Abstract
We investigate a portfolio optimization problem under the threat of a market crash, where the interest rate of the bond is modeled as a Vasicek process, which is correlated with the stock price process. We adopt a non-probabilistic worst-case approach for the height and time of the market crash. On a given time horizon [0; T], we then maximize the investor’s expected utility of terminal wealth in the worst-case crash scenario. Our main result is an explicit characterization of the worst-case optimal portfolio strategy for the class of HARA (hyperbolic absolute risk aversion) utility functions.
Highlights
Since Merton [1] published his pioneering work on continuous time portfolio optimization, there has been a vast stream of literature on generalization methods, models and tasks in this area
As Korn and Wilmott [2] proposed, we model the market crash as an uncertain event (τ, l), where the [0, T ] ∪ {∞}-valued stopping time τ stands for the crash time and l ∈ [0, l∗ ] denotes the crash size. l∗ ∈ [0, 1) is the maximal crash size, which is assumed to be given
In the two sections, we provide an explicit solution of the worst-case optimization problem by applying the following three ideas, which have already been successfully applied in the case of constant interest rates
Summary
Since Merton [1] published his pioneering work on continuous time portfolio optimization, there has been a vast stream of literature on generalization methods, models and tasks in this area. There, as a non-standard feature, a market crash is modeled as an uncertain event, rather than a risky event, without any assumptions on the distributions of the crash height and crash time They introduced the notion of a worst-case optimal portfolio under the threat of such a crash. A new martingale approach that is based on interpreting the worst-case problem as a controller vs stopper game has been introduced by Seifried in [5], where the method is applied to a worst-case portfolio problem for rather general asset price dynamics. A related paper to our current one is [10], where a worst-case optimization model over an infinite time horizon with logarithmic utility was considered, but where the interest rate is stochastic only after the market crash.
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