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 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.
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References
Abramowitz, M.A, Stegan, I.A.: Handbook of Mathematical Functions. Dover, Mineola (1972)
Bensoussan, A., Crouhy, M., Proth, J.M.: Mathematical Theory of Production Planning. North-Holland, Amsterdam (1983)
Chang, J.S.K, Chang, C., Shi, M.: A market-based martingale valuation approach to optimum inventory control in a doubly stochastic jump-diffusion economy. J. Oper. Res. Soc. 66, 405–420 (2015)
Dixit, A.K., Pindyck, R.S.: Decision Making Under Uncertainty. Princeton University Press, Princeton (1994)
Eberly, J.C., Van Mieghem, J.A.: Multi-factor investment under uncertainty. J. Econ. Theory 75(8), 345–387 (1997)
Gietzmann, M.B., Ostaszewski, A.J.: Hedging the purchase of direct inputs in an inflationary environment. Manag. Account. Res. 10, 61–84 (1999)
Gietzmann, M.B., Ostaszewski, A.J.: Predicting firm value: the superiority of q-theory over residual income. Account. Bus. Res. 34(4), 379–382 (2004)
Hull, J.C.: Options, Futures, and Other Derivatives, 10th edn. Pearson Educational Limited, Harlow (2017)
Kendall, M.G., Stuart, A.: The Advanced Theory of Statistics, vol. 1. Griffin & Co, London (1963)
Kouvelis, P., Zhan, P., Qing D.: Integrated commodity management and financial hedging: a dynamic mean-variance analysis. Prod. Oper. Manag. 27(6), 1052–1073 (2018)
Musiela, M., Rutkowski, M.: Martingale Methods in Financial Modelling. Springer, Berlin (1997)
Ostaszewski, A.J.: Subdominant eigenvalue location and the robustness of dividend policy irrelevance. In: J. Brzdȩk, D. Popa, T.M. Rassias (eds.), Ulam Type Stability (this volume). https://doi.org/10.1007/978-3-030-28972-0, chapter 13
Patel, J.K., Read, C.R.: Handbook of the Normal Distribution. Marcel Dekker, New York (1982)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Scarf, H.: The optimality of (S, s) policies in the dynamic inventory problem. In: Arrow, Karli, Suppes (eds.), Mathematical Methods in the Social Sciences, 1959 First Stamford Symposium. University Press, Stanford (1960)
Acknowledgement
It is a pleasure to thank Alain Bensoussan for his very helpful advice and encouragement.
Postscript
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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Davies, R.O., Ostaszewski, A.J. (2019). Optimal Forward Contract Design for Inventory: A Value-of-Waiting Analysis. In: Brzdęk, J., Popa, D., Rassias, T. (eds) Ulam Type Stability . Springer, Cham. https://doi.org/10.1007/978-3-030-28972-0_4
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