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The Pricing of Short-Lived Options When Price Uncertainty Is Log-Symmetric Stable

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  • J. Huston McCulloch

Abstract

The well-known option pricing formula of Black and Scholes depends upon the assumption that price fluctuations are log-normal. However, this formula greatly underestimates the value of options with a low probability of being exercised if, as appears to be more nearly the case in most markets, price fluctuations are in fact symmetrics table or log-symmetric stable. This paper derives a general formula for the value of a put or call option in a general equilibrium, expected utility maximization context. This general formula is found to yield the Black-Scholes formula for a wide variety of underlying processes generating log-normal price uncertainty. It is then used to derive the value of a short-lived option for certain processes that generate log-symmetric stable price uncertainty. Our analysis is restricted to short-lived options for reasons of mathematical tractability. Nevertheless, the formula is useful for evaluating many types of risk.

Suggested Citation

  • J. Huston McCulloch, 1978. "The Pricing of Short-Lived Options When Price Uncertainty Is Log-Symmetric Stable," NBER Working Papers 0264, National Bureau of Economic Research, Inc.
    • Handle: RePEc:nbr:nberwo:0264
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    References listed on IDEAS

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    1. 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.
    2. J. Huston McCulloch, 1978. "Interest Rate Risk and Capital Adequacy For Traditional Banks and Financial Intermediaries," NBER Working Papers 0237, National Bureau of Economic Research, Inc.
    3. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    4. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    5. Cornell, W Bradford & Dietrich, J Kimball, 1978. "The Efficiency of the Market for Foreign Exchange under Floating Exchange Rates," The Review of Economics and Statistics, MIT Press, vol. 60(1), pages 111-120, February.
    6. Dusak, Katherine, 1973. "Futures Trading and Investor Returns: An Investigation of Commodity Market Risk Premiums," Journal of Political Economy, University of Chicago Press, vol. 81(6), pages 1387-1406, Nov.-Dec..
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    1. J. Huston McCulloch, 1978. "Interest Rate Risk and Capital Adequacy For Traditional Banks and Financial Intermediaries," NBER Working Papers 0237, National Bureau of Economic Research, Inc.
    2. Climent Hernández José Antonio & Venegas Martínez Francisco, 2013. "Valuación de opciones sobre subyacentes con rendimientos a-estables," Contaduría y Administración, Accounting and Management, vol. 58(4), pages 119-150, octubre-d.

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