Abstract
In the framework of an incomplete financial market where the stock price dynamics are modeled by a continuous semimartingale (not necessarily Markovian), an explicit second-order expansion formula for the power investor’s value function—seen as a function of the underlying market price of risk process—is provided. This allows us to provide first-order approximations of the optimal primal and dual controls. Two specific calibrated numerical examples illustrating the accuracy of the method are also given.
Highlights
In an incomplete financial setting with noise governed by a continuous martingale and in which the investor’s preferences are modeled by a negative power utility function, we provide a second-order Taylor expansion of the investor’s value function with respect to perturbations of the underlying market price of risk process
Findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF)
Since we focus on model expansion in utility maximization in this paper, we point the reader to some of the most recent papers, namely [AMKS15], and [MKK15], and the references therein, for further information
Summary
In an incomplete financial setting with noise governed by a continuous martingale and in which the investor’s preferences are modeled by a negative power utility function, we provide a second-order Taylor expansion of the investor’s value function with respect to perturbations of the underlying market price of risk process. In the present paper we take the stability analysis one step further and provide a first-order Taylor expansion in an infinite-dimensional space of the market price of risk processes. The arguments in these papers rely on convexity and concavity properties in the expansion parameter (wealth and number of unspanned claims) This is not the case in the present paper; when seen as a function of the underlying market price of risk process, the investor’s value function is neither convex nor concave and a more delicate, local, analysis needs to be performed. Our approximation technique turns out to be applicable and our error bounds are quite tight in the relevant parameter ranges
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