### Abstract

We construct closed-form asymptotic formulas for the Green's function of parabolic equations (e.g., Fokker-Planck equations) with variable coefficients in one space dimension. More precisely, let u(t, x) = Gt(x, y)f(y)dy be the solution of δtu-(au''+bu'+cu) = 0 for t > 0, u(0, x) = f(x). Then we find computable approximations G^{[n]} _{t} of G_{t}. The approximate kernels are derived by applying the Dyson-Taylor commutator method that we have recently developed for short-time expansions of heat kernels on arbitrary dimension Euclidean spaces. We then utilize these kernels to obtain closed-form pricing formulas for European call options. The validity of such approximations to large time is extended using a bootstrap scheme. We prove explicit error estimates in weighted Sobolev spaces, which we test numerically and compare to other methods.

Original language | English (US) |
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Pages (from-to) | 901-934 |

Number of pages | 34 |

Journal | SIAM Journal on Financial Mathematics |

Volume | 2 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1 2011 |

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### All Science Journal Classification (ASJC) codes

- Numerical Analysis
- Finance
- Applied Mathematics

### Cite this

*SIAM Journal on Financial Mathematics*,

*2*(1), 901-934. https://doi.org/10.1137/100796832