The LIBOR Market Model (LMM)

The LMM is used when calculating the prices and risks of caps, swaptions, and Bermudan swaptions.

Moving the forward rate for a given period directly

The LIBOR market model (LMM) takes the forward rate, quoted as a simple rate for a fixed period, as its variable. The period is usually 3 months (0.25 years) or 6 months (0.5 years)¹. The rate for this period follows from bond prices. A zero-coupon bond pays just 1 at maturity. Write $Z_i(t)$ for the price at time $t$ of a zero-coupon bond maturing at $T_i$, and $\delta_i = T_{i+1} - T_i$ for the length of the period from date $T_i$ to date $T_{i+1}$. The forward rate $L_i(t)$ for that period is as follows².

$$L_i(t) = \frac{1}{\delta_i}\left( \frac{Z_i(t)}{Z_{i+1}(t)} - 1 \right)$$

$Z_i(t)/Z_{i+1}(t)$ is the value at $T_{i+1}$ of 1 deposited at $T_i$, with interest, and it equals $1 + \delta_i L_i(t)$. The value of $L_i(t)$ is fixed when $T_i$ arrives.

Consider a hypothetical example in which, at $t = 0$, the zero-coupon bond maturing at $T_1$ has price $Z_1 = 0.99$ and the one maturing at $T_2 = T_1 + 0.5$ years has price $Z_2 = 0.98$. Then $\delta_1 = 0.5$ and $L_1(0) = (0.99/0.98 - 1)/0.5 = 2.04\%$ (a different value from the later worked example).

A short-rate model moves the instantaneous interest rate (the short rate), which cannot be observed directly in the market, and derives bond prices from its movement. The LMM moves the rates for a given period that can be observed in the market, and builds bond prices from the forward rates³. The LMM's variables are the rates that underlie caplets and similar products. The model can therefore move the same quantity that Black's formula uses⁴.

The LIBOR in the name is an example of the rate the model moves. The same form also applies to Euribor⁵.

The model equation

Fix in advance the finite set of dates that the forward rates move over: $0 = T_0 < T_1 < \dots < T_{M+1}$. The period lengths $\delta_i$ differ slightly from one another because of day-count conventions. The prices of many products, including caps, swaps, swaptions, and Bermudan swaptions, depend only on the rates at this finite set of dates².

The figure below lays out these dates and the times at which the values of $L_i$ are fixed, for the same three forward rates (at half-year steps) as in the later worked example.

Figure description

L1 in the figure is $L_1$ in the text, and T1 is $T_1$ (figures cannot contain equations, so this note uses the figure's notation). As time passes, fewer forward rates are not yet fixed. Today, L1, L2, and L3 are all not yet fixed; $L_i$ is fixed at $T_i$, and its period ends at $T_{i+1}$. The spacing in the figure does not represent the length of time. The filled dots mark the times at which a value is fixed.

The LMM assumes that each forward rate, for $i = 1, \dots, M$, obeys the stochastic differential equation below⁶.

$$dL_i(t) = \mu_i(t)\, L_i(t)\, dt + L_i(t)\, \sigma_i(t)^{\top} dW(t), \qquad 0 \le t \le T_i$$

$W(t)$ is a $d$-dimensional Brownian motion. $\sigma_i(t)$ is a $d$-dimensional vector, and its length $\|\sigma_i(t)\|$ is the volatility of the rate of change of $L_i$. $\mu_i(t)$ is the drift (the expected movement of the rate of change). Both may depend on the information known up to time $t$, including the current set of forward rates⁶. The covariance of the rates of change of two forward rates $L_i$ and $L_j$ is $\sigma_i^{\top}\sigma_j$. The correlation divides this by the product of the standard deviations, as follows⁷.

$$\rho_{ij}(t) = \frac{\sigma_i(t)^{\top}\sigma_j(t)}{\|\sigma_i(t)\|\,\|\sigma_j(t)\|}$$

The volatility $\sigma_i$ can be chosen freely, but the drift $\mu_i$ cannot. The next section shows the form that the no-arbitrage condition fixes.

