Zero-Coupon Bond Pricing with Constant and Stochastic Rates
Summary
The note explains why two zero-coupon bond pricing formulas appear in finance. With a constant short rate, the bond price is the exponential discount factor for the rate multiplied by time to maturity. When rates vary randomly, pricing instead takes the risk-neutral expected value of discounting over the entire path of short rates, conditional on current information.
In Ho-Lee and Hull-White models, this conditional expectation can be expressed in an exponential-affine form involving functions of time and the current short rate. Thus, the formulas reflect different assumptions rather than conflicting pricing rules: constant rates give the simpler expression, while the affine form applies in certain stochastic-rate models. The note offers a conceptual connection but does not derive the model-specific functions or discuss calibration, market data, or numerical evidence.
Key ideas
- A zero-coupon bond price is the risk-neutral expected discount factor over the remaining life of the bond.
- With a constant short rate, this expectation reduces to exponential discounting by the rate and time to maturity.
- Time-varying rates require accounting for the accumulated short rate along future paths.
- Ho-Lee and Hull-White models allow the bond price to be written in an exponential-affine form using the current short rate.
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# Why Bond pricing formula is changed?
# Why Bond pricing formula is changed?
When I first learn about finance, a bond with continuous yield was priced via
$$Z = e^{-rT},$$ where $r$ is the yield, $T$ the time to maturity.
But, when I learned about stochastic interest rate models like Ho-Lee and Hull & White, etc. the formula changed to
$$Z= e^{A-Br},$$ where $A$, $B$ are functions.
What is the reason for it? Is there any reference which deal with this? I need explanation that is intuitive.
## Answer by Kevin (score 1)
https://quant.stackexchange.com/a/53938
In general, the time $t$ price of a zero-coupon bond maturing at time $T$ is given by \begin{align*} P(t,T) &= \mathbb{E}^\mathbb{Q}\left[\exp\left(-\int_t^T r_s\mathrm{d}s\right) \Bigg|\mathcal{F}_t\right]. \end{align*} Here, $r_t$ is the short rate, i.e. the cost for borrowing time from $t$ until $t+\mathrm{d}t$. This formula follows directly from the absence of arbitrage.
If $r_t\equiv r$ is constant, then \begin{align*} P(t,T)=e^{-r(T-t)}, \end{align*} which is the first formula you mentioned.
If $r_t$ is time-varying, you have to used the above equation with the conditional expectations. As it happens, in the Ho-Lee and the Hull-White models (and others too), this conditional expectation can be written as \begin{align*} P(t,T)=e^{A(t,T)+r_tB(t,T)}, \end{align*} for suitable functions $A,B$, see here.
So both equations you mentioned are special cases of a more general formula. The first special case assumes constant interest rates, the second one assumes normally distributed short rates.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.