Finding the Short Rate from a Time-Dependent Forward Curve

Summary

The document asks how to obtain a short or spot rate from a specified time-dependent forward-rate curve. It proposes evaluating the curve on the diagonal, where the maturity date equals the observation time, and asks whether that substitution is sufficient or whether another step is needed. The expression includes an initial forward curve, a time-dependent drift term involving volatility, and a Brownian-motion term.

No answer, derivation, or supporting evidence is included, so the document does not establish a final rate formula or clarify the rate conventions being used. In particular, it leaves open whether the notation defines an instantaneous forward rate and how the resulting diagonal value should be interpreted as the short rate. The item is a focused mathematical question rather than a worked term-structure model.

Key ideas

• The question concerns extracting a short rate from a time-dependent forward curve.
• It proposes evaluating the forward curve at maturity equal to current time.
• The stated curve contains an initial term structure, a volatility-dependent drift, and a Brownian component.
• The document supplies no derivation or answer, so the proposed substitution is not validated there.

Tags

# Term Structure and short rates


# Term Structure and short rates












If I have a term structure/yield curve given by: $$f(t, T) = f(0, T) + σ^2t(T − \frac{t}{2}) + σB_t $$

and want to find the short/spot rate $r_t$, is this simply:

$$f(t,t) = f(0,t) + \sigma^2t(t-\frac{t}{2}) + \sigma B_t$$

Can anything further be done?

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