TY - JOUR
T1 - Explicit Density Approximations for Local Volatility Models Using Heat Kernel Expansions
AU - Taylor, Stephen
AU - Glasgow, Scott
AU - Taylor, James
AU - Vecer, Jan
N1 - Publisher Copyright:
© 2015, Springer Science+Business Media New York.
PY - 2016/9/1
Y1 - 2016/9/1
N2 - 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.
AB - 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.
KW - CEV model
KW - Heat kernel expansion
KW - Local volatility
KW - Short rate models
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U2 - 10.1007/s11009-015-9463-6
DO - 10.1007/s11009-015-9463-6
M3 - Article
AN - SCOPUS:84941670239
SN - 1387-5841
VL - 18
SP - 847
EP - 867
JO - Methodology and Computing in Applied Probability
JF - Methodology and Computing in Applied Probability
IS - 3
ER -