January 26, 2025
Instantaneous short (repo) rate (f_t).
Stochastic discount D_t = \exp(-\int_0^t f_s\,ds)P.
D_t(\Omega) = \exp(-\int_0^t f(s)\,ds).
D_t(u)P = (D_u/D_t)|_{\mathcal{A}_t} = \exp(-\int_t^u f_t(s)\,ds)P
Futures \phi_t(u)P = f_u P|_{\mathcal{A}_t}.