Forward rate market models, financial engineering for quants lesson

A product whose payoff depends on several future rate fixings at once, such as a coupon linked to two consecutive half-year rates, needs the joint behavior of several simple forward rates, for example the rates for the half-year periods that start in one year and in one and a half years. Each of these rates has no drift when values are measured in units of the bond that pays on its own payment date, but a simulation of all of them has to run under one common measure. This lesson defines the simple forward rate as a bond ratio, builds the lognormal market model under the own forward measures, derives the drift that each rate acquires under the terminal measure, and explains why a backward-looking compounded overnight rate is a different contract.

Key terms

Simple forward rate

Definition Fix tenor dates $0 \le T_0 < T_1 < \dots < T_N$ and the accrual fractions $\delta_i > 0$ of the periods $[T_i, T_{i+1}]$ under the contract’s day count, for $i = 0, \dots, N-1$ . In a single-curve market of default-free zero-coupon bonds with positive prices $P(t,T)$ , the simple forward rate of period $i$ at time $t \le T_i$ is

\begin{aligned} &L_i(t) \\[0.4em] &= \frac{1}{\delta_i}\Bigl(\frac{P(t,T_i)}{P(t,T_{i+1})} - 1\Bigr). \end{aligned}

Equivalently, the accrual factor is a bond ratio,

1 + \delta_i L_i(t) = \frac{P(t,T_i)}{P(t,T_{i+1})} .

At $t = T_i$ the rate becomes the fixing $L_i(T_i) = \bigl(1/P(T_i,T_{i+1}) - 1\bigr)/\delta_i$ , the simple rate for lending over the period, which is set at the start of the period and paid at its end. Before that, $L_i(t)$ is the fair fixed rate of a forward rate agreement on the period, as in Interest rate swaps and forward rate agreements. The accrual factor is always positive, so $L_i(t) > -1/\delta_i$ , while the rate itself is negative whenever $P(t,T_{i+1}) > P(t,T_i)$ . A rate index with credit or basis effects that is projected from its own curve is not given by this bond ratio.

In plain words The simple forward rate is the rate that can be locked in today for lending over a future period, read off from two discount factors. Its accrual factor says how much one unit lent at the start of the period grows to by the end.

Example On the constructed curve $P(0,1) = 0.96186$ , $P(0,1.5) = 0.943$ and $P(0,2) = 0.92$ with half-year accruals, the accrual factors are $0.96186/0.943 = 1.02$ and $0.943/0.92 = 1.025$ , so $L_0(0) = 0.04$ for the period $[1, 1.5]$ and $L_1(0) = 0.05$ for $[1.5, 2]$ . If instead $P(0,1) = 1.002$ and $P(0,1.5) = 1.004$ , the accrual factor is about $0.998008$ and the forward rate is about $-0.003984$ , negative but above $-1/0.5 = -2$ .

Own forward measure

Definition Let $\mathbb{Q}$ be a pricing measure with the money-market account $B$ as numeraire, and assume that the discounted bond prices $P(t,T_i)/B_t$ and $P(t,T_{i+1})/B_t$ are true martingales under $\mathbb{Q}$ . The own forward measure of $L_i$ is the $T_{i+1}$ -forward measure $\mathbb{Q}^{T_{i+1}}$ , whose numeraire is the bond that pays at the end of the period, with density

\begin{aligned} &\frac{d\mathbb{Q}^{T_{i+1}}}{d\mathbb{Q}}\Big\vert_{\mathcal{F}_t} \\[0.4em] &= \frac{P(t,T_{i+1})}{B_t\,P(0,T_{i+1})} . \end{aligned}

Under it, the price of a traded asset without intermediate payments, divided by $P(t,T_{i+1})$ , is a martingale whenever the discounted price of the asset is a true martingale under $\mathbb{Q}$ . The ratio $P(t,T_i)/P(t,T_{i+1})$ is such a price for $t \le T_i$ , and subtracting one and dividing by the fixed number $\delta_i$ keeps the martingale property, so

