# Optimal Trading with a Trailing Stop

## TL;DR

This paper develops an optimal trading strategy framework incorporating trailing stops within a linear diffusion model, providing analytic solutions and numerical methods for optimal buy and sell timing under path-dependent conditions.

## Contribution

It introduces a comprehensive optimal double stopping approach for trading with trailing stops, including both acquisition and liquidation strategies, under a general diffusion setting.

## Key findings

- Optimal liquidation strategy prior to trailing stop is derived.
- Using a sell limit order with trailing stop is proven optimal.
- Numerical methods for computing optimal trading regions are developed.

## Abstract

Trailing stop is a popular stop-loss trading strategy by which the investor will sell the asset once its price experiences a pre-specified percentage drawdown. In this paper, we study the problem of timing buy and then sell an asset subject to a trailing stop. Under a general linear diffusion framework, we study an optimal double stopping problem with a random path-dependent maturity. Specifically, we first derive the optimal liquidation strategy prior to a given trailing stop, and prove the optimality of using a sell limit order in conjunction with the trailing stop. Our analytic results for the liquidation problem is then used to solve for the optimal strategy to acquire the asset and simultaneously initiate the trailing stop. The method of solution also lends itself to an efficient numerical method for computing the the optimal acquisition and liquidation regions. For illustration, we implement an example and conduct a sensitivity analysis under the exponential Ornstein-Uhlenbeck model.