Edoardo Berton

In an arbitrage-free simple market, we demonstrate that for a class of state-dependent exponential utilities, there exists a unique prediction of the random risk aversion that ensures the consistency of optimal strategies across any time horizon. Our solution aligns with the theory of forward performances, with the added distinction of identifying, among the infinite possible solutions, the one for which the profile remains optimal at all times for the market-adjusted system of preferences adopted.

We consider a family of conditional nonlinear expectations defined on the space of bounded random variables and indexed by the class of all the sub-sigma-algebras of a given underlying sigma-algebra. We show that if this family satisfies a natural consistency property, then it collapses to a conditional certainty equivalent defined in terms of a state-dependent utility function. This result is obtained by embedding our problem in a decision theoretical framework and providing a new characterization of the Sure-Thing Principle. In particular we prove that this principle characterizes those preference relations which admit consistent backward conditional projections. We build our analysis on state-dependent preferences for a general state space as in Wakker and Zank (1999) and show that their numerical representation admits a continuous version of the state-dependent utility. In this way, we also answer positively to a conjecture posed in the aforementioned paper.

In this study, we propose a new formula for spread option pricing with the dependence of two assets described by a copula function. The advantage of the proposed method is that it requires only the numerical evaluation of a one-dimensional integral. Any univariate stock price process, admitting an affine characteristic function, can be used in our formula to get an efficient numerical procedure for computing spread option prices. In the numerical analysis we present a comparison with Monte Carlo simulation methods to assess the performance of our approach, assuming that the univariate stock price follows three widely applied models: Variance Gamma, Heston's Stochastic Volatility and Affine Heston Nandi GARCH(1,1) model.