Finance and Stochastics

, Volume 6, Issue 1, pp 63–90 | Cite as

Stochastic volatility, jumps and hidden time changes

  • Hélyette Geman
  • Dilip B. Madan
  • Marc Yor
Original Paper


Stochastic volatility and jumps are viewed as arising from Brownian subordination given here by an independent purely discontinuous process and we inquire into the relation between the realized variance or quadratic variation of the process and the time change. The class of models considered encompasses a wide range of models employed in practical financial modeling. It is shown that in general the time change cannot be recovered from the composite process and we obtain its conditional distribution in a variety of cases. The implications of our results for working with stochastic volatility models in general is also described. We solve the recovery problem, i.e. the identification the conditional law for a variety of cases, the simplest solution being for the gamma time change when this conditional law is that of the first hitting time process of Brownian motion with drift attaining the level of the variation of the time changed process. We also introduce and solve in certain cases the problem of stochastic scaling. A stochastic scalar is a subordinator that recovers the law of a given subordinator when evaluated at an independent and time scaled copy of the given subordinator. These results are of importance in comparing price quality delivered by alternate exchanges.

Key words: Brownian subordination, variance gamma and normal inverse Gaussian processes, variance swap, quadratic variation, filtering 
JEL Classification: C10, G12, G13 
Mathematics Subject Classification (1991): 60G07, 60G35 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Hélyette Geman
    • 1
  • Dilip B. Madan
    • 2
  • Marc Yor
    • 3
  1. 1.University of Paris Dauphine and ESSEC, Finance Department, 95021 Cergy-Pontoise, France (e-mail: FR
  2. 2.Robert H. Smith School of Business, Van Munching Hall, University of Maryland, College Park, MD 20742, USA ( US
  3. 3.Laboratoire de probabilités et Modeles aléatoires, Université Pierre et Marie Curie, 4, Place Jussieu, 75252 Paris Cedex, France (e-mail: FR

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