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 VB 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 VB enjoys a monotonicity property in the correlation between tradable and nontradable assets.

Convergence results for the indifference value based on the stability of BSDEs

“…This has already been pointed out in Section 5 of Frei and Schweizer [4] by saying that V H t is monotonic in |ρ| only when the measureP from (4.2), which depends via W on ρ, is kept fixed. Proposition 3 of Frei and Schweizer [4] gives a result analogous to parts 2) and 3) of Proposition 4.3 when |q| and |ρ| can be separated by a constant. Proposition 4.3 shows that this additional assumption is superfluous and generalises Proposition 3 of [4].…”

“…We refer to Sections 4.1 and 4.2 of Frei and Schweizer [4] for an overview of the related literature and a thorough explanation of the standing assumption (4.5), which corresponds to case (I) in [4]. For case (II) in [4], results analogous to those in Section 4.2 can be derived if ρ is predictable for the filtration generated byŶ . The standing assumption (4.5) allows us to give a BSDE characterisation for V H in the Y-filtration.…”

“…However, since ρ still appears in (4.9), we cannot simply say that V H t is monotonic in |ρ|. This has already been pointed out in Section 5 of Frei and Schweizer [4] by saying that V H t is monotonic in |ρ| only when the measureP from (4.2), which depends via W on ρ, is kept fixed. Proposition 3 of Frei and Schweizer [4] gives a result analogous to parts 2) and 3) of Proposition 4.3 when |q| and |ρ| can be separated by a constant.…”

“…While the methods in Frei and Schweizer [4] and Frei et al [5] give bounds for V H t , we approximate in the subsequent sections V H t by approaching ρ by piecewise constant processes. Definition 4.2.…”

“…However, the situation is different for general ρ. With stochastic and/or time-dependent ρ, one knows for h t only bounds and a structural formula where one parameter is not explicitly determined; see Frei and Schweizer [4]. In view of this lack of explicit results for h t , the goal of this paper is to give an explicit sequence that converges to h t .…”