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On the Stochastic Volatility in the Generalized Black-Scholes-Merton Model

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  • Roman V. Ivanov

    (Laboratory of Control under Incomplete Information, V.A. Trapeznikov Institute of Control Sciences of RAS, Profsoyuznaya 65, 117997 Moscow, Russia)

Abstract

This paper discusses the generalized Black-Scholes-Merton model, where the volatility coefficient, the drift coefficient of stocks, and the interest rate are time-dependent deterministic functions. Together with it, we make the assumption that the volatility, the drift, and the interest rate depend on a gamma or inverse-gamma random variable. This model includes the models of skew Student’s t- and variance-gamma-distributed stock log-returns. The price of the European forward-start call option is derived from the considered models in closed form. The obtained formulas are compared with the Black-Scholes formula through examples.

Suggested Citation

  • Roman V. Ivanov, 2023. "On the Stochastic Volatility in the Generalized Black-Scholes-Merton Model," Risks, MDPI, vol. 11(6), pages 1-23, June.
    • Handle: RePEc:gam:jrisks:v:11:y:2023:i:6:p:111-:d:1167116
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    Cited by:

    1. Rajeev Rajaram & Nathan Ritchey, 2023. "Simultaneous Exact Controllability of Mean and Variance of an Insurance Policy," Mathematics, MDPI, vol. 11(15), pages 1-16, July.

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