Optimal Forward Contract Design for Inventory: A Value-of-Waiting Analysis

  • Roy O. Davies
  • Adam J. OstaszewskiEmail author


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 à 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 (inflationary) drift. Expected revenue maximization identifies an optimal ‘strike price’ for the expansion option to be exercised and reveals the underlying martingale in a truncated (censored) commodity price. The paper establishes comparative statics of the censor, using sensitivity analysis on the related censor functional equation; key here is that the censor, as a function of the drift and volatility of price, is the solution of a functional equation. Asymptotic approximation allows a tractable analysis of the optimal timing.


Value of waiting Censor functional equation Optimal forward contract Optimal exercise price Optimal timing Comparative statics Asymptotic approximation Martingale 

Mathematics Subject Classification (2010)

Primary 91B32 91B38 39B22; Secondary 91G80 49J55 49K40 



It is a pleasure to thank Alain Bensoussan for his very helpful advice and encouragement.


Harold Wilson (1916–1995, British prime minister 1964–1970 and 1974–1976) famously always emphasized the importance of keeping his options open.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematics and Actuarial ScienceUniversity of LeicesterLeicesterUK
  2. 2.Department of MathematicsLondon School of EconomicsLondonUK

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