### 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.

Original language | English (US) |
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Article number | P06010 |

Journal | Journal of Statistical Mechanics: Theory and Experiment |

Volume | 2008 |

Issue number | 6 |

DOIs | |

State | Published - Jun 1 2008 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Statistics and Probability
- Statistics, Probability and Uncertainty

### Keywords

- Financial instruments and regulation
- Models of financial markets
- Risk measure and management
- Stochastic processes (experiment)

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## Cite this

*Journal of Statistical Mechanics: Theory and Experiment*,

*2008*(6), [P06010]. https://doi.org/10.1088/1742-5468/2008/06/P06010