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An Option Pricing Model with Memory

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  • Flavia Sancier
  • Salah Mohammed

Abstract

We obtain option pricing formulas for stock price models in which the drift and volatility terms are functionals of a continuous history of the stock prices. That is, the stock dynamics follows a nonlinear stochastic functional differential equation. A model with full memory is obtained via approximation through a stock price model in which the continuous path dependence does not go up to the present: there is a memory gap. A strong solution is obtained by closing the gap. Fair option prices are obtained through an equivalent (local) martingale measure via Girsanov's Theorem and therefore are given in terms of a conditional expectation. The models maintain the completeness of the market and have no arbitrage opportunities.

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  • Flavia Sancier & Salah Mohammed, 2017. "An Option Pricing Model with Memory," Papers 1709.00468, arXiv.org.
    • Handle: RePEc:arx:papers:1709.00468
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    References listed on IDEAS

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    1. Baxter,Martin & Rennie,Andrew, 1996. "Financial Calculus," Cambridge Books, Cambridge University Press, number 9780521552899, January.
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    6. Cox, John C. & Ross, Stephen A., 1976. "The valuation of options for alternative stochastic processes," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 145-166.
    7. Kazmerchuk, Yuriy & Swishchuk, Anatoliy & Wu, Jianhong, 2007. "The pricing of options for securities markets with delayed response," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 75(3), pages 69-79.
    8. Scott, Louis O., 1987. "Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 22(4), pages 419-438, December.
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    Cited by:

    1. Li, Xiaoyue & Mao, Xuerong & Song, Guoting, 2024. "An explicit approximation for super-linear stochastic functional differential equations," Stochastic Processes and their Applications, Elsevier, vol. 169(C).

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