The futures–forward gap under Vasicek rates

Under $\mathbb{Q}$, $dS_t/S_t = r_t\,dt + \sigma\,dW^S_t$, $dr_t = \kappa(\theta - r_t)\,dt + \eta\,dW^r_t$, $d\langle W^S, W^r\rangle_t = \rho\,dt$. (a) Show that $I_T = \int_0^T r_s\,ds$ is Gaussian with $\operatorname{Var}(I_T) = \eta^2\int_0^T B(s,T)^2ds$, where $B(s,T) = (1 - e^{-\kappa(T-s)})/\kappa$, and that $\operatorname{Cov}(W^S_T, I_T) = \rho\eta\int_0^T B(s,T)\,ds$. (b) Using Prop. 7.6, show that $F_0/G_0 = \exp\left(\operatorname{Cov}(\ln S_T, I_T)\right) = \exp\left(\eta^2 I_2 + \rho\sigma\eta I_1\right)$ with $I_1 = \int_0^T B\,ds$ and $I_2 = \int_0^T B^2ds$. (c) Derive the closed forms $I_1 = (T - B_0)/\kappa$ and $I_2 = \left(T - 2B_0 + (1 - e^{-2\kappa T})/(2\kappa)\right)/\kappa^2$, with $B_0 = B(0,T)$. (d) Evaluate $F_0/G_0 - 1$ for (i) $\sigma = 0.15$, $\kappa = 0.3$, $\eta = 0.01$, $\rho = 0.3$, $T = 0.25$, and (ii) $\sigma = 0.25$, $\kappa = 0.3$, $\eta = 0.02$, $\rho = \pm 0.9$, $T = 2$. Express (i) in Nifty points at 24,700 and compare it with the day-to-day noise in the end-of-day Nifty basis (7.3, 7.8.2).