This paper is concerned with the study of insurance related derivatives on financial markets that are based on non-tradable underlyings, but are correlated with tradable assets. We calculate exponential utility-based indifference prices, and corresponding derivative hedges. We use the fact that they can be represented in terms of solutions of forward-backward stochastic differential equations (FBSDE) with quadratic growth generators. We derive the Markov property of such FBSDE and generalize results on the differentiability relative to the initial value of their forward components. In this case the optimal hedge can be represented by the price gradient multiplied with the correlation coefficient. This way we obtain a generalization of the classical 'delta hedge' in complete markets.
Deep Dive into Pricing and hedging of derivatives based on non-tradable underlyings.
This paper is concerned with the study of insurance related derivatives on financial markets that are based on non-tradable underlyings, but are correlated with tradable assets. We calculate exponential utility-based indifference prices, and corresponding derivative hedges. We use the fact that they can be represented in terms of solutions of forward-backward stochastic differential equations (FBSDE) with quadratic growth generators. We derive the Markov property of such FBSDE and generalize results on the differentiability relative to the initial value of their forward components. In this case the optimal hedge can be represented by the price gradient multiplied with the correlation coefficient. This way we obtain a generalization of the classical ‘delta hedge’ in complete markets.
In recent years more and more financial instruments have been created which are not derived from exchange traded securities. For instance in 1999 the Chicago Mercantile Exchange introduced weather futures contracts, the payoffs of which are based on average temperatures at specified locations. Another example of derivatives with non-tradable underlyings are catastrophe futures based on an insurance loss index regulated by an independent agency or simply derivatives based on equity indices such as S&P or DAX.
Financial or insurance derivatives of this type are impossible to perfectly hedge, since it is impossible to trade the underlying variable that carries independent uncertainty. To circumvent this problem, in practice one looks for a tradable asset that is correlated to the non-tradable underlying of the derivative. Even though investing in the correlated asset cannot provide a total hedge of the derivative, and a non-hedgeable basis risk remains, it is better than not hedging at all.
In the following we will investigate utility-based pricing principles for derivatives based on non-tradable underlyings. Moreover we will show how the derivatives can be partially hedged by investing in correlated assets. We present explicit hedging strategies that optimize the expected utility of a portfolio of such derivatives. To this end we will establish some structure and smoothness properties of indifference prices such as the Markov property and differentiability with respect to the underlyings. Once these properties are established, we can explicitly describe the optimal hedging strategies in terms of the price gradient and correlation coefficients. This way we obtain a generalization of the classical delta hedge of the Black-Scholes model.
The hedging of claims based on non-tradable underlyings has already been studied by many authors, see for example [HH02], [Hen02], [MZ04], [Dav06], [Mon04], [AIP07]. As a common feature of all these papers, optimal hedging strategies are derived with standard stochastic control techniques. The essential components of this analytical approach consist in a formulation of the optimization problem in terms of HJB partial differential equations, and the use of a verification theorem and uniqueness result in order to obtain a representation of the indifference price and the optimal control strategy. We instead employ an approach with a stochastic focus. It starts with the well-known observation that the maximal expected exponential utility may be computed by appealing to the martingale optimality principle which leads to a description of price and optimal hedging strategy in terms of a forward-backward stochastic differential equation (FBSDE) with a nonlinearity of quadratic type (see [REK00], [HIM05]). This immediately implies that the utility indifference price resp. hedge is equal to the difference of initial states resp. control processes of two FBSDE with a quadratic nonlinearity in the generator. The forward component is given by a Markov process describing the non-tradable underlying. The main mathematical contribution of this paper is that it provides simple sufficient conditions for general FBSDE with quadratic nonlinearity to satisfy a Markov property, and -for the BSDE component -to be differentiable with respect to the initial condition of the forward equation. The techniques for proving differentiability of BSDE with quadratic nonlinearity have been developed independently in [BC07] and [AIDR07]. Unfortunately, the setup of both papers is not general enough to cover the BSDE needed to calculate exponential indifference prices. Therefore, a slight generalization of these differentiability results is given in the last section of this paper.
As a consequence of the explicit description of indifference prices and hedges in terms of the solution processes of the FBSDE, and in view of the smoothness results mentioned, it is straightforward to describe optimal hedging strategies in terms of the indifference price gradient and the correlation coefficients explicitly. An economics related contribution of the paper is that the framework presented allows to refine the results obtained for example in [MZ04], [Dav06]. Firstly, no longer we need to impose any restrictions on the coefficients of the diffusion modeling the tradable asset price. More importantly, the BSDE techniques allow to deal with multidimensional underlyings and traded assets. In the approach based on the HJB equation, a solution of the PDE is obtained by using an exponential Hopf-Cole transformation that in general seems to require that there exists only one traded asset. In practice many derivatives are based on more than one underlying, such as spread options or basket options. In order to illustrate how to hedge with more than one asset, we will study in more detail so-called crack spreads, which are written for instance on the difference of crude oil futures and kerosene prices (see Example 1.2 and 4.9).
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