L_i(t) = \mathbb{E}^{\mathbb{Q}^{T_{i+1}}}\bigl[L_i(T_i) \mid \mathcal{F}_t\bigr]

for $0 \le t \le T_i$ . A payment $X$ at $T_{i+1}$ that is integrable under $\mathbb{Q}^{T_{i+1}}$ has the time- $t$ value $P(t,T_{i+1})\,\mathbb{E}^{\mathbb{Q}^{T_{i+1}}}[X \mid \mathcal{F}_t]$ . Each forward rate has its own measure, and in general a forward rate is not a martingale under the forward measure of a different payment date.

In plain words Measure every value in units of the bond that pays when the forward rate’s interest is paid. In those units the forward rate has no drift, so its expected fixing is today’s forward rate. Such expectations are pricing weights for one payment date, not forecasts of the rate.

Example On the constructed curve the own forward measure of $L_0$ is $\mathbb{Q}^{1.5}$ , and $\mathbb{E}^{\mathbb{Q}^{1.5}}[L_0(1)] = 0.04$ in every model with these martingale properties. A forward rate agreement with notional $1{,}000{,}000$ , accrual $0.5$ and fixed rate $0.035$ , which pays $1{,}000{,}000 \cdot 0.5\,\bigl(L_0(1) - 0.035\bigr)$ at time $1.5$ to the party that receives floating, is worth

\begin{aligned} &1{,}000{,}000 \cdot 0.5 \\[0.4em] &\quad \times 0.943 \cdot 0.005 \\[0.4em] &= 2357.5 \end{aligned}

to that party, whatever the volatility of the rate.

Forward rate market model

Definition A lognormal forward rate market model on the tenor dates takes positive initial forward rates $L_i(0)$ from today’s curve and, for each $i$ , a deterministic volatility vector $\lambda_i(t) \in \mathbb{R}^d$ on $[0, T_i]$ with $\int_0^{T_i} \lVert \lambda_i(t) \rVert^2\,dt < \infty$ , and assumes

\begin{gathered} \frac{dL_i(t)}{L_i(t)} = \lambda_i(t) \cdot dW^{i+1}_t , \\[0.4em] 0 \le t \le T_i , \end{gathered}

where $W^{i+1}$ is a $d$ -dimensional Brownian motion under the own forward measure $\mathbb{Q}^{T_{i+1}}$ . Each forward rate is then a driftless geometric Brownian motion under its own measure, and its fixing is lognormal,

\begin{gathered} L_i(T_i) = L_i(0)\,e^{-v_i^2/2 + v_i Z}, \\[0.4em] v_i^2 = \int_0^{T_i} \lVert \lambda_i(t) \rVert^2\,dt , \end{gathered}

with $Z$ standard normal under $\mathbb{Q}^{T_{i+1}}$ and $v_i \ge 0$ . The forward-rate correlation of $L_i$ and $L_j$ at time $t$ is

\rho_{ij}(t) = \frac{\lambda_i(t) \cdot \lambda_j(t)}{\lVert \lambda_i(t) \rVert\,\lVert \lambda_j(t) \rVert}

when both volatilities are nonzero. In scalar form, with $\sigma_i(t) = \lVert \lambda_i(t) \rVert$ and one Brownian motion per rate, $\lambda_i \cdot \lambda_j = \sigma_i \sigma_j \rho_{ij}$ , and the matrix $(\rho_{ij})$ must be symmetric and positive semidefinite with ones on its diagonal. A change to an equivalent measure alters drifts but not quadratic covariation, so the correlations are the same under every measure of the model. The equations for different rates are written under different measures, and the next two cards put them under one. The lognormal specification keeps every forward rate positive, so it cannot describe a negative forward rate without a shift, a normal specification or a different model.

