Optimal Purchasing Policy For Mean-Reverting Items in a Finite Horizon

TL;DR

This paper develops an optimal purchasing policy for items with mean-reverting prices over a finite horizon, using a dynamic programming approach to determine a time-dependent threshold for purchase decisions.

Contribution

It introduces a novel threshold-based policy for mean-reverting price models and provides an efficient algorithm to compute it, including explicit crossing time and overshoot equations.

Findings

Optimal policy is a time-varying threshold function.

Algorithm efficiently computes the threshold using dynamic programming.

Threshold characteristics depend on time, costs, and price process parameters.

Abstract

In this research we study a finite horizon optimal purchasing problem for items with a mean reverting price process. Under this model a fixed amount of identical items are bought under a given deadline, with the objective of minimizing the cost of their purchasing price and associated holding cost. We prove that the optimal policy for minimizing the expected cost is in the form of a time-variant threshold function that defines the price region in which a purchasing decision is optimal. We construct the threshold function with a simple algorithm that is based on a dynamic programming procedure that calculates the cost function. As part of this procedure we also introduce explicit equations for the crossing time probability and the overshoot expectation of the price process with respect to the threshold function. The characteristics and dynamics of the threshold function are analyzed with respect to time, holding cost, and different parameters of the price process, and yields meaningful practical insights, as well as theoretical insights.