Asymptotics for the Laplace transform of the time integral of the geometric Brownian motion

Abstract

We present an asymptotic result for the Laplace transform of the time integral of the geometric Brownian motion F(θ,T)=E[e−θXT] with XT=∫0TeσWs+(a−12σ2)sds, which is exact in the limit σ2T→0 at fixed σ2θT2 and aT. This asymptotic result is applied to pricing zero coupon bonds in the Dothan model of stochastic interest rates. The asymptotic result provides an approximation for bond prices which is in good agreement with numerical evaluations in a wide range of model parameters. As a side result we obtain the asymptotics for Asian option prices in the Black-Scholes model, taking into account interest rates and dividend yield contributions in the σ2T→0 limit.