The drift follows from the no-arbitrage condition

Under the risk-neutral measure (see Risk-Neutral Valuation), a bond price divided by the price of a reference asset (the numeraire) is a martingale, a stochastic process whose expected future value equals its current value⁸. The LMM sets the drift of each forward rate so that this condition holds⁶. The formula for the drift of the same $L_i$ changes with the choice of numeraire.

The LMM does not use an account that accumulates the short rate as the numeraire. That would require the value of the short rate at every point in time, which weakens the point of moving the forward rates directly⁹. Two choices are common instead⁸.

• Under the spot measure, the numeraire is an asset that invests 1 in the bond maturing at $T_1$ and, at each maturity, rolls into the next bond.
• Under a forward measure, the numeraire is the bond maturing at a chosen date.

Under the spot measure, the drift of $L_i$ is as follows¹⁰.

$$\mu_i(t) = \sum_{j=\varphi(t)}^{i} \frac{\delta_j L_j(t)\, \sigma_i(t)^{\top}\sigma_j(t)}{1 + \delta_j L_j(t)}$$

$\varphi(t)$ is the number of the date right after time $t$ (the smallest $i$ with $t < T_i$). The sum includes only those $L_j$ whose values are not yet fixed and for which $j \le i$.

Under the forward measure that takes the bond maturing at the last date $T_{M+1}$ as numeraire, the Brownian motion is replaced by $W^{M+1}(t)$, and the drift is as follows¹⁰.

$$\mu_i(t) = -\sum_{j=i+1}^{M} \frac{\delta_j L_j(t)\, \sigma_i(t)^{\top}\sigma_j(t)}{1 + \delta_j L_j(t)}$$

The range of the sum is reversed and the sign changes. For $i = M$ the sum is empty and the drift is 0. In both formulas, the drift depends on the current values of the forward rates.

Deep diveHow the drift formula is derived

Let $D_i(t)$ be a bond price divided by the numeraire of the spot measure. By the no-arbitrage condition, $D_i$ is a martingale, so it can be written as $dD_i = D_i\,\eta_i^{\top} dW$ for some $d$-dimensional process $\eta_i(t)$. Applying Ito's lemma to $\ln D_i$, the volatility term determines $\eta_i$.

$$\eta_i(t) = -\sum_{j=\varphi(t)}^{i-1} \frac{\delta_j L_j(t)\, \sigma_j(t)}{1 + \delta_j L_j(t)}$$

Comparing the drift terms gives $\mu_i(t) = -\sigma_i(t)^{\top}\eta_{i+1}(t)$. Substituting the $\eta_{i+1}$ above gives the drift formula under the spot measure. Comparing the drift terms directly is messy, so induction is cleaner¹⁰.

Worked example: the size of the drift

Consider a hypothetical example in which the discount rate is 2% a year, continuously compounded (a flat curve), and $Z_i(t)$ is $e^{-0.02(T_i - t)}$. The period is 6 months ($\delta_i = 0.5$) and the dates are $T_i = 0.5\,i$ years. The three forward rates $L_1$, $L_2$, and $L_3$ at $t = 0$ then all equal $(e^{0.01} - 1) / 0.5 = 2.0100\%$. Volatility is 20% for all three, and the correlation is 0.9 for every pair. $\delta_j L_j/(1 + \delta_j L_j)$ is 0.009950 for each¹¹.

The drift of $L_2$ under the spot measure is $0.009950 \times 0.2 \times 0.2 \times (0.9 + 1) = 0.000756$, or 0.0756% a year. The correlation of $L_2$ with itself is 1 and with $L_1$ is 0.9, so the two terms for $j = 1, 2$ are added. For $L_1$ under the forward measure that takes the bond maturing at $T_4 = 2$ years as numeraire, the two terms for $j = 2, 3$ are added, giving $-0.009950 \times 0.2 \times 0.2 \times (0.9 + 0.9) = -0.000716$.

The drift is a proportion of $L_i$. In terms of the level of the rate, the spot-measure drift of $L_1$ is 0.0008% a year (0.08bp). The drift has opposite signs under the two measures. A Monte Carlo simulation can be run under whichever measure you like¹².

