# Random walk model from the point of view of algorithmic trading

## TL;DR

This paper explores the random walk model in algorithmic trading, providing an exact solution for static order execution costs and analyzing the implications of price randomness on order strategies.

## Contribution

It offers a novel exact solution for static order execution costs under the random walk assumption and examines the impact on optimal limit order placement.

## Key findings

- All limit levels have equal execution costs in a random walk market.
- Derived estimations for limit order risk and execution probability.
- No optimal limit level exists for order execution in this model.

## Abstract

Despite the fact that an intraday market price distribution is not normal, the random walk model of price behaviour is as important for the understanding of basic principles of the market as the pendulum model is a starting point of many fundamental theories in physics. This model is a good zero order approximation for liquid fast moving markets where the queue position is less important than the price action. In this paper we present an exact solution for the cost of the static passive slice execution. It is shown, that if a price has a random walk behaviour, there is no optimal limit level for an order execution: all levels have the same execution cost as an immediate aggressive execution at the beginning of the slice. Additionally the estimations for the risk of a limit order as well as the probability of a limit order execution as functions of the slice time and standard deviation of the price are derived.