectively, lognormal and arbitrary dynamics. Our results show that a nonlinear pricing rule emerges with certainty equivalent characteristics, yielding the price as a nonlinear expectation of the derivative’s payoff under the appropriate pricing measure. The latter is a martingale measure that minimizes its relative to the historical measure entropy. Copyright Springer-Verlag Berlin/Heidelberg 2004Incomplete markets, indifference prices, nonlinear asset pricing,

Indifference valuation in incomplete binomial models

“…Indeed, for this class of utilities, certain additive scaling properties with respect to the wealth argument facilitate the solution of the underlying optimization problems and, in turn, the construction of exponential indifference prices. There is a plethora of results for continuous-time models, derived either using duality theory or PDE techniques for Markovian models (see, among others, [4], [13], [9], [17] and [23]). In some simple cases -specifically, when the nested model is complete -indifference prices can be constructed explicitly ( [18]).…”

Inf-convolution of risk measures and optimal risk transfer

“…The optimal structure of a contract depending on the non-tradable risk and its price are determined. Several authors, notably El Karoui and Rouge (2000), Becherer (2001), Davis (2001) and Musiela and Zariphopoulou (2004), have been interested in these new products. As is usual in finance, they focus on the pricing rule of the contracts.…”

Robust portfolio selection under exponential preferences

“…Let us recall that the methodology based on the Cole-Hopf transformations was developed by Zariphopoulou [20], [21], Musiela and Zariphopoulou [10], and Pham [13]. We should also mention that HJBI equations were successfully applied to robust optimal investment problems by Mataramvura and Øksendal [9], Øksendal and Sulem [12], [11], and Talay and Zhang [18], but to our knowledge this is the first time they are used in models with an untradable risk factor to obtain an explicit formula for a robust optimal strategy.…”

“…The rigorous result together with necessary assumptions on the market coefficients is given later. Calculations of the type presented below are commonly used in the existing literature (see Musiela and Zariphopoulou [10] or Benth and Karlsen [1]) and are not given here with all details. Notice that if V xx < 0 then the maximum over π in (4.1) is well defined.…”