# Abstracts

**Optimal Investment with Transaction Costs and Stochastic Volatility**

*Maxim Bichuch (Worcester Polytechnic Institute, USA)*

Thursday June 5, 11:30-12:00 | session 7.1 | Transaction Costs | room AB

Two major financial market frictions are transaction costs and uncertain volatility, and we analyze their joint impact on the problem of portfolio optimization. When volatility is constant, the transaction costs optimal investment problem has a long history, especially in the use of asymptotic approximations when the cost is small. Under stochastic volatility, but with no transaction costs, the Merton problem under general utility functions can also be analyzed with asymptotic methods. Here, we look at the long-run growth rate problem when both frictions are present, using separation of time scales approximations. This leads to perturbation analysis of an eigenvalue problem. We find the first term in the asymptotic expansion in the time scale parameter, of the optimal long-term growth rate, and of the optimal strategy, for fixed small transaction costs.

**Rebalancing with Linear and Quadratic Trading Costs**

*Ren Liu (ETH Zürich, Switzerland)*

Thursday June 5, 12:00-12:30 | session 7.1 | Transaction Costs | room AB

We consider a market consisting of one safe and one risky asset with constant investment opportunities. With a nontrivial bid-ask spread and linear price impact, i.e. linear and quadratic trading costs, we derive optimal rebalancing policies for representative investors with constant relative risk aversion and a long horizon.

**Small-Cost Asymptotics for the Long-Term Growth Rate in the Heston Model**

*Yaroslav Melnyk (University of Kaiserslautern, Germany)*

Thursday June 5, 12:30-13:00 | session 7.1 | Transaction Costs | room AB

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.