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Pricing rule based on non-arbitrage arguments for random volatility and volatility smile

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  • Nikolai Dokuchaev

Abstract

We consider a generic market model with a single stock and with random volatility. We assume that there is a number of tradable options for that stock with different strike prices. The paper states the problem of finding a pricing rule that gives Black-Scholes price for at-money options and such that the market is arbitrage free for any number of tradable options, even if there are two Brownian motions only: one drives the stock price, the other drives the volatility process. This problem is reduced to solving a parabolic equation.

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  • Nikolai Dokuchaev, 2002. "Pricing rule based on non-arbitrage arguments for random volatility and volatility smile," Papers math/0205120, arXiv.org.
  • Handle: RePEc:arx:papers:math/0205120
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    File URL: http://arxiv.org/pdf/math/0205120
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    References listed on IDEAS

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    1. Dokuchaev, Nikolai & Yu Zhou, Xun, 2001. "Optimal investment strategies with bounded risks, general utilities, and goal achieving," Journal of Mathematical Economics, Elsevier, vol. 35(2), pages 289-309, April.
    2. Elyès Jouini, 2003. "Market imperfections , equilibrium and arbitrage," Post-Print halshs-00167131, HAL.
    3. Johnson, Herb & Shanno, David, 1987. "Option Pricing when the Variance Is Changing," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 22(02), pages 143-151, June.
    4. Hull, John C & White, Alan D, 1987. " The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    5. 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.
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