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First-order calculus and option pricing

Author

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  • Peter Carr

    (New York University, Courant Institute, 251 Mercer Street, New York, NY 10012, USA)

Abstract

The modern theory of option pricing rests on It? calculus, which is a second-order calculus based on the quadratic variation of a stochastic process. One can instead develop a first-order stochastic calculus, which is based on the running minimum of a stochastic process, rather than its quadratic variation. We focus here on the analog of geometric Brownian motion (GBM) in this alternative stochastic calculus. The resulting stochastic process is a positive continuous martingale whose laws are easy to calculate. We show that this analog behaves locally like a GBM whenever its running minimum decreases, but behaves locally like an arithmetic Brownian motion otherwise. We provide closed form valuation formulas for vanilla and barrier options written on this process. We also develop a reflection principle for the process and use it to show how a barrier option on this process can be hedged by a static postion in vanilla options.

Suggested Citation

  • Peter Carr, 2014. "First-order calculus and option pricing," Journal of Financial Engineering (JFE), World Scientific Publishing Co. Pte. Ltd., vol. 1(01), pages 1-19.
    • Handle: RePEc:wsi:jfexxx:v:01:y:2014:i:01:n:s2345768614500093
    DOI: 10.1142/S2345768614500093
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    Cited by:

    1. Lingjiong Zhu, 2015. "Short maturity options for Azéma–Yor martingales," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 2(04), pages 1-32, December.
    2. Avram, Florin & Vu, Nhat Linh & Zhou, Xiaowen, 2017. "On taxed spectrally negative Lévy processes with draw-down stopping," Insurance: Mathematics and Economics, Elsevier, vol. 76(C), pages 69-74.
    3. Florin Avram & Danijel Grahovac & Ceren Vardar-Acar, 2019. "The W , Z / ν , δ Paradigm for the First Passage of Strong Markov Processes without Positive Jumps," Risks, MDPI, vol. 7(1), pages 1-15, February.
    4. Wang, Wenyuan & Chen, Ping & Li, Shuanming, 2020. "Generalized expected discounted penalty function at general drawdown for Lévy risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 91(C), pages 12-25.

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