April 25, 2024
SOFR is the Secured Overnight Financing Rate. Quants should be overjoyed that there is a highly liquid market instrument that is a proxy for the continously compounded risk-free forward rate, f_t. The reality is that SOFR can spike, as it did in September 2019.
D_t = \exp(-\int_0^t f_s\,ds).
D_t(u) = E_t[D_u]/D_t = \exp(-\int_t^u f_t(s)\,ds).
Term SOFR is an average of backward looking rates, similar to Fed Fund Futures settelment quotes.
S_t(t - h, t) = (1/h)\int_{t - h}^t f_s\,ds
Forward rate at t over [t, t + h].
F_t(t, t + h) = (1/h)(1/D_t(t + h) - 1) \approx (1/h)\int_t^{t + h} f_t(s)\,ds
f_t = r(t) + \sigma B_t.
f_t(u) = r(t) - \sigma^2 (u - t)^2/2 + \sigma B_t_.
\int_t^u B_s\,ds = \int_t^u s\,dB_s + (u - t) B_t.