Advanced Interest Rate Models: HJM Framework & LMM Guide

In the previous chapter, we explored short-rate models like Vasicek, CIR, and Hull-White, which describe the evolution of the instantaneous short rate $r(t)$. While these models provide elegant closed-form solutions for bond prices and some derivatives, they face a fundamental limitation: starting from a single factor (the short rate), they derive the entire term structure of interest rates. This approach often struggles to match the rich dynamics observed in real yield curves and can produce unrealistic implied volatility structures for interest rate derivatives. This gap between simple models and market reality led to frameworks that better capture interest rate behavior.

The Heath-Jarrow-Morton (HJM) framework and the LIBOR Market Model (LMM) provide a significant advance in interest rate modeling. Rather than modeling a latent short rate and deriving forward rates, these frameworks model forward rates directly. This approach aligns model variables with market observables. The HJM framework, developed by David Heath, Robert Jarrow, and Andrew Morton in 1992, models the evolution of the entire instantaneous forward rate curve. By treating the forward curve as the fundamental object, HJM unified different modeling approaches. The LMM, developed by Brace, Gatarek, and Musiela (1997) and Jamshidian (1997), models discrete forward LIBOR rates directly, which are the actual rates quoted in the market for caps, floors, and swaptions. This practical orientation made LMM the workhorse of the derivatives trading desk.

These advanced models enabled you to price complex interest rate derivatives consistently with observed market prices. While short-rate models require calibration tricks to match market data, HJM and LMM frameworks naturally incorporate the current yield curve and can be calibrated directly to liquid derivative prices. Matching market prices at initialization and flexible volatility structures make these frameworks essential for pricing derivatives. This chapter develops both frameworks from first principles, demonstrating their mathematical foundations and practical implementation.

The HJM framework represents a fundamental conceptual shift from short-rate modeling. Instead of specifying dynamics for a single rate and deriving forward rates, HJM directly models the evolution of the entire forward rate curve. Modeling the observed rate continuum directly is better than deriving it from a simple process. This section develops the framework step by step, building from the definition of instantaneous forward rates through to the crucial no-arbitrage condition that constrains all valid HJM models.

Recall from our discussion of the term structure of interest rates in Part II that the instantaneous forward rate $f(t, T)$ represents the rate at time $t$ for instantaneous borrowing at future time $T$. This concept captures the market's expectation, adjusted for risk, of what very short-term borrowing costs will be at some future date. The instantaneous forward rate relates to zero-coupon bond prices through a fundamental relationship that connects the entire forward curve to tradeable securities:

where:

• $P(t, T)$: price at time $t$ of a zero-coupon bond maturing at $T$
• $f(t, u)$: instantaneous forward rate at time $t$ for maturity $u$
• $r(t) = f(t, t)$: short rate, representing the instantaneous forward rate for immediate borrowing

This exponential relationship reveals a deep connection: the bond price is determined by accumulating the forward rates along the path from the current time to maturity. Intuitively, you can think of buying a zero-coupon bond as locking in a series of instantaneous forward borrowing rates from today until maturity. The integral in the exponent sums up all these infinitesimal forward rates, and the exponential converts this accumulated rate into a discount factor.

The key insight of HJM is that specifying the dynamics of $f(t, T)$ for all maturities $T$ simultaneously determines the entire yield curve evolution. This means we can model the term structure as a single infinite-dimensional object evolving through time. However, not all specifications are arbitrage-free. If we were free to choose any dynamics for forward rates, we could inadvertently create money machines by trading in bonds of different maturities. The key insight of HJM is the precise restriction on forward rate dynamics that ensures no arbitrage, thereby identifying exactly which models are economically meaningful.

