Exponential utility indifference valuation in two Brownian settings with stochastic correlation

Abstract

We study the exponential utility indifference valuation of a contingent claim B in an incomplete market driven by two Brownian motions. The claim depends on a nontradable asset stochastically correlated with the traded asset available for hedging. We use martingale arguments to provide upper and lower bounds, in terms of bounds on the correlation, for the value V B of the exponential utility maximization problem with the claim B as random endowment. This yields an explicit formula for the indifference value b of B at any time, even with a fairly general stochastic correlation. Earlier results with constant correlation are recovered and extended. The reason why all this works is that, after a transformation to the minimal martingale measure, the value V B enjoys a monotonicity property in the correlation between tradable and nontradable assets.

A Multidimensional Exponential Utility Indifference Pricing Model with Applications to Counterparty Risk

“…The last assertion of the above proposition tells us that the higher the idiosyncratic volatility σ 2 P of the traded asset (or as a proportion of total volatility), the worse it is as a hedging instrument, and the lower the price one is willing to pay. This generalizes the monotonicity obtained in the one dimensional non-traded asset model (see, for example, Henderson [21] and Frei and Schweizer [16] in a non-Markovian model with stochastic correlation).…”

“…This trick results in an explicit formula for the exponential utility indifference price. Subsequent generalizations of the model from Tehranchi [45], Frei and Schweizer [16] and [17] showed that the exponential utility indifference value can still be written in a closed-form expression similar to that known for the Brownian setting, although the structure of the formula can be much less explicit. On the other hand, Davis [14] used the duality to derive an explicit formula for the optimal hedging strategy (see also Monoyios [39]), and Becherer [4] showed that the dual pricing formula exists even in a general semimartingale setting.…”

Utility Indifference Valuation for Non-smooth Payoffs with an Application to Power Derivatives

“…This is mainly motivated by the recent literature on structural models for electricity markets, which aim at describing electricity prices as a result of the interaction of some underlying structural factors that can be either exchanged on a financial markets (like fuels) or not (like demand and fuel capacities), and which are often supposed to have simple Gaussian dynamics. In our framework the payoff is supposed to be a function of both traded and nontraded assets, contrarily to most of the literature where the payoff depends only on the nontraded assets which are assumed to be correlated to the traded ones, so that one usually works directly with the correlation of the traded assets with the payoff to be hedged (see, for example, [He02], [Be06], [AID10], [FS08], [IRR12]). An exception is [SZ04], where the payoff considered depends on both types of assets in a bidimensional stochastic volatility framework where the payoff is assumed to be smooth and bounded.…”

Pseudo linear pricing rule for utility indifference valuation

“…The well known one dimensional non-traded assets model is an exception, and in a Markovian framework with a derivative written on a single non-traded asset, and partial hedging in a financial asset, Henderson and Hobson [23], Henderson [22], and Musiela and Zariphopoulou [33] use the Cole-Hopf transformation (or distortion power ) to linearize the nonlinear partial differential equations (PDEs for short) for the value function. Subsequent generalizations of the model from Tehranchi [35], Frei and Schweizer [20] [21] have shown the exponential utility indifference price can still be written in a closed-form expression similar to that known for the Brownian setting, although the structure of the formula can be much less explicit. On the other hand, Davis [13] uses the duality to derive an explicit formula for the optimal hedging strategy, and Becherer [5] shows that the dual pricing formula exists even in a general semimartingale setting.…”