This paper considers exponential utility indifference pricing for a multidimensional non-traded assets model subject to inter-temporal default risk, and provides a semigroup approximation for the utility indifference price. The key tool is the splitting method, whose convergence is proved based on the Barles-Souganidis monotone scheme, and the convergence rate is derived based on Krylov's shaking the coefficients technique. We apply our methodology to study the counterparty risk of derivatives in incomplete markets.
The purpose of this article is to consider exponential utility indifference pricing in a multidimensional non-traded assets setting subject to intertemporal default risk, which is motivated by our study of counterparty risk of derivatives in incomplete markets. Our interest is in pricing and hedging derivatives written on assets which are not traded. The market is incomplete as the risks arising from having exposure to non-traded assets cannot be fully hedged. We take a utility indifference approach whereby the utility indifference price for the derivative is the cash amount the investor is willing to pay such that she is no worse off in expected utility terms than she would have been without the derivative.
There has been considerable research in the area of exponential utility indifference valuation, but despite the interest in this pricing and hedging approach, there have been relatively few explicit formulas derived. 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 [22], Henderson [20], and Musiela and Zariphopoulou [40] used the Cole-Hopf transformation (or distortion power ) to linearize the non-linear PDE for the value function. 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.
As soon as one of the assumptions made in the one dimensional non-traded assets model breaks down, explicit formulas are no longer available. For example, if the option payoff depends also on the traded asset, Sircar and Zariphopoulou [42] developed bounds and asymptotic expansions for the exponential utility indifference price. In an energy context, we may be interested in partially observed models and need filtering techniques to numerically compute expectations (see Carmona and Ludkovski [11] and Chapter 7 of [10]). If the utility function is not exponential, Henderson [20] and Kramkov and Sirbu [33] developed expansions in small quantity for the utility indifference price under power utility.
In this paper, we study exponential utility indifference valuation in a multidimensional setting subject to intertemporal default risk with the aim of developing a pricing methodology. The main economic motivation for us to develop the multidimensional framework is to consider the counterparty default risk of options traded in over-the-counter (OTC) markets, often called vulnerable options. The credit crisis has brought to the forefront the importance of counterparty default risk as numerous high profile defaults lead to counterparty losses. In response, there have been many recent studies (see, for example, Bielecki et al [5] and Brigo et al [9]) addressing in particular the counterparty risk of credit default swaps (CDS). In contrast, there is relatively little recent work on counterparty risk for other derivatives, despite OTC options being a sizable fraction of the OTC derivatives market. 1 The option holder faces both price risk arising from the fluctuation of the assets underlying her option and counterparty default risk that the option writer does not honor her obligations. Default occurs either when the assets of the counterparty are below its liabilities at maturity (the structural approach) or when an exogenous random event occurs (the reduced-form approach), so intertemporal default is considered. In our setting, the assets of the counterparty and the assets underlying the option may be non-traded and thus a multidimensional non-traded assets model naturally arises.
Our use of the utility indifference approach is motivated by its recent use in credit risk modeling where the concern is the default of the reference name rather than the default of the counterparty. Utility based pricing has also been utilized by Bielecki and Jeanblanc [6], Sircar and Zariphopolou [43] and recently Jiao et al [27] [28] in an intensity based setting. Several authors have applied it in modeling of defaultable bonds where the problem remains one dimensional, see in particular Leung et al [35], Jaimungal and Sigloch [25], and Liang and Jiang [36]. In contrast, options subject to counterparty risk are a natural situation where two or more dimensions arise.
Our first contribution is the derivation of a reaction-diffusion partial differential equation (PDE) in Theorem 2.3 to characterize the utility
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