# Stop-loss and Leverage in optimal Statistical Arbitrage with an application to Energy market

## TL;DR

This paper develops an optimal statistical arbitrage strategy incorporating stop-loss and leverage, using mean-reverting processes and applied to energy futures, with analytical formulas for strategy performance.

## Contribution

It introduces a novel method to analytically derive optimal leverage and entry/exit levels in high-frequency trading with stop-loss, based on mean-reverting models.

## Key findings

- Optimal strategies depend on probabilities and expected passage times.
- Analytical expressions for expected first-exit times are derived for Ornstein-Uhlenbeck processes.
- Long-run returns are expressed as functions of stop-loss levels.

## Abstract

In this paper we develop a statistical arbitrage trading strategy with two key elements in hi-frequency trading: stop-loss and leverage. We consider, as in Bertram (2009), a mean-reverting process for the security price with proportional transaction costs; we show how to introduce stop-loss and leverage in an optimal trading strategy. We focus on repeated strategies using a self-financing portfolio. For every given stop-loss level we derive analytically the optimal investment strategy consisting of optimal leverage and market entry/exit levels. First we show that the optimal strategy a' la Bertram depends on the probabilities to reach entry/exit levels, on expected First-Passage-Times and on expected First-Exit-Times from an interval. Then, when the underlying log-price follows an Ornstein-Uhlenbeck process, we deduce analytical expressions for expected First-Exit-Times and we derive the long-run return of the strategy as an elementary function of the stop-loss. Following industry practice of pairs trading we consider an example of pair in the energy futures' market, reporting in detail the analysis for a spread on Heating-Oil and Gas-Oil futures in one year sample of half-an-hour market prices.