Methodology and Computing in Applied Probability

, Volume 18, Issue 3, pp 847–867 | Cite as

Explicit Density Approximations for Local Volatility Models Using Heat Kernel Expansions

  • Stephen TaylorEmail author
  • Scott Glasgow
  • James Taylor
  • Jan Vecer


Heat kernel perturbation theory is a tool for constructing explicit approximation formulas for the solutions of linear parabolic equations. We review the crux of this perturbative formalism and then apply it to differential equations which govern the transition densities of several local volatility processes. In particular, we compute all the heat kernel coefficients for the CEV and quadratic local volatility models; in the later case, we are able to use these to construct an exact explicit formula for the processes’ transition density. We then derive low order approximation formulas for the cubic local volatility model, an affine-affine short rate model, and a generalized mean reverting CEV model. We finally demonstrate that the approximation formulas are accurate in certain model parameter regimes via comparison to Monte Carlo simulations.


Heat kernel expansion Local volatility CEV model Short rate models 


35K08 91G20 


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Stephen Taylor
    • 1
    Email author
  • Scott Glasgow
    • 2
  • James Taylor
    • 2
  • Jan Vecer
    • 3
    • 4
  1. 1.Hutchin Hill CapitalNew YorkUSA
  2. 2.BYU Department of MathematicsProvoUSA
  3. 3.Vysoka skola aplikovanecho pravaPrague 4Czech Republic
  4. 4.Faculty of Mathematics and PhysicsCharles UniversityPrague 8Czech Republic

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