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Explicit Density Approximations for Local Volatility Models Using Heat Kernel Expansions

Author

Listed:
  • Stephen Taylor

    (Hutchin Hill Capital)

  • Scott Glasgow

    (BYU Department of Mathematics)

  • James Taylor

    (BYU Department of Mathematics)

  • Jan Vecer

    (Vysoka skola aplikovanecho prava
    Charles University)

Abstract

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.

Suggested Citation

  • Stephen Taylor & Scott Glasgow & James Taylor & Jan Vecer, 2016. "Explicit Density Approximations for Local Volatility Models Using Heat Kernel Expansions," Methodology and Computing in Applied Probability, Springer, vol. 18(3), pages 847-867, September.
  • Handle: RePEc:spr:metcap:v:18:y:2016:i:3:d:10.1007_s11009-015-9463-6
    DOI: 10.1007/s11009-015-9463-6
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    References listed on IDEAS

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    4. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
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