Standard Article

Moment Explosions

  1. Peter K. Friz1,2,3,
  2. Martin Keller Ressel4,5

Published Online: 15 MAY 2010

DOI: 10.1002/9780470061602.eqf08005

Encyclopedia of Quantitative Finance

Encyclopedia of Quantitative Finance

How to Cite

Friz, P. K. and Ressel, M. K. 2010. Moment Explosions. Encyclopedia of Quantitative Finance. .

Author Information

  1. 1

    University of Cambridge, Cambridge, UK

  2. 2

    Austrian Academy of Science, Linz, Austria

  3. 3

    TU Berlin and WIAS Berlin, Berlin, Germany

  4. 4

    Vienna University of Technology, Vienna, Austria

  5. 5

    ETH Zurich, Zurich, Switzerland

Publication History

  1. Published Online: 15 MAY 2010

Abstract

Let (St)t ≥ 0 be the discounted price process in a stochastic volatility model. A moment explosion takes place, if the moment inline image of some given order u ∈ ℝ becomes infinite (“explodes”) after some finite time T*(u). Moment explosions are closely related to the shape of the implied volatility surface, where they can be used to obtain approximations for deep in-the-money and out-of-the-money strikes. Furthermore, moment explosions may lead to infinite prices of derivatives with superlinear payoff, and to the breakdown of error estimates for numerical approximation schemes. Comparison results for parabolic partial differential equations (PDEs) combined with an exponentially affine ansatz for the solution, allow to link explosion times to the blow-up time of ordinary differential equations of the (generalized) Riccati type.

Keywords:

  • moment explosion;
  • stochastic volatility;
  • Heston model;
  • SABR model;
  • affine stochastic volatility model;
  • implied volatility surface;
  • smile asymptotics;
  • (generalized) Riccati equation