Bond Valuation Using the Short Rate

Let r(t) be the deterministic risk-free interest rate defined for times t \in [0,T] , where T is the maturity date of the bond. We make the assumption that the interest rate is not an independent state variable but is a known function of time. Hence the bond price here can be assumed to be a function of only time. Let B(t) be the price of the bond at time t , and c(t) be the price of the coupon payment at time t . There is a boundary condition where we know explicitly the value of the bond at the maturity, as this is just the par value: B(T) = P .

Let dt be an infinitesimal time increment. The change in the value of the bond over this infinitesimal time interval is \frac{dB}{dt}dt and the corresponding coupon received over this time is c(t)dt . By the no-arbitrage principle this amount must equal to the risk-free interest return on the same value on the same infinitesimal time increment: r(t)B(t)dt , hence (after dividing through by dt ):

\frac{dB}{dt} + c(t) = r(t)B(t),\quad t < T

But this is an ordinary different equation. To solve we create the integrating factor:

I=e^{\int_t^T r(s)ds}

and multiply both side by this integrating factor:

\frac{d}{dt}\left(B(t) e^{\int_t^T r(s)ds}\right) = -c(t)e^{\int_t^T r(s)ds}

Integrating both sides from t to T and using the boundary condition we obtain:

B(t) = e^{-\int_t^T r(s)ds}\left( P+\int_t^T c(u)e^{\int_u^T r(s)ds}du\right)

From this we see that the coupon amount c(u)du received over the time period [u,u+du] will grow to the amount c(u)e^{\int_u^T r(s)ds} by the maturity time T . The whole thing inside the brackets represents the future value of all cashflows (including the face value). The component on the outside of the brackets represents the discount factor that is used to discount all cashflows back to the present time.

The Yield Curve

As above, let dt be an infinitesimal time increment. The change in the value of the bond over this infinitesimal time interval is \frac{dB}{dt}dt and there are no coupon payments so c(t) = 0 for all time t . By the no-arbitrage principle this amount must equal to the risk-free interest return on the same value on the same infinitesimal time increment: r(t)B(t)dt , hence (after dividing through by dt ):

\begin{array}{rcl} \frac{dB}{dt} &=& rB \\ \frac{dB}{B} &=& rdt \quad \mbox{Dividng by B and multiplying by dt} \\ \ln(dB) &=& rdt \quad \mbox{replacing differential with natural logarithm.} \\ \int_t^T \ln(dB) &=& \int_t^T rdt \quad\mbox{Integrating both sides.} \\ \ln(B(T,T)) - \ln(B(t,T)) &=& r(T-t) \\ \ln(B(t,T)) - \ln(B(T,T)) &=& -r(T-t) \\ \ln\left(\frac{B(t,T)}{B(T,T)}\right) &=& -r(T-t) \\ \ln(B(t,T)) &=& -r(T-t) \quad\mbox{Using the fact that } B(T,T) = 1 \mbox{ at maturity.} \\ B(t,T) &=& e^{-r(T-t)} \quad\mbox{Taking exponentials of both sides.} \end{array}

Re-arranging this formula for the rate gives the formula for the yield to maturity or YTM:

r = -\frac{1}{T-t} \ln B(t,T)

If you plot the value of r for different times t you obtain the yield curve

Bond Valuation Using the Forward Rate

Consider the price of a forward contract at time t where the buyer of this forward agrees to purchase a zero-coupon bond with maturity T_2 at a later date T_1 < T_2. We define the forward rate, f , as seen by the market at time t for the future period of time between [T_1,T_2] by

f(t;T_1,T_2)=-\frac{1}{T_2-T_1}\ln\frac{B(t,T_2)}{B(t,T_1)}

The forward rate is thus a rate of interest over a future time period implied by today’s zero-coupon bond market price.

Instantaneous Forward Rates

In the limit as we decrease the length of time between maturity dates T_2 - T_1 \rightarrow 0 , we can arrive at a formula for the instantaneous forward rate:

f(t,T) = -\frac{1}{B(t,T)}\frac{\partial}{\partial T}B(t,T)

Note our use of f for both versions of the forward rate. Now, integrating this formula between times t and T we obtain the formula for the instantaneous bond pricing formula:

B(t,T) = \exp\left(-\int_t^T f(t,u)du\right)

This equation says that the price of a bond can be recovered from the knowledge of the term structure of the forward rate, or equivalently, if you know the instantaneous forward rates for each future time up to maturity time T then you can derive today’s price of a bond over the same time period with the same maturity date.