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The large-maturity smile for the Stein–Stein model

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  • Forde, Martin

Abstract

We compute the large-maturity smile for the correlated Stein–Stein stochastic volatility model dSt=StYtdWt1,dYt=κ(θ−Yt)dt+σdWt2, dWt1dWt2=ρdt, using the known closed-form solution for the characteristic function of the log stock price given in Schöbel and Zhu (1999). The Stein–Stein model is not covered by the results in Forde and Kumar (submitted for publication) and Jacquier et al. (2013) because the volatility fails to satisfy the sublinear growth condition in Forde and Kumar (submitted for publication) and is not an affine model.11We thank Rohini Kumar for insightful comments.

Suggested Citation

  • Forde, Martin, 2014. "The large-maturity smile for the Stein–Stein model," Statistics & Probability Letters, Elsevier, vol. 91(C), pages 145-152.
    • Handle: RePEc:eee:stapro:v:91:y:2014:i:c:p:145-152
    DOI: 10.1016/j.spl.2014.04.009
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    References listed on IDEAS

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    1. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," The Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-752.
    2. Antoine Jacquier & Aleksandar Mijatović, 2014. "Large Deviations for the Extended Heston Model: The Large-Time Case," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 21(3), pages 263-280, September.
    3. Jim Gatheral & Antoine Jacquier, 2011. "Convergence of Heston to SVI," Quantitative Finance, Taylor & Francis Journals, vol. 11(8), pages 1129-1132.
    4. Martin Forde & Antoine Jacquier, 2011. "The large-maturity smile for the Heston model," Finance and Stochastics, Springer, vol. 15(4), pages 755-780, December.
    5. Antoine Jacquier & Martin Keller-Ressel & Aleksandar Mijatovic, 2011. "Large deviations and stochastic volatility with jumps: asymptotic implied volatility for affine models," Papers 1108.3998, arXiv.org.
    6. Martin Forde & Andrey Pogudin, 2013. "The Large-Maturity Smile For The Sabr And Cev-Heston Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 16(08), pages 1-20.
    7. Martin Forde & Antoine Jacquier & Aleksandar Mijatovic, 2009. "Asymptotic formulae for implied volatility in the Heston model," Papers 0911.2992, arXiv.org, revised May 2010.
    8. L. Rogers & M. Tehranchi, 2010. "Can the implied volatility surface move by parallel shifts?," Finance and Stochastics, Springer, vol. 14(2), pages 235-248, April.
    9. Forde, Martin, 2011. "Large-time asymptotics for an uncorrelated stochastic volatility model," Statistics & Probability Letters, Elsevier, vol. 81(8), pages 1230-1232, August.
    10. Martin Forde, 2011. "Exact Pricing And Large-Time Asymptotics For The Modified Sabr Model And The Brownian Exponential Functional," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 14(04), pages 559-578.
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