Some of the popular traded fixed income instruments are bonds, swaps, futures and forwards.

One of the major differences in modelling interest rates as opposed to equities is that the current interest rate is a function of the maturity of the bond, i.e. it is a time-indexed curve as opposed to a number.

The initial instantaneous forward rate curve $f(0,T)$ can be inferred from market bond prices $P(0,T)$ and other fixed income instruments. However, the instruments are only available for certain tenors, and the interpolation of curve to other points is a central problem in rate curve modelling.

Heath-Jarrow-Morton framework provides restrictions on modelling of the diffusions of instantaneous forward rates so that they are arbitrage-free.

The framework assumes that the price process of zero coupon bond of each maturity follows the following diffusion in risk neutral measure:

where ${\nu }_{(}t,T)$ can be any ${ℱ}_t$ adapted process.

Note that we have already imposed the no-arbitrage condition on bond prices by setting the drift term to be $r(t)P(t,T)$. $v(t,T)$ specifies a volatility curve indexed with respect to the maturity of the bond T and is called the volatility term structure.

Now let’s see what this form of bond diffusion implies for the instantaneous forward rate.

We have therefore derived the drift of the instantaneous forward rate curve in terms of the volatility term structure.

If $\sigma (t,T)$ is deterministic then the instantaneous forward rates have Gaussian distribution, and the bond prices have log-normal distribution. The models in this case are called Gaussian HJM models.

Note that we have used a single Brownian function to generate diffusion of the entire instantaneous forward rate curve. This implies, in Gaussian case, that different points on instantaneous forward curve move in same direction in proportion to their volatility, on action of Brownian motion, and hence are perfectly correlated. This correlation can be relaxed by using multiple Brownian motions with different volatility term structure coeffecients.

Consider one-factor HJM diffusion. Instrument pricing is much more tractable if we restrict the volatility term-structure to be deterministic, separable and of the form:

Specifically, in Hull-White model we have time-independent $\alpha (s)=\alpha$ and $\sigma (t)=\sigma$.

In this case,

Using Ito isometry, $f(t,T)$ therefore is normally distributed as

where $N(\mu ,{\sigma }^2)$ is a normal distribution with mean $\mu$ and variance ${\sigma }^2$.

We then have

Therefore,

and the bond price

is log-normally distributed.

The Hull-White model can be specified completely in terms of the short rate diffusion as well, and is therefore called a short-rate model. Let us see how this re-parametrization works.

Notice here that the stochastic evolution of both ${\int }_t^{T}f(t,s)𝑑s$ and $r(t)$ is determined by the term ${\int }_0^t{e}^{\alpha u}𝑑W_u$ and therefore the bond yields $\ln P(t,T)$ can be specified as a linear function of $r(t)$ with coeffecients of this linear relationship being deterministic functions of t and T. Models with such linear relationship between short rate and yield term structure are known as affine term structure models.

where $A_i(t,T)$ and $B_i(t,T)$ are deterministic functions of t and T.

Getting back to short rate diffusion specification, we have

where we have introduced the parameter $\theta (t)$ which is specified of terms of other deterministic parameters $\alpha$, $\sigma$ and initial instantaneous forward rate curve $f(0,\cdot )$. From this specification, we have another interpretation of $\alpha$ as the mean-reversion strength of short rate.

In this section we will work through calculations to compute the difference between LIBOR futures and LIBOR forwards. LIBOR rates were recently decommisioned but the calculations are instructive and the logic can be transferred to pricing of futures and forwards on other benchmark rates.

Notice that $L(t,T,T+\delta )$ is a ratio of two prices plus a constant, and scaled by a constant. Therefore it is martingale in measure with $P(t,T)$ as numeraire i.e. T-forward measure. Therefore

is known as forward LIBOR rate.

Eurodollar futures are settled in market at value $1-L(T,T+\delta )$ but for ease of comparison we will take futures value at T $\text{Fut}(T)$ to be $L(T,T+\delta )$. We can get the true futures value from $1-\text{Fut}(0)$.

Futures contracts on LIBOR are different from forwards in two major ways.

1. 1.

   Futures contracts are daily margined. That is, the buyer does not pay for the contract initially, except for initial margin which is usually much smaller than the futures price. Then the movement in futures prices are settled daily; if the futures price goes up, the buyer has cash put in his margin account and if the futures price goes down, the buyer has cash withdrawn from this account.

