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Closed-form asymptotics and numerical approximations of 1D parabolic equations with applications to option pricing. (English) Zbl 1242.35131

Summary: 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 \(u(t,x)=\int\mathcal{G}_t(x,y)f(y)dy\) be the solution of \(\partial_tu-(au''+bu'+cu)=0\) for \(t>0, u(0,x)=f(x)\). Then we find computable approximations \(\mathcal{G}_t^{[n]}\) of \(\mathcal{G}_t\). 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.

MSC:

35K08 Heat kernel
35K10 Second-order parabolic equations
35K65 Degenerate parabolic equations
91G20 Derivative securities (option pricing, hedging, etc.)
91G80 Financial applications of other theories
91G60 Numerical methods (including Monte Carlo methods)
35A08 Fundamental solutions to PDEs
35Q84 Fokker-Planck equations
35C05 Solutions to PDEs in closed form
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