Esscher transform has been introduced in the context of insurance pricing, in Esscher (1936)*. Several applications have been proposed later on in insurance (see here for an example). Gerber and Shiu (1994, here) proved the interest of this transformation in financial derivative pricing to derive risk neutral probabilities. In standard models (Black-Scholes
option-pricing formula, the pure-jump option-pricing formula, and the binomial option-pricing formula), they prove - and the paper is extremely pedagogical - that standard changes of measure to obtain a risk neutral measure (so that prices are martingales) can be seen as an Esscher transform (see also the survey by Paul Embrechts).
Bühlmann (1980, here) proved that the Esscher transform can be obtained as the price in an equilibrium model. And wanted here to stress here this interpretation (this will be one part that I will discuss next week in Montpelier, see here). Note that the paper has been revised by Shaun Wang at the AFIR (Actuarial approach for Financial Risks) conference in Cancun in 2002 (here).
Consider agents with exponential utility functions facing risk . Then the equilibrium premium for some risk is
This can be related to some exponential tilting (as defined by Shaun Wang), i.e.
so that .
Remark if where is the systematic risk and the diversifiable risk, then
Esscher, Fredrik (1932). On the probability function in the collective theory of risk, Scandinavian Actuarial Journal 15, 175-195. If anyone has a pdf version of the original paper, I'd be glad to read it.