# How to replicate the future instantaneous short rate?

Suppose we have an interest rate model $$R(t)=\alpha(t)d(t)+\sigma d\tilde{W}(t)$$, where the brownian motion is under the risk neutral measure. Suppose $$S(t)$$ is the price at time $$t$$ for a contract that pays $$R(T)$$ at time $$T$$, where $$0\leq t\leq T$$. Here is how we price this contract: $$S(t)=\tilde{\mathbb{E}}_t[e^{-\int_t^TR(u)du}R(T)]=-\tilde{\mathbb{E}}_t[\frac{\partial}{\partial T}e^{-\int_t^TR(u)du}]=-\frac{\partial}{\partial T}B(t,T)$$ where $$B(t,T)$$ is the price of a zero coupon bond at time $$t$$ with maturity $$T$$. I understand how we come to the price. My question is how can we replicate this contract $$S(t)$$? Do we trade between the short rate and the zero coupon bond?

I took the liberty of modifying the title of your question, as it is not the zero coupon you want to replicate but the future value of the short rate, $$S(T) = R(T)$$ in your notation.
You have already given the answer yourself, namely $$S(t) = \lim_{\epsilon \rightarrow 0} \frac{B(t,T) - B(t, T+\epsilon)}{\epsilon}$$ In practice it is not possible to trade an infinitely tight zero coupon calendar spread, hence Libor rates are traded.
Also, note that the running instantaneous short rate $$R(t)$$ cannot be traded.
• Oh yes that was what I meant, thanks for changing the title. So theoretically what does the replicating strategy look like? I mean the process $\Delta(t)$ of the number of $B(t,T)$ at time $t$. It seems $S(t)$ is a limit, so I am confused even just in theory what this means. In addition, could you expand upon how Libor rates work? Thanks very much! May 31, 2020 at 13:23
• The static replicating portfolio would be long $1/\epsilon$ units of the $T$ bond and short the same number of units of the $T+\epsilon$ bond, where $\epsilon$ is a "very small" number, eg 1E-23400087564. The limit notation in my answer is just another way to write the derivative of the zero coupon wrt $T$. Clearly this would lead to theoretically infinite number of long and short bonds, which just cannot be done in practice. The Libor rate definition is probably somewhere in your textbook, or otherwise googling it should give you the expression for it in terms of zero coupon bonds. May 31, 2020 at 13:40