Instantaneous Short Rates and Forward Rates in HJM Models

Summary

The document clarifies the distinction between an instantaneous short rate and an instantaneous forward rate. In standard notation, a forward rate at time t for maturity T describes the continuously compounded rate over an infinitesimal period beginning at T. The short rate is the special case where that forward maturity coincides with the current time, so it is represented by the forward curve at T equal to t.

Heath-Jarrow-Morton models specify the evolution of forward rates, while a short-rate model specifies the current instantaneous rate directly. The answer gives an example: choosing a particular maturity-dependent forward-rate volatility structure leads to the Hull-White short-rate dynamics, with mean reversion toward a time-varying level. This illustrates how a short-rate model can arise within the broader forward-rate framework. The text notes that HJM models are general and require additional specifications; it offers a conceptual relationship and one construction rather than a full derivation or practical calibration guidance.

Key ideas

• An instantaneous forward rate applies at time t to an infinitesimal future accrual period starting at maturity T.
• The instantaneous short rate is the forward rate with maturity equal to the current time.
• HJM models describe forward-rate dynamics across maturities.
• A suitable forward-rate volatility specification can produce Hull-White short-rate dynamics.
• The general HJM framework needs further choices before it becomes a specific usable model.

Tags

# what's the difference between instantaneous short rate and instantaneous forward rate?


# what's the difference between instantaneous short rate and instantaneous forward rate?












In the short rate models, sometimes it models the instantaneous short rate and sometimes it models the instantaneous forward rate. Does instantaneous short rate = F(0, t + tau) and instantaneous forward rate = F(0, t, t + tau)?

## Answer by Kurt G. (score 3)

https://quant.stackexchange.com/a/67746

In more standard notation the instantaneous forward rate is written as $f(t,T)$, that is, the continuously compounded interest rate seen at $t$ for the infinitesimal interest period $[T,T+dt]\,.$ Heath Jarrow & Morton decided to model those forward rates. Likewise, one can model the constant maturity forward rates $f(t,t+\tau)\,.$ These models are very general (almost too general to be useful in practice) and need further specifications. For example: when the volatility of the forward rate $f(t,T)$ is chosen to be $\sigma(t)\exp(-\int_t^T\lambda(s)\,ds)$ you will get the Hull-White model of the short rate $r(t)=f(t,t)$ which satisfies the SDE $$ dr(t)=\lambda(t)(\theta(t)-r(t))\,dt+\sigma(t)\,dW_t\,. $$ HJM's work showed that the no arbitrage properties of many interest rate models can be studied under a common framework.

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