Random constant Gaussian interest-rate mixture (${\rho }_{\mathrm{eff}}={\rho }_0-{\tau }^{2}T/2,{\sigma }_{\mathrm{eff}}^2={\sigma }^2+{\tau }^{2}T$)

Average constant-rate Black-Scholes model prices of a fixed terminal payoff over $\rho \sim N({\rho }_0,{\tau }^2)$. Tilting this Gaussian by the discount factor shifts its mean by $-{\tau }^{2}T$; combining the resulting log-price variance with stock variance gives the displayed effective parameters. Thus the average equals the Black-Scholes price with those parameters, whenever the expectation is well defined. This is an average of constant-parameter prices, rather than a specification of an adapted stochastic short-rate market.