The p-optimal martingale measure and its asymptotic relation with the minimal-entropy martingale measure
The p-optimal martingale measure and its asymptotic relation with the minimal-entropy martingale measure
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DOI：
10.2307/3318433
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发表时间：
1999-04
期刊：
影响因子：
1.5
通讯作者：
P. Grandits
中科院分区：
文献类型：
--
作者：
P. Grandits
近年来，寻找随机过程的鞅测度的问题已在数学金融领域得到应用，例如用于评估欧式看涨期权的著名布莱克-斯科尔斯公式可以被视为随机变量相对于贴现股票价格过程的（在本例中是唯一的）鞅测度的期望值。一般来说，随机过程没有唯一的鞅测度。因此，人们面临着选择合适的鞅测度的问题。非常流行的可能性是所谓的最小鞅测度，它是由 F611mer 和 Schweizer（1991）引入的，或者是方差最优测度（Schweizer 1995；Delbaen 和 Schachermayer 1996；Delbaen 等人 1997）。后者的特征在于，在该过程的所有带符号鞅测度中，相对于原始测度，最小化新测度的 Radon-Nikodym 导数的 L2 范数。前者在本地展示了此功能（更准确的描述请参见 F611mer 和 Schweizer (1991)）。另一种可能性是最小熵鞅测度。 Frittelli (1996) 已经证明，对于有界过程，始终存在唯一的鞅测度，它可以最小化原始测度和鞅测度之间的相对熵。此外，如果相对熵是有限的，则这两个测度是等价的。有关方差最优和最小熵度量的经济解释，请参阅 Delbaen 等人。 (1997) 并分别参见Frittelli (1996) 以及Platen 和Rebolledo (1995)。本文的目的是在有限范围内的离散时间内找到这两个概念之间的联系。事实证明，缺失的环节是由鞅测度给出的，我们称之为 p 最优，其特征是最小化 I' 范数，而不是
In recent years the problem of finding martingale measures for a stochastic process has found applications in the field of mathematical finance, e.g. the famous Black-Scholes formula for evaluating a European call option can be seen as the expectational value of a random variable with respect to the (in this case unique) martingale measure for the discounted stock price process. In general there is no unique martingale measure for a stochastic process. So one is confronted with the problem of choosing a proper martingale measure. Very popular possibilities are the so-called minimal martingale measure, which has been introduced by F611mer and Schweizer (1991), or the variance-optimal measure (Schweizer 1995; Delbaen and Schachermayer 1996; Delbaen et al. 1997). The latter is characterized by minimizing the L2 norm of the Radon-Nikodym derivative of the new measure with respect to the original measure among all signed martingale measures for the process. The former exhibits this feature locally (for a more exact description see F611mer and Schweizer (1991)). Another possibility is the minimal-entropy martingale measure. It has been shown by Frittelli (1996) that for a bounded process a unique martingale measure, which minimizes relative entropy between the original measure and the martingale measure, always exists. In addition, if the relative entropy is finite, the two measures are equivalent. For an economic interpretation of the variance-optimal and minimal-entropy measures see Delbaen et al. (1997) and see Frittelli (1996) and Platen and Rebolledo (1995) respectively. The aim of this paper is to find a connection between these two concepts in discrete time with a finite horizon. It turns out that the missing link is given by martingale measures, which we call p optimal and which are characterized by minimizing the I' norm instead of