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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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    1. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    2. 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.
    3. 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.
    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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