Abstracts

In this paper we conduct an asymptotic analysis of the long-term growth rate (LTG rate) in the Heston stochastic volatility model under Morton-Pliska transaction costs (proportional to wealth) with respect to the cost parameter $\epsilon$. Using a dynamic programming approach we determine the leading order in the expansions of the value function ($\epsilon$) and of the LTG rate ($\epsilon^{1/2}$) with respect to the small cost parameter, compute the leading-order coefficients, define a trading strategy which maximizes the LTG rate at the leading order and prove a corresponding rigorous verification result.
We show that the LTG rate and the value function in the problem under consideration are characterized by a system of quasi-variational inequalities (QVIs) in the sense that a solution to the QVIs defines the no-trading region and optimal decisions. The QVIs corresponding to the problem under consideration cannot be solved in closed form. However, by introducing the concept of local growth rates and proposing an educated guess for the expansions of the LTG rate and the value function we are able to obtain a system QVIs for the leading-order coefficients. This system of leading-order QVIs is of lower dimension and admits a closed-form solution that provides a natural candidate for an optimal strategy at the leading order.
Our main result shows rigorously that this candidate is indeed optimal at the leading order: First, we establish an asymptotic upper bound for the LTG rate. Second, we prove an associated lower bound and thus verify optimality of our candidate strategy at the leading order. At the same time this verification result implies that the guesses for the expansions of the LTG rate and the value function are correct. Finally we provide numerical illustrations of asymptotically optimal trading strategies.