In plain words The market model treats each forward rate like a lognormal stock price with its own volatility, valued in units of its own payment bond. The correlations say how the rates of different periods move together. The price of a single caplet does not depend on them, while products that depend on several fixings jointly do.

Example With $d = 2$ , take $\lambda_0 = 0.2\,(1, 0)$ and $\lambda_1 = 0.25\,(0.6, 0.8)$ , in units per square root year. Then $\sigma_0 = 0.2$ , $\sigma_1 = 0.25$ and

\begin{aligned} \lambda_0 \cdot \lambda_1 &= 0.2 \cdot 0.25 \cdot 0.6 \\[0.4em] &= 0.03 \end{aligned}

per year, so $\rho_{01} = 0.6$ . For $L_0$ with $T_0 = 1$ , $v_0^2 = 0.04$ , so under $\mathbb{Q}^{1.5}$ the fixing $L_0(1)$ has mean $0.04$ and median $0.04\,e^{-0.02} \approx 0.039208$ .

Terminal measure

Definition The terminal measure $\mathbb{Q}^N = \mathbb{Q}^{T_N}$ is the forward measure of the bond that matures on the last tenor date, assuming $P(t,T_N)/B_t$ is a true martingale under $\mathbb{Q}$ . For $t \le T_i$ the density of the own measure $\mathbb{Q}^{T_{i+1}}$ with respect to $\mathbb{Q}^N$ is the bond ratio $P(t,T_{i+1})/P(t,T_N)$ divided by its value at time $0$ . That ratio is the price of a traded bond in units of the numeraire, so it is a positive martingale under $\mathbb{Q}^N$ , and it telescopes into the accrual factors of the later forward rates,

\begin{aligned} &\frac{P(t,T_{i+1})}{P(t,T_N)} \\[0.4em] &= \prod_{j=i+1}^{N-1} \bigl(1 + \delta_j L_j(t)\bigr), \end{aligned}

an empty product equal to $1$ when $i = N-1$ . In the market model each factor has the proportional Brownian loading $\delta_j L_j \lambda_j / (1 + \delta_j L_j)$ , and the loadings of a product add, so the density has the loading

\gamma_i(t) = \sum_{j=i+1}^{N-1} \frac{\delta_j L_j(t)\,\lambda_j(t)}{1 + \delta_j L_j(t)} .

With the density convention of the lesson on numeraires and forward measures, Girsanov’s theorem gives

W^{i+1}_t = W^N_t - \int_0^t \gamma_i(s)\,ds ,

where $W^N$ is a Brownian motion under the terminal measure. Only the rates paid after $T_{i+1}$ enter, not the earlier ones.

In plain words One numeraire, the longest bond, serves all the rates at once. Moving from a rate’s own measure to the terminal measure reweights the paths by the ratio of the two bonds, and that ratio depends only on the forward rates of the periods after the rate’s own payment date.

Example For the constructed two-period curve, $N = 2$ and the terminal measure is $\mathbb{Q}^{2}$ , which is already the own measure of $L_1$ . The density from $\mathbb{Q}^{2}$ to $\mathbb{Q}^{1.5}$ , the own measure of $L_0$ , is $\bigl(1 + 0.5\,L_1(t)\bigr)/1.025$ , since $P(0,1.5)/P(0,2) = 0.943/0.92 = 1.025$ . Paths on which the later rate $L_1$ is high get more weight under $\mathbb{Q}^{1.5}$ than under $\mathbb{Q}^{2}$ . At time $0$ the loading is $\gamma_0 = (0.025/1.025)\,\lambda_1 \approx 0.024390\,\lambda_1$ .