Under the HJM framework, the instantaneous forward rate evolves according to a stochastic differential equation that allows both deterministic trends and random fluctuations:

where:

• $\alpha(t, T)$: drift of the forward rate for maturity $T$
• $\sigma(t, T)$: volatility function (can be a vector for multiple factors)
• $W(t)$: Brownian motion under the real-world probability measure

The functions $\alpha$ and $\sigma$ can depend on the current time $t$, the maturity $T$, and potentially on the entire current forward rate curve $\{f(t, u) : u \geq t\}$. This flexibility is both a strength and a challenge. On one hand, it allows us to capture rich dynamics including level shifts, slope changes, and curvature movements in the yield curve. On the other hand, unrestricted choices of drift and volatility can lead to arbitrage opportunities or economically implausible behavior. The HJM framework addresses this by providing a precise mathematical condition that separates valid models from invalid ones.

Under the risk-neutral measure $\mathbb{Q}$, the drift $\alpha(t, T)$ is not free to choose. It is completely determined by the volatility function $\sigma(t, T)$ through the HJM drift condition:

where:

• $\alpha(t, T)$: forward rate drift under the risk-neutral measure
• $\sigma(t, T)$: volatility function of the forward rate

This result states that once you specify how forward rates fluctuate randomly (through the volatility function), the average direction they move (the drift) is automatically determined. There is no freedom to choose both independently. Defining the cumulative volatility $\Sigma(t, T) = \int_t^T \sigma(t, s) \, ds$, this condition can be written compactly as:

where:

• $\alpha(t, T)$: forward rate drift under risk-neutral measure
• $\sigma(t, T)$: forward rate volatility function
• $\Sigma(t, T) = \int_t^T \sigma(t, s) \, ds$: cumulative volatility

The cumulative volatility $\Sigma(t, T)$ has a natural interpretation: it measures the total volatility accumulated from the current forward rate maturity out to the bond maturity. This quantity appears naturally because it represents the volatility of the zero-coupon bond price itself. The drift condition says that forward rates must drift upward by an amount equal to their own volatility times the bond price volatility.

To understand why this condition must hold, let's derive it step by step. The derivation illuminates the deep connection between forward rate dynamics and bond price behavior, and shows why the no-arbitrage requirement imposes such specific constraints. The zero-coupon bond price satisfies:

where:

• $P(t, T)$: price of a zero-coupon bond maturing at $T$
• $f(t, u)$: instantaneous forward rate

Applying Itô's lemma requires determining the dynamics of the log-price, since the bond price is an exponential function of an integral. Let $X(t) = \ln P(t, T) = -\int_t^T f(t, u) \, du$. Using the Leibniz integral rule for stochastic processes, which tells us how to differentiate an integral when both the integrand and limits change, the differential is:

where:

• $X(t)$: log-price of the zero-coupon bond
• $f(t, t) = r(t)$: instantaneous short rate
• $\alpha(t, u)$: drift of the forward rate
• $\Sigma(t, T) = \int_t^T \sigma(t, s) \, ds$: cumulative volatility
• $W(t)$: Brownian motion

The first term $f(t, t) \, dt = r(t) \, dt$ arises because as time advances, the lower limit of integration moves, and the forward rate at that point "drops out" of the integral. The second term captures how the forward rates within the integral are themselves changing. Notice that the stochastic term simplifies beautifully: integrating the volatility function from $t$ to $T$ gives exactly the cumulative volatility $\Sigma(t, T)$.

Since $P(t, T) = e^{X(t)}$, Itô's lemma gives $dP/P = dX + \frac{1}{2}(dX)^2$. The additional term $\frac{1}{2}(dX)^2$ is the Itô correction that accounts for the fact that exponentiating a stochastic process is not the same as exponentiating its drift. Substituting $dX$ and noting that $(dX)^2 = \Sigma(t, T)^2 dt$ (since only the Brownian motion term contributes to the quadratic variation), the bond price dynamics are:

where:

• $r(t)$: instantaneous short rate
• $\alpha(t, u)$: forward rate drift
• $\Sigma(t, T) = \int_t^T \sigma(t, s) \, ds$: volatility of the zero-coupon bond
• $W(t)$: Brownian motion

This equation tells us exactly how the bond price evolves. The drift consists of three parts: the short rate $r(t)$, a negative contribution from the integrated forward rate drifts, and a positive convexity adjustment $\frac{1}{2}\Sigma(t, T)^2$. The volatility of the bond price is simply the cumulative forward rate volatility.

