= Random constant Gaussian interest-rate mixture
{title2=$\rho_{\rm eff}=\rho_0-\tau^2T/2,\quad\sigma_{\rm eff}^2=\sigma^2+\tau^2T$}
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 $-\tau^2T$; 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.
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