Terminal-measure drift

Definition Substituting the Girsanov shift into the own-measure equation of the market model gives, for $i = 0, \dots, N-1$ and $0 \le t \le T_i$ ,

\begin{aligned} &\frac{dL_i(t)}{L_i(t)} = \mu_i^N(t)\,dt \\[0.4em] &\qquad + \lambda_i(t) \cdot dW^N_t , \end{aligned}

with the terminal-measure drift

\begin{aligned} &\mu_i^N(t) \\[0.4em] &= -\sum_{j=i+1}^{N-1} \frac{\delta_j L_j}{1 + \delta_j L_j}\,\lambda_i \cdot \lambda_j , \end{aligned}

where every quantity on the right is taken at time $t$ . In scalar form $\lambda_i \cdot \lambda_j = \sigma_i \sigma_j \rho_{ij}$ , so the drift is built from the forward-rate correlations. The last rate $L_{N-1}$ has an empty sum and no drift, because the terminal measure is its own measure, and when every $\lambda_i \cdot \lambda_j$ is positive all the other rates have negative drifts. The drift depends on the current levels of the later rates, so the system is coupled, and in general the rates other than $L_{N-1}$ are not lognormal under $\mathbb{Q}^N$ . For bounded deterministic $\lambda_i$ , the equations for $\ln L_i$ have bounded Lipschitz coefficients, so the system has a unique solution and every rate stays positive. Replacing $L_j(t)$ by $L_j(0)$ in the weights, called freezing the drift, gives an approximation, not the model. The formula is for classical single-curve term rates that are fixed at the start of their periods.

In plain words Under the terminal measure each rate is pulled by its covariance with the rates paid after it, each weighted by how strongly its accrual factor moves the bond ratio. The rate paid last needs no correction, because the terminal bond is its own payment bond.

Example Take the constructed values $L_0 = 0.04$ , $L_1 = 0.05$ , $\delta_1 = 0.5$ , $\sigma_0 = 0.2$ , $\sigma_1 = 0.25$ and $\rho_{01} = 0.6$ . The weight is $0.025/1.025 \approx 0.024390$ and $\lambda_0 \cdot \lambda_1 = 0.03$ per year, so the proportional drift of $L_0$ under $\mathbb{Q}^{2}$ is

\begin{aligned} \mu_0^N &\approx -0.024390 \cdot 0.03 \\[0.4em] &\approx -0.000731707 \end{aligned}

per year. Multiplying by the level $0.04$ gives an absolute drift of about $-0.0000292683$ per year, roughly $-0.29$ basis points per year, a change in the rate itself rather than a proportional change. The rate $L_1$ has no drift, and with $\rho_{01} = 0$ the drift of $L_0$ would vanish as well. Freezing the drift approximates $\mathbb{E}^{\mathbb{Q}^{2}}[L_0(1)]$ by $0.04\,e^{-0.000731707} \approx 0.039971$ , slightly below the own-measure expectation $0.04$ .

Backward-looking overnight rate

Definition Let the accrual period $[T_i, T_{i+1}]$ contain the business days $k = 1, \dots, m$ , each with an overnight fixing $r_k$ that applies for $d_k$ calendar days, and let the day-count basis be $360$ with $D = d_1 + \dots + d_m$ and $\delta_i = D/360$ . The backward-looking compounded overnight rate $R_i$ of the period is defined by

\begin{aligned} &1 + \delta_i R_i \\[0.4em] &= \prod_{k=1}^{m} \Bigl(1 + \frac{r_k\,d_k}{360}\Bigr) . \end{aligned}

It is built from the fixings of the whole period, so it is known only at the end of the period, or a few days earlier with a lookback or an observation shift, and its payment may come later still. As a random variable it is measurable with respect to $\mathcal{F}_{T_{i+1}}$ , not $\mathcal{F}_{T_i}$ . Day counts, holiday calendars, lookbacks, observation shifts and payment lags are part of the contract’s definition. In an idealized single-curve model in which overnight lending accrues continuously at the short rate, so that $1 + \delta_i R_i = B_{T_{i+1}}/B_{T_i}$ , and discounted bonds are true martingales, the time- $t$ value of the payment $\delta_i R_i$ at $T_{i+1}$ is, for $t \le T_i$ ,

