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Martingale methods for pricing inventory penalties under continuous replenishment and compound renewal demands

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  • Junmin Shi

    ()

  • Michael Katehakis

    ()

  • Benjamin Melamed

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Abstract

This paper addresses the problem of inventory penalty pricing under the risk-neutral valuation principle. The underlying production-inventory system has a constant replenishment rate and a compound renewal demand stream (i.e., iid demand interarrival times are independent of iid demand sizes), and is subject to underage and overage penalties. Our pricing approach treats the penalties as a series of perpetual American options, and constructs auxiliary martingale processes in term of the inventory process. We provide a necessary and sufficient martingale condition for general compound renewal demands. Explicit expressions of penalty functions for underage and overage are obtained for the case where demand arrivals follow a Poisson process. Copyright Springer Science+Business Media, LLC 2013

Suggested Citation

  • Junmin Shi & Michael Katehakis & Benjamin Melamed, 2013. "Martingale methods for pricing inventory penalties under continuous replenishment and compound renewal demands," Annals of Operations Research, Springer, vol. 208(1), pages 593-612, September.
  • Handle: RePEc:spr:annopr:v:208:y:2013:i:1:p:593-612:10.1007/s10479-012-1130-5
    DOI: 10.1007/s10479-012-1130-5
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    References listed on IDEAS

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    1. Hans U. Gerber & Hlias S. W. Shiu, 1996. "Martingale Approach To Pricing Perpetual American Options On Two Stocks," Mathematical Finance, Wiley Blackwell, vol. 6(3), pages 303-322, July.
    2. Ernst, Ricardo & Powell, Stephen G., 1995. "Optimal inventory policies under service-sensitive demand," European Journal of Operational Research, Elsevier, vol. 87(2), pages 316-327, December.
    3. John D. Stowe & Tie Su, 1997. "A Contingent-Claims Approach to the Inventory-Stocking Decision," Financial Management, Financial Management Association, vol. 26(4), Winter.
    4. Relph, Geoff & Barrar, Peter, 2003. "Overage inventory--how does it occur and why is it important?," International Journal of Production Economics, Elsevier, vol. 81(1), pages 163-171, January.
    5. B. L. Schwartz, 1970. "Optimal Inventory Policies in Perturbed Demand Models," Management Science, INFORMS, vol. 16(8), pages 509-518, April.
    6. Gerber, Hans U. & Shiu, Elias S.W., 1994. "Martingale Approach to Pricing Perpetual American Options," ASTIN Bulletin, Cambridge University Press, vol. 24(2), pages 195-220, November.
    7. Benjamin L. Schwartz, 1966. "A New Approach to Stockout Penalties," Management Science, INFORMS, vol. 12(12), pages 538-544, August.
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    Cited by:

    1. repec:spr:annopr:v:261:y:2018:i:1:d:10.1007_s10479-017-2563-7 is not listed on IDEAS
    2. repec:eee:jomega:v:83:y:2019:i:c:p:181-198 is not listed on IDEAS
    3. Yonit Barron, 2016. "Performance analysis of a reflected fluid production/inventory model," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 83(1), pages 1-31, February.
    4. repec:eee:jomega:v:79:y:2018:i:c:p:133-145 is not listed on IDEAS
    5. M. Güler & Taner Bilgiç & Refik Güllü, 2015. "Joint pricing and inventory control for additive demand models with reference effects," Annals of Operations Research, Springer, vol. 226(1), pages 255-276, March.
    6. repec:spr:annopr:v:241:y:2016:i:1:d:10.1007_s10479-014-1679-2 is not listed on IDEAS
    7. repec:spr:annopr:v:271:y:2018:i:2:d:10.1007_s10479-018-2764-8 is not listed on IDEAS
    8. Yonit Barron, 2016. "Performance analysis of a reflected fluid production/inventory model," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 83(1), pages 1-31, February.
    9. Taras Bodnar & Nestor Parolya & Wolfgang Schmid, 2015. "A closed-form solution of the multi-period portfolio choice problem for a quadratic utility function," Annals of Operations Research, Springer, vol. 229(1), pages 121-158, June.
    10. Arda Yenipazarli, 2015. "A road map to new product success: warranty, advertisement and price," Annals of Operations Research, Springer, vol. 226(1), pages 669-694, March.
    11. Vorotnikova, Ekaterina, 2016. "Optimal Storage Capacity Allocation in Grain Merchandizing," 2016 Annual Meeting, February 6-9, 2016, San Antonio, Texas 230128, Southern Agricultural Economics Association.

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