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Option pricing under stochastic volatility: the exponential Ornstein-Uhlenbeck model

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  • Josep Perello
  • Ronnie Sircar
  • Jaume Masoliver

Abstract

We study the pricing problem for a European call option when the volatility of the underlying asset is random and follows the exponential Ornstein-Uhlenbeck model. The random diffusion model proposed is a two-dimensional market process that takes a log-Brownian motion to describe price dynamics and an Ornstein-Uhlenbeck subordinated process describing the randomness of the log-volatility. We derive an approximate option price that is valid when (i) the fluctuations of the volatility are larger than its normal level, (ii) the volatility presents a slow driving force toward its normal level and, finally, (iii) the market price of risk is a linear function of the log-volatility. We study the resulting European call price and its implied volatility for a range of parameters consistent with daily Dow Jones Index data.

Suggested Citation

  • Josep Perello & Ronnie Sircar & Jaume Masoliver, 2008. "Option pricing under stochastic volatility: the exponential Ornstein-Uhlenbeck model," Papers 0804.2589, arXiv.org, revised May 2008.
    • Handle: RePEc:arx:papers:0804.2589
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    Cited by:

    1. Oliver Pfante & Nils Bertschinger, 2019. "Information-Theoretic Analysis Of Stochastic Volatility Models," Advances in Complex Systems (ACS), World Scientific Publishing Co. Pte. Ltd., vol. 22(01), pages 1-21, February.
    2. Michela Pelizza & Klaus R. Schenk-Hoppé, 2020. "Pricing Defaulted Italian Mortgages," JRFM, MDPI, vol. 13(2), pages 1-14, February.
    3. Andrea Barletta & Elisa Nicolato & Stefano Pagliarani, 2019. "The short‐time behavior of VIX‐implied volatilities in a multifactor stochastic volatility framework," Mathematical Finance, Wiley Blackwell, vol. 29(3), pages 928-966, July.
    4. Oliver Pfante & Nils Bertschinger, 2016. "Uncertainty Estimates in the Heston Model via Fisher Information," Papers 1610.04760, arXiv.org, revised Oct 2016.
    5. Oliver Pfante & Nils Bertschinger, 2016. "Volatility Inference and Return Dependencies in Stochastic Volatility Models," Papers 1610.00312, arXiv.org.
    6. Frontczak, Robert & Rostek, Stefan, 2015. "Modeling loss given default with stochastic collateral," Economic Modelling, Elsevier, vol. 44(C), pages 162-170.
    7. Oliver Pfante & Nils Bertschinger, 2019. "Volatility Inference And Return Dependencies In Stochastic Volatility Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(03), pages 1-44, May.
    8. Giacomo Bormetti & Valentina Cazzola & Guido Montagna & Oreste Nicrosini, 2008. "Probability distribution of returns in the exponential Ornstein-Uhlenbeck model," Papers 0805.0540, arXiv.org, revised Oct 2008.
    9. S. Kuchuk-Iatsenko & Y. Mishura & Y. Munchak, 2016. "Application of Malliavin calculus to exact and approximate option pricing under stochastic volatility," Papers 1608.00230, arXiv.org.
    10. Nils Bertschinger & Oliver Pfante, 2015. "Inferring Volatility in the Heston Model and its Relatives -- an Information Theoretical Approach," Papers 1512.08381, arXiv.org.
    11. Kim, See-Woo & Kim, Jeong-Hoon, 2019. "Variance swaps with double exponential Ornstein-Uhlenbeck stochastic volatility," The North American Journal of Economics and Finance, Elsevier, vol. 48(C), pages 149-169.
    12. Yunhong Lyu & Sévérien Nkurunziza, 2023. "Inference in generalized exponential O–U processes," Statistical Inference for Stochastic Processes, Springer, vol. 26(3), pages 581-618, October.
    13. Giacomo Bormetti & Valentina Cazzola & Danilo Delpini, 2009. "Option pricing under Ornstein-Uhlenbeck stochastic volatility: a linear model," Papers 0905.1882, arXiv.org, revised May 2010.
    14. Sergii Kuchuk-Iatsenko & Yuliya Mishura, 2015. "Pricing the European call option in the model with stochastic volatility driven by Ornstein--Uhlenbeck process. Exact formulas," Papers 1510.01848, arXiv.org.

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