Why caplets match Black's formula

An interest rate cap is a bundle of period-by-period options called caplets. A caplet with a notional amount of 1 pays, at time $T_{i+1}$, $\delta_i$ times the amount by which $L_i(T_i)$, fixed at time $T_i$, exceeds the strike rate $K$. Taking the bond maturing at the payment date $T_{i+1}$ as numeraire, the price is the price of that bond times the expected payoff⁴.

In the market, this price is expressed with the formula of the Black model. The formula is in Caps and Floors and is not repeated here. Its discount factor $P(0, T+\delta)$ is $Z_{i+1}(0)$ on this page, its volatility $\sigma_B$ is $\bar\sigma_i$, defined in the next paragraph, and its forward rate $F$ is $L_i(0)$¹³.

Black's formula began as a framework for quoting prices in the market, not as a price derived from the no-arbitrage condition of a specific model. The LMM shows this formula as a result of an arbitrage-free model¹⁴.

Under the forward measure that takes the bond maturing at the payment date $T_{i+1}$ as numeraire, the drift of $L_i$ is 0. This is the same as the last forward rate having zero drift under the terminal forward measure in the previous section. Choosing $T_{i+1}$ as the last date again makes this hold for every forward rate¹⁵.

$$dL_i(t) = L_i(t)\, \sigma_i(t)^{\top} dW^{i+1}(t)$$

If $\sigma_i(t)$ is a function of time only (deterministic), $L_i(T_i)$ is lognormally distributed, and the variance of $\ln L_i(T_i)$ is $\int_0^{T_i}\|\sigma_i(s)\|^2 ds$. Set the volatility of the Black model to $\bar\sigma_i^2 = \frac{1}{T_i}\int_0^{T_i}\|\sigma_i(s)\|^2 ds$, and the caplet price equals the Black formula in Caps and Floors¹³. In other words, the volatility in Black's formula is the square root of the time average of the squared LMM volatility. For example, if volatility is 25% for the first 0.75 years and 15% for the remaining 0.75 years, then $\bar\sigma_i^2 = (0.25^2 \times 0.75 + 0.15^2 \times 0.75)/1.5 = 0.0425$ and $\bar\sigma_i = 20.6155\%$.

In the market, a cap is quoted by adding up the prices of its caplets at the same volatility¹³. In the LMM, each caplet matches Black's formula under the forward measure of its own payment date. If you put everything under a single measure, the forward rates other than the last one are not martingales and keep a drift. Replacing $L_j(t)$ in the drift with the initial value $L_j(0)$ gives a lognormal approximation¹⁶.

Worked example: the price of one caplet

Consider a hypothetical example in which a caplet is written on $L_3$ from the previous example. It pays, 2 years from now, the amount by which the 6-month rate fixed 1.5 years from now exceeds the strike rate of 2.00%. The notional amount is 10 billion yen, the volatility $\bar\sigma_3$ is a constant 20%, and the discount factor to year 2 is $e^{-0.04} = 0.960789$¹⁷. $\Phi$ is the standard normal cumulative distribution function, and $d_1$ and $d_2$ are the values of the Caps and Floors formula with $\sigma_B = \bar\sigma_3$ and $F = L_3(0)$.

At the initial values, the Black formula price and the Monte Carlo price agree within the standard error, and the same holds when switching to the terminal forward measure. The buyer pays 9,631,905 yen. If the rate fixed 1.5 years from now is 2.50%, the buyer receives 10 billion yen × 0.5 × 0.50% = 25 million yen 2 years from now, which exceeds the price paid. If the rate is 2.00% or lower, the buyer receives nothing, and the price paid is a full loss. The seller's profit and loss is the opposite.

What correlation and calibration fit

Caplet prices depend only on the level of each forward rate's volatility, and correlation does not enter. The price of a swaption, on the other hand, also depends on the correlations between forward rates¹⁸. In calibration, which fits the model's parameters to market prices, the level of volatility is therefore fitted to market caplets. For the correlations, a common choice is to fit them to swaption prices or to historical correlations¹⁸. The procedure, the objective function, and overfitting are covered in the calibration article.

