More on Interest Rates

When it comes to predicting future interest rates, the Heath-Jarrow-Morton framework is considered a standard approach. It focuses on modeling the evolution of the interest rate curve (instantaneous forward rate curve in particular). The equation itself is a rather evolved derivation, incorporating bond prices, forward rates, risk free rates, the Wiener process, Leibniz's rule, and Fubini's Theorem. While the details of this calculation are a bit outside the scope of discussion here, the equation can ultimately be described as:

${\displaystyle \text{df}(\text{t},\text{u})=\left({\boldsymbol {\Sigma }}(\text{t},\text{u})\int _{\text{t}}^{\text{u}}{\boldsymbol {\Sigma }}(\text{t},\text{s})^{\text{T}}\text{ds}\right)\text{dt}+{\boldsymbol {\Sigma }}(\text{t},\text{u})\text{dW}_{\text{t}}}$

For the sake of this discussion, it suffices to say that the input of existing yield curves is useful in projected future interest rates under a number of varying perspectives.