Replicating a Future Short Rate with a Bond Calendar Spread

Summary

The document considers a contract that pays the instantaneous short rate at a future date and relates its price to zero-coupon bond prices. Under risk-neutral valuation, the expected discounted future short rate can be expressed as the negative maturity derivative of the bond price. This identifies a theoretical replication using the change in bond prices as maturity changes.

The proposed construction is the limit of a position in two bonds with nearby maturities, scaled by the maturity gap. Such an infinitely narrow calendar spread is not practically tradable, so traded Libor rates are mentioned as a practical substitute. The answer also distinguishes the future payoff from the running instantaneous short rate, which it says cannot itself be traded. The document offers a conceptual replication argument, without specifying a rate model calibration, hedge ratios for a finite spread, or the risks introduced by using traded instruments.

Key ideas

• The future short-rate claim is priced as the expected discounted short rate under risk-neutral valuation.
• Its price equals the negative maturity derivative of the zero-coupon bond price.
• A limiting spread between nearby maturity bonds provides a theoretical replication.
• An infinitely narrow bond spread cannot be traded, so practical replication requires traded rate instruments.
• The running instantaneous short rate is not itself tradable in the setup described.

Tags

# How to replicate the future instantaneous short rate?


# 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?

## Answer by user34971 (score 2, accepted)

https://quant.stackexchange.com/a/54559

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.