# Optimal forward contract design for inventory: a value-of-waiting analysis

## TL;DR

This paper applies financial option techniques to optimize forward contract design in inventory management, determining optimal delivery timing and quantities using a stochastic commodity price model.

## Contribution

It introduces a simplified model that combines inventory management with option valuation, providing explicit optimal strategies and comparative statics analysis.

## Key findings

- Identifies optimal delivery time and amount for continuous-time inventory processes.
- Derives an optimal strike price for the expansion option based on expected revenue.
- Analyzes the impact of price drift and volatility on contract design through comparative statics.

## Abstract

A classical inventory problem is studied from the perspective of embedded options, reducing inventory-management to the design of optimal contracts for forward delivery of stock (commodity). Financial option techniques \`{a} la Black-Scholes are invoked to value the additional `option to expand stock'. A simplified approach which ignores distant time effects identifies an optimal `time to deliver' and an optimal `amount to deliver' for a production process run in continuous time modelled by a Cobb-Douglas revenue function. Commodity prices, quoted in initial value terms, are assumed to evolve as a geometric Brownian process with positive drift. Expected revenue maximization identifies an optimal `strike price' for the expansion option to be exercised, and uncovers the underlying martingale in a truncated (censored) commodity price. The paper establishes comparative statics of the censor in terms of drift and volatility, and uses asymptotic approximation for a tractable analysis of the optimal timing.