\begin{aligned} &B_t\,\mathbb{E}^{\mathbb{Q}}\Bigl[\frac{1}{B_{T_i}} - \frac{1}{B_{T_{i+1}}} \mid \mathcal{F}_t\Bigr] \\[0.4em] &= P(t,T_i) - P(t,T_{i+1}) , \end{aligned}

so the conditional expectation of $R_i$ under $\mathbb{Q}^{T_{i+1}}$ given $\mathcal{F}_t$ equals the term forward rate $L_i(t)$ . The two rates are nevertheless different random variables. The term rate is fixed at $T_i$ , while the compounded rate keeps changing through the accrual period, so an option on it has a different payoff and a longer period of uncertainty, and the market model dynamics, written for $t \le T_i$ , do not describe it during the period. With collateral and several curves, its forward rate has to come from the relevant curve rather than from this bond ratio.

In plain words A term rate is agreed at the start of its period. A backward-looking rate compounds the overnight rates that are actually realized during the period, so it is known only once the period is over.

Example In a constructed week the overnight fixings from Monday to Friday are $0.0400$ , $0.0410$ , $0.0405$ , $0.0395$ and $0.0400$ , and the Friday fixing applies for three calendar days, so $D = 7$ . Compounding gives $R \approx 0.0401544$ , slightly above the day-weighted average $0.281/7 \approx 0.0401429$ . Neither number is known before the last fixing of the week.

Check yourself

The constructed bond prices are $P(0,1) = 0.96096$ , $P(0,2) = 0.924$ and $P(0,3) = 0.88$ , with annual accruals $\delta_0 = \delta_1 = 1$ for the periods $[1,2]$ and $[2,3]$ . Find $L_0(0)$ and $L_1(0)$ , the weight of $L_1$ in the terminal-measure drift of $L_0$ , and that drift at time $0$ when $\sigma_0 = 0.18$ , $\sigma_1 = 0.22$ and $\rho_{01} = 0.75$ . Which rate is driftless under the terminal measure?

Show the answer

The forward rates are $L_0(0) = 0.96096/0.924 - 1 = 0.04$ and $L_1(0) = 0.924/0.88 - 1 = 0.05$ . The terminal measure is $\mathbb{Q}^{3}$ , and the weight of $L_1$ is

\begin{aligned} \frac{\delta_1 L_1}{1 + \delta_1 L_1} &= \frac{0.05}{1.05} = \frac{1}{21} \\[0.4em] &\approx 0.047619 , \end{aligned}

which also equals $1 - P(0,3)/P(0,2) = 1 - 0.88/0.924$ . With $\lambda_0 \cdot \lambda_1 = 0.18 \cdot 0.22 \cdot 0.75 = 0.0297$ per year, the proportional drift of $L_0$ is $-0.0297/21 \approx -0.00141429$ per year, and the absolute drift is $0.04$ times that, about $-0.0000565714$ per year, or $-0.57$ basis points per year. The rate $L_1$ is paid at $T_2 = 3$ , so the terminal measure is its own measure and it has no drift there.

Three forward rates with half-year accruals have $L_0 = 0.03$ , $L_1 = 0.035$ and $L_2 = 0.04$ , volatilities $\sigma_0 = 0.2$ , $\sigma_1 = 0.22$ and $\sigma_2 = 0.24$ , and correlations $\rho_{01} = 0.9$ , $\rho_{02} = 0.8$ and $\rho_{12} = 0.9$ . Find the proportional drifts of all three rates under the terminal measure $\mathbb{Q}^{T_3}$ . A colleague sums over the earlier rates $j < i$ instead and concludes that $L_0$ is driftless while $L_2$ has the largest drift. Explain the mistake.

