### Abstract

Three processes reflecting persistence of volatility are initially formulated by evaluating three Lévy processes at a time change given by the integral of a mean-reverting square root process. The model for the mean-reverting time change is then generalized to include non-Gaussian models that are solutions to Ornstein-Uhlenbeck equations driven by one-sided discontinuous Lévy processes permitting correlation with the stock. Positive stock price processes are obtained by exponentiating and mean correcting these processes, or alternatively by stochastically exponentiating these processes. The characteristic functions for the log price can be used to yield option prices via the fast Fourier transform. In general mean-corrected exponentiation performs better than employing the stochastic exponential. It is observed that the mean-corrected exponential model is not a martingale in the filtration in which it is originally defined. This leads us to formulate and investigate the important property of martingale marginals where we seek martingales in altered filtrations consistent with the one-dimensional marginal distributions of the level of the process at each future date.

Original language | English (US) |
---|---|

Pages (from-to) | 345-382 |

Number of pages | 38 |

Journal | Mathematical Finance |

Volume | 13 |

Issue number | 3 |

DOIs | |

State | Published - Jul 2003 |

### Fingerprint

### Keywords

- Lévy marginal
- Leverage
- Martingale marginal
- OU equation
- Static arbitrage
- Stochastic exponential
- Variance gamma

### ASJC Scopus subject areas

- Accounting
- Finance
- Social Sciences (miscellaneous)
- Economics and Econometrics
- Applied Mathematics

### Cite this

*Mathematical Finance*,

*13*(3), 345-382. https://doi.org/10.1111/1467-9965.00020

**Stochastic volatility for Lévy processes.** / Carr, Peter; Geman, Hélyette; Madan, Dilip B.; Yor, Marc.

Research output: Contribution to journal › Article

*Mathematical Finance*, vol. 13, no. 3, pp. 345-382. https://doi.org/10.1111/1467-9965.00020

}

TY - JOUR

T1 - Stochastic volatility for Lévy processes

AU - Carr, Peter

AU - Geman, Hélyette

AU - Madan, Dilip B.

AU - Yor, Marc

PY - 2003/7

Y1 - 2003/7

N2 - Three processes reflecting persistence of volatility are initially formulated by evaluating three Lévy processes at a time change given by the integral of a mean-reverting square root process. The model for the mean-reverting time change is then generalized to include non-Gaussian models that are solutions to Ornstein-Uhlenbeck equations driven by one-sided discontinuous Lévy processes permitting correlation with the stock. Positive stock price processes are obtained by exponentiating and mean correcting these processes, or alternatively by stochastically exponentiating these processes. The characteristic functions for the log price can be used to yield option prices via the fast Fourier transform. In general mean-corrected exponentiation performs better than employing the stochastic exponential. It is observed that the mean-corrected exponential model is not a martingale in the filtration in which it is originally defined. This leads us to formulate and investigate the important property of martingale marginals where we seek martingales in altered filtrations consistent with the one-dimensional marginal distributions of the level of the process at each future date.

AB - Three processes reflecting persistence of volatility are initially formulated by evaluating three Lévy processes at a time change given by the integral of a mean-reverting square root process. The model for the mean-reverting time change is then generalized to include non-Gaussian models that are solutions to Ornstein-Uhlenbeck equations driven by one-sided discontinuous Lévy processes permitting correlation with the stock. Positive stock price processes are obtained by exponentiating and mean correcting these processes, or alternatively by stochastically exponentiating these processes. The characteristic functions for the log price can be used to yield option prices via the fast Fourier transform. In general mean-corrected exponentiation performs better than employing the stochastic exponential. It is observed that the mean-corrected exponential model is not a martingale in the filtration in which it is originally defined. This leads us to formulate and investigate the important property of martingale marginals where we seek martingales in altered filtrations consistent with the one-dimensional marginal distributions of the level of the process at each future date.

KW - Lévy marginal

KW - Leverage

KW - Martingale marginal

KW - OU equation

KW - Static arbitrage

KW - Stochastic exponential

KW - Variance gamma

UR - http://www.scopus.com/inward/record.url?scp=0038742720&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0038742720&partnerID=8YFLogxK

U2 - 10.1111/1467-9965.00020

DO - 10.1111/1467-9965.00020

M3 - Article

AN - SCOPUS:0038742720

VL - 13

SP - 345

EP - 382

JO - Mathematical Finance

JF - Mathematical Finance

SN - 0960-1627

IS - 3

ER -