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.

  • 2
    $\begingroup$ In the first formula, which is always true, $r$ is the interest rate (or YTM) for the term of the bond (i.e. the 7 year interest rate in the case of a 7 year bond). In the second formula $r_t$ is the instantaneous i.r., or Spot Rate, which is the (annualized) rate for lending money for a short period if time, such as 1 day. You can use two different letters (such as $y$ and $r_t$) if you want. $\endgroup$
    – noob2
    May 7 '20 at 14:23

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.


Not the answer you're looking for? Browse other questions tagged or ask your own question.