Using Monte Carlo simulation

Prices of caps, floors, and swaptions can be obtained from analytical formulas or approximations. The prices of other products are generally found by Monte Carlo simulation. In models where volatility itself moves randomly, even caps and swaptions need Monte Carlo simulation¹². The paper from the Institute for Monetary and Economic Studies of the Bank of Japan also says that the LMM is often implemented by Monte Carlo simulation and that its computational load is relatively heavy¹⁹.

One example of the procedure is as follows.

1. Fix the dates and the $L_i(0)$ obtained from the market curve.
2. Fix the volatilities $\sigma_i(t)$ and the correlations (the calibrated values).
3. Choose either the spot measure or the terminal forward measure, and compute the drift $\mu_i$ from the $L_j$ at the start of each step.
4. Update $\ln L_i$ using the step size $\Delta$ and a $d$-dimensional standard normal random vector $G$.
5. Divide the product's payoff by the numeraire of the chosen measure, and average over the paths.

The update formula in step 4 applies Ito's lemma to $\ln L_i$ and treats the drift and volatility as constant within the step (an example of the Euler method).

$$L_i(t+\Delta) = L_i(t) \exp\left\{ \left(\mu_i(t) - \tfrac{1}{2}\|\sigma_i(t)\|^2\right)\Delta + \sqrt{\Delta}\, \sigma_i(t)^{\top} G \right\}$$

The simulation uses a discretization such as the Euler method. Only a finite number of forward rates move, so even a fine step size is feasible to compute¹². Instead of discretizing the forward rates directly, you can also construct the bond prices divided by the numeraire as discrete-time martingales¹².

A product with several exercise dates, such as a Bermudan swaption, requires deciding, path by path, whether to exercise on each exercise date. This decision needs a technique such as the least-squares Monte Carlo method of Longstaff and Schwartz, which makes the computational load heavier still¹⁹. The product is covered in Bermudan Swaptions.

Assumptions and limits

LMM prices depend on how volatility and correlation are set and on the assumption that forward rates move in a nearly lognormal way. If the assumptions fail, the price departs from the market and the results read differently (model risk). The main limits are as follows.

If volatility is a function of time only, the Black volatility of a caplet is the same for every strike. In the market, this was roughly true until the mid-1990s, but afterward the volatility came to differ by strike, a pattern known as the smile²⁰. To handle this smile (see Volatility Smiles and Surfaces), extensions were developed in which volatility itself is stochastic. There are also CEV-type, shifted lognormal, lognormal mixture, and jump extensions²⁰. However, adding features to fit the smile makes calibration to market prices more complicated¹⁹.

Black's formulas for caplets and swaptions cannot both hold in one model. If the volatility of the forward rates is a function of time only, the volatility of the swap rate is not. The LMM matches Black's formula for caplets and agrees with Black's formula for swaptions only approximately²¹. How to prepare for changes in volatility also differs by model, so the strengths and weaknesses of each model matter²².

Notes

1. Martin Haugh, "Market Models" (Columbia University, Term Structure Models: IEOR E4710, 2010), page 2. LIBOR is a simple rate, and the period is usually $\delta = 1/4$ (3 months) or $1/2$ (6 months). ↩

2. Haugh, "Market Models", page 5, equation (9). The source's symbols $Z_t^n$, $L_n(t)$, and $n$ correspond to $Z_i(t)$, $L_i(t)$, and $i$ on this page. The reason for fixing a finite set of dates, and the difference in period lengths, are in the same place. ↩ ↩²

3. Haugh, "Market Models", the opening of page 1. Short-rate models and HJM models target the unobservable instantaneous rate, and market models directly model observable rates such as LIBOR or swap rates. ↩

4. Haugh, "Market Models", page 2 (the caplet price takes the form of an expectation with the bond maturing at the payment date as numeraire) and page 8 (the Black formula for caplets is derived from equation (25)). ↩ ↩²