Under the risk-neutral measure $\mathbb{Q}$, all traded assets must have a drift equal to the risk-free rate $r(t)$. This is the fundamental requirement of risk-neutral pricing: when we discount at the short rate, all asset prices become martingales. Therefore, the term in the parentheses must equal $r(t)$:

where:

• $r(t)$: instantaneous short rate
• $\alpha(t, u)$: forward rate drift
• $\Sigma(t, T)$: cumulative volatility

Simplifying this equation by canceling $r(t)$ from both sides:

where:

• $\alpha(t, u)$: forward rate drift
• $\Sigma(t, T)$: cumulative volatility

This equation says that the integrated drift must equal half the squared cumulative volatility. To extract the drift at a specific maturity, we differentiate both sides with respect to $T$. Differentiating both sides with respect to $T$ yields the HJM drift condition. Using the chain rule for the term $\frac{1}{2}\Sigma(t, T)^2$:

where:

• $\alpha(t, T)$: forward rate drift
• $\Sigma(t, T)$: cumulative volatility
• $\sigma(t, T)$: forward rate volatility function

On the left side, the Fundamental Theorem of Calculus tells us that differentiating the integral gives us the integrand evaluated at the upper limit. On the right side, we apply the chain rule: the derivative of $\frac{1}{2}u^2$ with respect to $u$ is $u$, and we multiply by the derivative of $\Sigma(t, T)$ with respect to $T$, which is $\sigma(t, T)$ by the Fundamental Theorem again.

This restriction ensures that the discounted bond price is a martingale under $\mathbb{Q}$, which is equivalent to the absence of arbitrage. This result applies universally: any arbitrage-free model of forward rates must satisfy this condition, regardless of the specific volatility structure chosen.

The HJM framework encompasses all short-rate models as special cases. By choosing specific volatility functions, we recover familiar models that we studied in the previous chapter. This unification provides important insight: the different short-rate models are not fundamentally different approaches to interest rate modeling but rather different choices of volatility structure within the HJM framework.

Ho-Lee Model: Setting $\sigma(t, T) = \sigma$ (constant) gives a volatility that does not depend on either time or maturity. This is the simplest possible choice. The drift becomes:

where:

• $\alpha(t, T)$: forward rate drift
• $\sigma$: constant volatility parameter
• $T - t$: time to maturity

The forward rate dynamics become:

where:

• $f(t, T)$: instantaneous forward rate
• $dW^\mathbb{Q}(t)$: Brownian motion increment under the risk-neutral measure

Notice that the drift increases linearly with maturity. This reflects the convexity adjustment: longer-maturity forward rates must drift upward faster to compensate for the risk in holding longer-dated bonds. The constant volatility means that all forward rates fluctuate by the same amount, leading to parallel shifts in the yield curve.

Hull-White Model: Setting the volatility function to:

where:

• $\sigma(t, T)$: volatility function of the forward rate
• $\sigma$: short rate volatility parameter
• $\kappa$: mean reversion parameter
• $T - t$: time to maturity

This specification recovers the Hull-White model. The mean reversion parameter $\kappa$ controls how volatility decreases with maturity. Economically, this captures the intuition that short-term rates are more volatile than long-term rates because the latter represent averages of many future short rates. As maturity increases, the individual rate fluctuations average out, reducing overall volatility.

The Ho-Lee drift grows without bound as maturity increases, which can lead to unrealistic long-term behavior. Forward rates at very long maturities would drift upward at an accelerating pace, potentially reaching implausibly high levels. The Hull-White specification, with its exponentially decaying volatility, produces a bounded drift that stabilizes at longer maturities. This behavior aligns better with economic intuition: long-term rates should not exhibit explosive dynamics.

