Prices of Asian options under stochastic interest rates (Q2505261)

The authors consider a stock process of the form \[ dS_t= r_t S_t dt+ \sigma_t S_t dB_r\qquad (0\leq t\leq T), \] where the interest rate process \((r_t)\) is a Hull-White (extended Vasicek) model \[ dr_t= (\alpha_t r_t+ \beta_t)\,dt+ \gamma_t dZ_t\qquad (0\leq t\leq T) \] such that \[ \rho(dB_t, dZ_t)= \rho\,dt\qquad (0\leq t\leq T). \] In Section 2 a Black-Scholes formula for the call option price is obtained. Based on the process \[ J_t= \exp\Biggl({1\over t} \int^t_0\log S_u\,du\Biggr) \] (continuously sampled geometric average) the authors derive in Section 3 a pricing formula for an Asian call option of the form \((J_T- K)^+\).