Show the answer

The correlation matrix has eigenvalues of about $0.0658$ , $0.2$ and $2.7342$ , all positive, so it is a valid correlation matrix. The weights are $0.5 \cdot 0.035/1.0175 \approx 0.0171990$ for $L_1$ and $0.5 \cdot 0.04/1.02 \approx 0.0196078$ for $L_2$ . With $0.2 \cdot 0.22 \cdot 0.9 = 0.0396$ and $0.2 \cdot 0.24 \cdot 0.8 = 0.0384$ , the two terms of the drift of $L_0$ are $0.0171990 \cdot 0.0396 \approx 0.00068108$ and $0.0196078 \cdot 0.0384 \approx 0.00075294$ , so

\begin{aligned} \mu_0^N &\approx -0.00068108 \\[0.4em] &\quad - 0.00075294 \\[0.4em] &\approx -0.00143402 , \end{aligned}

and $\mu_1^N = -0.0196078 \cdot 0.04752 \approx -0.000931765$ , using $0.22 \cdot 0.24 \cdot 0.9 = 0.04752$ , all per year. The rate $L_2$ has $\mu_2^N = 0$ . The colleague’s sum would give $0$ , about $-0.000585$ and about $-0.001385$ , which reverses the pattern. The density from the terminal measure to the own measure of $L_i$ is the ratio $P(t,T_{i+1})/P(t,T_3)$ , the product of the accrual factors of the periods after $T_{i+1}$ . The earlier rates do not appear in it, and the last rate, whose own measure is the terminal measure, has no drift at all.

Use the idealized single-curve model of the last card with the constructed curve $P(0,1) = 0.96186$ and $P(0,1.5) = 0.943$ . Find the value of the payment $1{,}000{,}000 \cdot 0.5\,R_0$ at time $1.5$ , where $R_0$ is the compounded overnight rate of $[1, 1.5]$ , and compare it with the term-rate payment $1{,}000{,}000 \cdot 0.5\,L_0(1)$ . Show that $\mathbb{E}^{\mathbb{Q}^{1.5}}[R_0 \mid \mathcal{F}_1] = L_0(1)$ , and explain why a caplet on $R_0$ with strike $K$ is worth at least as much as a caplet on $L_0$ with the same strike and payment date. What is wrong with pricing it by Black’s formula with time to fixing $1$ ?

Show the answer

The value of $\delta_0 R_0$ paid at time $1.5$ is $P(0,1) - P(0,1.5) = 0.01886$ per unit of notional, so the payment is worth $18{,}860$ . The term-rate payment is worth $1{,}000{,}000 \cdot 0.5 \cdot 0.04 \cdot 0.943 = 18{,}860$ as well. Applying the value formula at $t = 1$ and dividing by $P(1,1.5)$ gives

\begin{aligned} &\delta_0\,\mathbb{E}^{\mathbb{Q}^{1.5}}[R_0 \mid \mathcal{F}_1] \\[0.4em] &= \frac{1 - P(1,1.5)}{P(1,1.5)} = \delta_0 L_0(1) . \end{aligned}

The function $x \mapsto (x - K)^{+}$ is convex, so the conditional Jensen inequality gives

\begin{aligned} &\mathbb{E}^{\mathbb{Q}^{1.5}}\bigl[(R_0 - K)^{+} \mid \mathcal{F}_1\bigr] \\[0.4em] &\ge \bigl(L_0(1) - K\bigr)^{+} . \end{aligned}

Taking expectations under $\mathbb{Q}^{1.5}$ and multiplying by $1{,}000{,}000 \cdot 0.5 \cdot P(0,1.5)$ shows that the caplet on $R_0$ is worth at least the term caplet. The two values are equal only when, almost surely, the conditional distribution of $R_0$ given the information at time $1$ puts all its mass on one side of $K$ . Black’s formula with time to fixing $1$ treats $R_0$ as if it were fixed at the start of the period. It ignores the uncertainty that the overnight fixings add between times $1$ and $1.5$ , and so it prices a different payoff.

Forward rate market models

Key terms

Simple forward rate

Own forward measure

Forward rate market model

Terminal measure

Terminal-measure drift

Backward-looking overnight rate

Check yourself

How this lesson connects

Builds on

Used later in