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Black–Scholes–Merton In Random Time: A New Stochastic Volatility Model With Path Dependence

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  • DMITRY OSTROVSKY

    (RBS Greenwich Capital, 600 Steamboat Road, Greenwich, CT 06830, USA)

Abstract

A generalized Black–Scholes–Merton economy is introduced. The economy is driven by Brownian motion in random time that is taken to be continuous and independent of Brownian motion. European options are priced by the no-arbitrage principle as conditional averages of their classical values over the random time to maturity. The prices are path dependent in general unless the time derivative of the random time is Markovian. An explicit self-financing hedging strategy is shown to replicate all European options by dynamically trading in stock, money market, and digital calls on realized variance. The notion of the average price is introduced, and the average price of the call option is shown to be greater than the corresponding Black–Scholes price for all deep in- and out-of-the-money options under appropriate sufficient conditions. The model is implemented in limit lognormal random time. The significance of its multiscaling law is explained theoretically and verified numerically to be a determining factor of the term structure of implied volatility.

Suggested Citation

  • Dmitry Ostrovsky, 2007. "Black–Scholes–Merton In Random Time: A New Stochastic Volatility Model With Path Dependence," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 10(05), pages 847-872.
    • Handle: RePEc:wsi:ijtafx:v:10:y:2007:i:05:n:s0219024907004421
    DOI: 10.1142/S0219024907004421
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    References listed on IDEAS

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    1. Bouchaud,Jean-Philippe & Potters,Marc, 2003. "Theory of Financial Risk and Derivative Pricing," Cambridge Books, Cambridge University Press, number 9780521819169, October.
    2. Benoit Mandelbrot & Adlai Fisher & Laurent Calvet, 1997. "A Multifractal Model of Asset Returns," Cowles Foundation Discussion Papers 1164, Cowles Foundation for Research in Economics, Yale University.
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