5. Haugh, "Market Models", footnote 2 on page 1. "These models apply equally well to Euribor rates." ↩

6. Haugh, "Market Models", page 6, equation (14). The Brownian motion is $d$-dimensional, $\mu_n(t)$ and $\sigma_n(t)$ are adapted processes that may depend on the current rates, and the drift is determined by no arbitrage and the positivity of bond prices. ↩ ↩² ↩³

7. The correlation formula is a definition computed from the diffusion term of equation (14), and the source does not write it in this form. The source uses the same quantity in the form $\sigma_i^{\top}\sigma_j$ (Haugh, "Market Models", page 7, equation (19)). ↩

8. Haugh, "Market Models", "Numeraire-EMM Pairs" on page 6. The spot measure uses the numeraire of equation (12), and the forward measure uses the bond maturing at $T$. The source gives one more, the swap measure. ↩ ↩²

9. Haugh, "Market Models", the opening of "3 Numeraires and Zero-Coupon Bond Prices" on page 5. ↩

10. Haugh, "Market Models", page 7, equations (19) and (20) (drift under the spot measure), and pages 7 to 8, equation (24) (drift under the forward measure with the bond maturing at $T_{M+1}$ as numeraire). $\eta$ corresponds to $\nu$ in equation (18). ↩ ↩² ↩³

11. $\delta L/(1+\delta L) = 1 - e^{-0.01} = 0.009950$. The sums in equations (20) and (24) are multiplied by the product of the volatilities and the correlations. The table is rounded to 4 decimal places. ↩

12. Haugh, "Market Models", "6 Monte-Carlo Simulation" on page 10. Monte Carlo simulation is generally needed, discretizations such as the Euler method are used, only a finite set of maturities moves, discrete-time martingales can be constructed (Glasserman's textbook), and the simulation can be run under "any EMM". ↩ ↩² ↩³ ↩⁴

13. Haugh, "Market Models", page 2, equation (3), and page 8, equation (26). The source's $\delta Z_t^{T+\delta}$, $\sigma$, and $L(t,T)$ correspond to $\delta_i Z_{i+1}(0)$, $\bar\sigma_i$, and $L_i(0)$ on this page. ↩ ↩² ↩³

14. Haugh, "Market Models", page 4. Black's formula was originally a framework for quoting market prices. Pages 8 to 9 say that Brace, Gatarek, and Musiela derived equation (3). ↩

15. Haugh, "Market Models", page 8, around equation (25). The source writes the case $n = M$, and $M$ can be chosen arbitrarily. Applying this to the other forward rates is the explanation in the text. ↩

16. Haugh, "Market Models", page 8, Remark 3. Under $P^{M+1}$, the forward rates with $n < M$ are not martingales, and replacing $L_j(t)$ in the drift with $L_j(0)$ gives a lognormal approximation. ↩

17. The discount factor is $e^{-0.02 \times 2}$. The table is rounded, and the price is computed from the unrounded values. ↩

18. Haugh, "Market Models", "Calibration" on page 9. Caplet prices depend only on the level, and swaption prices also depend on the correlations. The source describes a two-stage procedure. ↩ ↩²

19. Haruyasu Ota, "マルコフ関数モデルによる金利オプションの価格付けの実用化" (Putting the Markov-functional model into practice for pricing interest rate options), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (October 2010), "1. はじめに" (1. Introduction), page 132 (in Japanese). Implementation by Monte Carlo simulation, the least-squares Monte Carlo method for options with early exercise rights, and the added complexity of calibration when fitting the smile. ↩ ↩² ↩³

20. Haugh, "Market Models", "The Caplet Volatility Surface" on page 8. If volatility is deterministic, Black volatility does not depend on the strike; the skew since the mid-1990s; examples of extensions (CEV, shifted lognormal, lognormal mixtures, and jumps). ↩ ↩²

21. Haugh, "Market Models", page 10, Remark 5. If the volatility of the forward rates is deterministic, the volatility of the swap rate is not, so the two Black formulas cannot both hold in one model. ↩

22. Haugh, "Market Models", "7 Hedging and the Greeks" on page 11. ↩