Deriving Spot and Instantaneous Forward Rates from Zero-Coupon Prices
Summary
The document derives the relationship between a zero-coupon bond price and an interest rate that varies deterministically over time. Starting from the exponential of the negative integral of the short rate, it takes logarithms and differentiates with respect to maturity. The Leibniz rule gives the rate at the maturity endpoint, while the chain rule converts the derivative of the log price into the negative price derivative divided by price.
It then distinguishes this deterministic short-rate interpretation from the stochastic-rate setting. There, the same maturity derivative defines the instantaneous forward rate, and integrating that forward curve recovers the bond price. The instantaneous forward rate approaches the short rate as maturity approaches the observation time. The sign observation that bond prices usually fall with maturity is an intuition for positive rates, not a universal guarantee; the discussion does not cover market conventions, compounding choices, or noisy observed curves.
Key ideas
- Taking the logarithm of the deterministic zero-coupon price turns its integrated short rate into a maturity derivative problem.
- The Leibniz rule identifies the derivative of the integral with the short rate at maturity.
- The chain rule expresses the rate as the negative maturity derivative of bond price divided by bond price.
- With stochastic rates, the same price derivative defines the instantaneous forward rate rather than directly the future short rate.
- The instantaneous forward rate converges to the short rate as maturity approaches the observation time.
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Full text
# Interest rate equation from bond price?
# Interest rate equation from bond price?
If a zero coupon bond price at time $t$, with maturity $T$ ($t<T$), is denoted by
$B(t;T) = B(T;T) e^{(-\int_{t}^{T} r(s) ds)}$
where $r(t)$ is a known interest rate.
How does this transform to $r(T) = - \frac{1}{B(t;T)} \frac{\partial B(t;T)}{\partial T}$
I know that $B(T;T)=1$ and we can rearrange, but I don't understand how to obtain partial differentials from integrals?
## Answer by Kevin (score 2, accepted)
https://quant.stackexchange.com/a/46965
If interest rates are deterministic (i.e. time-dependent but non-ranom), then \begin{align} B(t,T) &= \exp\left( - \int_t^T r(s)\mathrm{d}s\right) \\ \Leftrightarrow \int_t^T r(u)\mathrm{d}u &= -\ln B(t,T). \end{align} Differentiating both sides with respect to $T$ according to the Leibniz rule yields \begin{align*} r(T) &= -\frac{\partial \ln B(t,T)}{\partial T}\\ &= -\frac{1}{B(t,T)} \frac{\partial B(t,T)}{\partial T}. \end{align*} The latter line uses the chain rule. Recall that $\frac{\partial B(t,T)}{\partial T}<0$ which gives you positive interest rates. After all, bond prices typically decrease with maturity.
If interest rates are stochastic, the first equation requires a conditional expectation. However, note that by definition, the instantaneous forward rate $f(t,T)$ satisfies \begin{align*} f(t,T) &= -\frac{\partial \ln B(t,T)}{\partial T} \\ &=-\frac{1}{B(t,T)}\frac{\partial B(t,T)}{\partial T}. \end{align*} The latter equality follows again from the chain rule. The first line allows you to write again \begin{align*} B(t,T) = \exp\left(-\int_t^T f(t,s)\mathrm{d}s\right). \end{align*} This holds even if interest rates are stochastic. Furthermore, note that $\lim\limits_{T\to t} f(t,T) = r(t)$, which is the instantaneous short rate.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.