# Abstracts

**Constant proportion strategies that deal with the lack of borrowing**

*Jurgen Vandenbroucke (University of Antwerp, Belgium)*

Thursday June 5, 11:30-12:00 | session 7.8 | Trading (Strategies) | room 1+2

This paper develops two alternative ways to deal with the lack of borrowing in constant proportion strategies linked to balanced funds. One variation focuses on the downside while the other focuses on the upside. The strategies are evaluated under various dynamics with respect to the mean, variance and correlation. The performance evaluation is based on stochastic dominance, partial moments and prospect theory such that the entire distribution is taken into account and asymmetries are accounted for. All performance criteria favor the alternative that focuses on the downside over the alternative that focuses on the upside. The alternatives are not stochastically dominated by the classic strategy. Performance measures based on partial moments favor the alternative that focuses on the downside. Prospect values favor the classic strategy.

**Optimal Discretization of Hedging Strategies with Market Trend**

*Jiatu Cai (University Paris Diderot, France)*

Thursday June 5, 12:00-12:30 | session 7.8 | Trading (Strategies) | room 1+2

We consider the hedging error due to discrete-time rebalancing of a given continuous hedging strategy, in the presence of a market trend, in the situation where the underlying asset price is not a martingale. A trader adjusts his position with regards to the benchmark. On one hand, the trader has to stay as close as possible to the continuous benchmark to replicate the option in question. On the other hand, he can also benefit from the market trend by deviating from the continuous benchmark. The problem is to choose the optimal rebalancing times under the mean-variance criterion. The problem admits in general no explicit solution and the existing numerical approaches are often computationally intensive. We propose in this paper an asymptotic approach in the spirit of [1] and [2], under which explicit (asymptotically) optimal strategies are available. We consider the process of suitably normalized hedging error. Within the proposed asymptotic framework we show the existence of a limit model for the hedging error. After [3], the problem can be reduced to a \emph{non-degenerate} Linear-Quadratic optimal control. Indeed, the choice of optimal rebalancing times is given by the hitting times of two barriers around the continuous strategy which are determined by the solution of the corresponding LQ problem. Under the Black-Scholes model, we obtain analytical expressions for the barriers and present several numerical results. A modified version of the Sharpe ratio, defined as the ratio of expected return and expected quadratic variation, is also considered. We give explicit expression for the (asymptotically) optimal strategies.

[1] Fukasawa M. (2011). Discretization error of stochastic integrals. The Annals of Applied Probability, 21(4), 1436-1465.

[2] Rosenbaum M. \& Tankov P. (2011). Asymptotically optimal discretization of hedging strategies with jumps. arXiv preprint arXiv:1108.5940.

[3] Zhou X. Y. \& Li D. (2000). Continuous-time mean-variance portfolio selection: A stochastic LQ framework. Applied Mathematics and Optimization, 42(1), 19-33.