Journal of Risk
ISSN:
1465-1211 (print)
1755-2842 (online)
Editor-in-chief: Farid AitSahlia
Need to know
- We show how existing results for optimal trade execution with linear temporary price impact/exponential resilience or proportional transaction costs can be adapted for when the drift of the asset is unknown, and not knowing the drift can lead to markedly different trading behaviour for agents.
- We use a well known filtering result to re-write the price process with the true drift replaced by its conditional expectation so the remaining term is a Brownian motion with respect to the price process filtration.
- We observe that an arithmetic Brownian motion with unknown (constant) drift is the continuation of a Brownian bridge process under the price process filtration.
- We warn against the common practice of using an OU process for the drift as it is almost impossible to estimate its parameters with low sample variance in practice.
Abstract
We show how existing results for optimal trading strategies with linear temporary price impact, exponential resilience or proportional transaction costs can be easily adapted for the more realistic situation in which the drift of the asset is unknown and we need to project to the observable filtration generated by the asset price process using results from nonlinear filtering theory. In particular, we observe that an arithmetic Brownian motion P with unknown (constant) drift μ is the continuation of a generalized bridge process under ƑP, with the true drift replaced with its unbiased estimate over a fixed time window.
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