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Deriving the Instantaneous Forward Rate from Discount Factors

Article Quant Q&A · Author: mathjacks
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Summary

The document shows how an instantaneous forward rate can be obtained from a discount factor by differentiating its logarithm with respect to maturity. It begins with a representation of the price at time t of a zero-coupon payment at T as the exponential of the negative integral of forward rates between those dates. Taking the maturity derivative recovers the forward rate at that maturity; at time zero, it is the negative maturity derivative of the log discount factor.

This is a compact calculus explanation, not a worked derivation of every step. For the stated constant-rate form, taking the logarithm and differentiating with respect to T gives the constant rate. The result relies on the discount factor and forward-rate integral being differentiable in maturity; the note does not discuss irregular curves, compounding conventions, or other market quoting details.

Key ideas

  • A discount factor can be represented using the integral of forward rates over time.
  • The instantaneous forward rate is the negative maturity derivative of the log discount factor.
  • With a constant continuously compounded rate, the discount factor’s log is linear in maturity.
  • The derivative relationship assumes sufficient smoothness in maturity.

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Fixed incomeDerivatives pricingStatistics
Full text
# Derive instantaneous forward rate


# Derive instantaneous forward rate












Given that $P(0,T)=e^{-RT}$, how does one get the formula for the instantaneous forward rate below? Specifically, how does one get to the partial derivative in the formula?

I'm sure the answer is obvious but I haven't been over my calculus in a while.

## Answer by DerekT (score 4)

https://quant.stackexchange.com/a/11329

You can start with $$P(t,T)=exp({-\int_t^T f_t(u).du})$$ then take derivative wrt to T $$R_F(0,T)=f_0(T)=-\frac{\partial} {\partial T}{ ln(P(0,T))} $$

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