2. 2.

   Futures contracts are settled at the maturity of futures contract, which is start of the LIBOR period T, as opposed to forwards which are settled at end of LIBOR period $T+\delta$.

The futures price $\text{Fut}(t)$ is such that price of future cashflows from daily margining is 0 i.e. for all t

where $\frac{\partial }{\partial s}\text{Fut}(s)ds$ is the cashflow from margin in the time interval $(s,s+ds)$. Differentiating with respect to t,

Integrating and using iterated condititioning, we get

which says that the current futures price is the expected settlement price of the future in risk neutral measure.

The future-forward basis is largely sensitive to the covariance between rate fixing and money market discount factor.

where ${\text{Cov}}^{\mathbb{P}}(X,Y)$ is the covariance of X and Y in risk neutral measure.

Note the sign. The covariance itself is negative: when rates rise the fixing $L(T,T,T+\delta )$ rises, while both the bond price $P(T,T+\delta )$ and the money market discount factor $D_T$ fall, so the Radon-Nikodym density $P(T,T+\delta )D_T/P(0,T+\delta )$ moves against the fixing. With the leading minus sign the basis is therefore positive, and the futures rate sits above the forward rate. This is the precise sense in which daily margining is worth something to the buyer of a future: the contract pays the fixing undiscounted, so the cash it throws off is largest exactly when it is worth least to reinvest.

We now compute the basis in Hull-White model.

We therefore only need to compute

We know that ${\int }_T^{T+\delta }f(T,s)𝑑s$ is normally distributed, say with mean $\mu$ and variance $V$ which were derived earlier in expression 5.1.

where we used the fact that $E[e^{\sigma W_t}]=E[e^{\frac{{\sigma }^{2}t}{2}}]$ that is the expecation of exponential of normal process with mean 0 is exponential of half of its variance. This can be verified by taking a log normal martingale process $d{X}_t/X_t=\sigma d{W}_t$ and using the property $E[X_T]=X_0$.

We now carry out the remaining algebra. Setting $t=T$ and replacing the upper limit $T$ by $T+\delta$ in expression 5.1, and abbreviating

so that $1-e^{-2\alpha T}=u(2-u)$ and $1-e^{-2\alpha \delta }=v(2-v)$, we get

It is convenient to name the part of the mean that is not already contained in today’s curve,

Since $P(0,T)=e^{-{\int }_0^{T}f(0,s)𝑑s}$, today’s curve gives

and therefore

which is the future-forward basis in the Hull-White model. Since $u,v\in (0,1)$ we have $2u+2v-uv>0$ and $u(2-u)>0$, so both $M$ and $V$ are strictly positive and the basis is strictly positive: the futures rate always sits above the forward rate, as the covariance argument above anticipated.

The Ho-Lee model is the $\alpha \to 0$ limit, in which the forward rate volatility $\sigma (t,T)=\sigma$ no longer decays with maturity. Using $u\to \alpha T$ and $v\to \alpha \delta$,

Both $M$ and $V$ are of order ${\sigma }^2{T}^2$ and hence small, so $e^{M+V/2}-1\approx M+V/2$; and $\delta L(0,T,T+\delta )\ll 1$, so the leading factor is approximately $1/\delta$. The Ho-Lee basis is therefore

From here on, we assume the notional to be 1. The prices can simply be scaled by notional to incorporate the notional.

Let ${\tau }_{i>0,i\le N_r}^r$ and ${\tau }_{i>0,i\le N_p}^p$ be the payment dates of receiver (fixed) leg and payer (floating) leg respectively with ${\tau }_0^r$ and ${\tau }_0^p$ being the swap start date. ${\delta }^r={\tau }_i^r-{\tau }_{i-1}^r$ is the periodicity of the receiver leg and ${\delta }^p={\tau }_i^p-{\tau }_{i-1}^p$ is the periodicity of the payer leg. The specified rate of the fixed leg is denoted as R.

The swap holder receives ${\delta }^{r}R$ at ${\tau }_{i>0}^r$ and pays ${\delta }^{p}L({\tau }_{i-1}^p,{\tau }_{i-1}^p,{\tau }_i^p)$ at ${\tau }_{i>0}^p$.

Price of float leg is

Price of fixed leg is then