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Should An American Option Be Exercised Earlier Or Later If Volatility Is Not Assumed To Be A Constant?

Author

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  • SONG-PING ZHU

    (School of Mathematics and Applied Statistics, University of Wollongong NSW 2522, Australia)

  • WEN-TING CHEN

    (School of Mathematics and Applied Statistics, University of Wollongong NSW 2522, Australia)

Abstract

In this paper, we present a correction to Merton (1973)'s well-known classical case of pricing perpetual American put options by considering the same pricing problem under a stochastic volatility model with the assumption that the volatility is slowly varying. Two analytic formulae for the option price and the optimal exercise price of a perpetual American put option are derived, respectively. Upon comparing the results obtained from our analytic approximations with those calculated by a spectral collocation method, it is shown that our current approximation formulae provide fast and reasonably accurate numerical values of both option price and the optimal exercise price of a perpetual American put option, within the validity of the assumption we have made for the asymptotic expansion. We shall also show that the range of applicability of our formulae is remarkably wider than it was initially aimed for, after the original assumption on the order of the "volatility of volatility" being somewhat relaxed. Based on the newly-derived formulae, the quantitative effect of the stochastic volatility on the optimal exercise strategy of a perpetual American put option has also been discussed. A most noticeable and interesting result is that there is a special cut-off value for the spot variance, below which a perpetual American put option priced under the Heston model should be held longer than the case of the same option priced under the traditional Black-Scholes model, when the price of the underlying is falling.

Suggested Citation

  • Song-Ping Zhu & Wen-Ting Chen, 2011. "Should An American Option Be Exercised Earlier Or Later If Volatility Is Not Assumed To Be A Constant?," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 14(08), pages 1279-1297.
    • Handle: RePEc:wsi:ijtafx:v:14:y:2011:i:08:n:s0219024911006851
    DOI: 10.1142/S0219024911006851
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    References listed on IDEAS

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    1. Jean-Pierre Fouque & Tracey Andrew Tullie, 2002. "Variance reduction for Monte Carlo simulation in a stochastic volatility environment," Quantitative Finance, Taylor & Francis Journals, vol. 2(1), pages 24-30.
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    Cited by:

    1. Tsekrekos, Andrianos E. & Yannacopoulos, Athanasios N., 2016. "Optimal switching decisions under stochastic volatility with fast mean reversion," European Journal of Operational Research, Elsevier, vol. 251(1), pages 148-157.
    2. Sobolev, Daphne, 2017. "The effect of price volatility on judgmental forecasts: The correlated response model," International Journal of Forecasting, Elsevier, vol. 33(3), pages 605-617.
    3. Chen Xiaoshan & Song Qingshuo, 2013. "American option of stochastic volatility model with negative Fichera function on degenerate boundary," Papers 1306.0345, arXiv.org.

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