Mean-Variance Hedging for General Claims
Mean-Variance Hedging for General Claims
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DOI：
10.1214/aoap/1177005776
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发表时间：
1992-02
影响因子：
1.8
通讯作者：
M. Schweizer
中科院分区：
文献类型：
--
作者：
M. Schweizer
0。导言。本文解决了一般未定权益具有均值-方差目标的连续时间套期保值问题。Duffie和Richardson(1991)处理了这个问题的一个特例，并为这项工作提供了动力。有两种资产的价格都是由具有随时间变化的随机系数的指数布朗运动来建模的。资产之间的回报率是相关的。在一个固定的时间，套期保值者面临随机损失，这可能完全取决于两种资产价格的整个演变。然而，为了对冲这一风险，只有一种资产可用。这意味着市场是不完整的，或有债权不能通过交易来复制。套期保值者的目标是最小化他的总期望二次成本，或者等价地最大化他对二次效用函数的终端财富的期望效用。第一节给出了精确的表述，第三节给出了解决方案。我们注意到，同样的论点也适用于任何N个有n个套期保值资产的驱动资产，其中1<；n<；N。我们对这个问题的处理遵循Duffie和Richardson(1991)的方法：我们证明了与正交投影的正规方程相关的内积是由一个具有显式解的常微分方程组定义的。这是通过为审议中的或有索赔选择一个适当的跟踪程序来实现的。与上述文章的本质区别在于两点：我们能够解决一般未定权益的套期保值问题，而不必从离散时间推理来猜测解。事实上，我们的方法表明，最优跟踪过程的自然选择是由与给定的未定权益相关的内在价值过程提供的。这个过程是根据用于套期保值的该资产价格的最小等价鞅度量来定义的。这两个概念在第2节中都有更详细的解释。我们在第4节中用一类例子来结束论文，其中显式
0. Introduction. In this paper, we solve the continuous-time hedging problem with a mean-variance objective for general contingent claims. A special case of this problem was treated by Duffie and Richardson (1991) and provided the motivation for this work. There are two assets whose prices are both modelled by exponential Brownian motions with time-dependent random coefficients. The rates of return between assets are correlated. At a fixed time, the hedger faces a random loss which may depend in full generality on the entire evolution of both asset prices. For the purpose of hedging against this risk, however, only one asset is available. This implies that markets are incomplete and contingent claims cannot be replicated by trading. The goal of the hedger is to minimize his total expected quadratic costs, or equivalently to maximize his expected utility from terminal wealth for a quadratic utility function. A precise statement is given in Section 1 and the solution is presented in Section 3. We remark that the same arguments would also work for any number N of driving assets with n hedging assets, where 1 < n < N. Our approach to this problem follows the method of Duffie and Richardson (1991): We show that the inner product associated with the normal equations for orthogonal projection is defined by an ordinary differential equation in time with an explicit solution. This is done by choosing a suitable tracking process for the contingent claim under consideration. The essential difference from the above paper lies in two points: We are able to solve the hedging problem for a general contingent claim and we do not have to conjecture the solution from discrete-time reasoning. In fact, our approach shows that the natural choice for the optimal tracking process is provided by the intrinsic value process associated to the given contingent claim. This process is defined in terms of the minimal equivalent martingale measure for that asset price which is used for hedging. Both of these concepts are explained in more detail in Section 2. We conclude the paper in Section 4 with a class of examples where explicit