To implement HJM in practice, we discretize time and simulate the forward rate curve evolution. Starting from today's observed forward curve $f(0, T)$, we evolve each point on the curve simultaneously. This parallel evolution captures how the entire term structure moves together while respecting the no-arbitrage constraint. The simulation proceeds by applying small increments to each forward rate, with the drift determined by the HJM condition and the random shock drawn from a standard normal distribution scaled by the volatility function.

The simulation shows how the forward rate curve evolves over time under HJM dynamics. The initial flat curve develops both level shifts and changes in shape, with uncertainty increasing over the simulation horizon. The widening confidence band reflects the accumulation of random shocks over time, while the upward drift of the median reflects the convexity adjustment built into the HJM drift condition. Note that as time passes, short-maturity rates "expire" and are no longer meaningful, which is why we shift the maturities when plotting the future distribution.

The key parameters for the HJM framework simulations are:

• f(0, T): Initial forward rate curve. The starting point for the evolution of the term structure, typically extracted from market bond prices or swap rates.
• σ(t, T): Volatility function. Determines the magnitude of random fluctuations and, via the drift condition, the trend of forward rates. This is the sole input needed to specify an HJM model.
• κ: Mean reversion parameter (in Hull-White specification). Controls how quickly volatility decays with maturity, influencing the correlation between forward rates of different tenors.
• dt: Time step size. Smaller steps reduce discretization error in the simulation but increase computational cost.

While the HJM framework provides a theoretically elegant approach to modeling forward rates, it works with instantaneous forward rates, which are not directly observable in markets. You cannot go to a broker and ask to trade the instantaneous forward rate for borrowing at exactly 2.5 years from now. You price caps, floors, and swaptions using discrete forward LIBOR rates for specific tenors (3-month, 6-month). The LIBOR Market Model (LMM), also known as the Brace-Gatarek-Musiela (BGM) model, directly models these market-observable rates. This alignment with market conventions makes LMM the natural choice for pricing and hedging interest rate derivatives.

Consider a set of tenor dates $T_0 < T_1 < \cdots < T_n$ with equal spacing $\delta = T_{i+1} - T_i$ (typically 0.25 or 0.5 years). The forward LIBOR rate $L_i(t)$ is the simply compounded rate at time $t$ for borrowing over the period $[T_i, T_{i+1}]$. Unlike the continuously compounded instantaneous forward rate in HJM, the LIBOR rate applies simple interest over a discrete period:

where:

• $\delta = T_{i+1} - T_i$: year fraction (accrual period)
• $P(t, T_i)$: price of a zero-coupon bond maturing at $T_i$
• $P(t, T_{i+1})$: price of a zero-coupon bond maturing at $T_{i+1}$

The formula expresses a simple arbitrage relationship: if you invest one dollar at the LIBOR rate from $T_i$ to $T_{i+1}$, you should receive $1 + \delta L_i(t)$ dollars. Alternatively, you could buy $1/P(t, T_{i+1})$ bonds maturing at $T_{i+1}$, which costs $P(t, T_i)/P(t, T_{i+1})$ in terms of bonds maturing at $T_i$. Equating these two strategies yields the formula above.

These rates are directly observable from the forward rate agreement (FRA) market and serve as the underlying rates for caps and floors. When a bank quotes a 6-month LIBOR rate starting in 1 year, that is exactly the forward rate $L_i(t)$ for the appropriate index $i$.

The key insight of LMM is that under the $T_{i+1}$-forward measure $\mathbb{Q}^{T_{i+1}}$, the forward LIBOR rate $L_i(t)$ is a martingale. This follows because $L_i(t)$ can be expressed as a ratio of asset prices (zero-coupon bonds), and ratios of tradeable assets are martingales under the numeraire corresponding to the denominator. This is a direct application of the change of numeraire theorem. When we use $P(t, T_{i+1})$ as the numeraire, any asset price divided by this numeraire becomes a martingale, and the forward rate is precisely such a ratio.

Under this forward measure, $L_i(t)$ has zero drift and follows:

where:

• $\sigma_i(t)$: volatility of the $i$-th forward rate
• $W_i^{T_{i+1}}(t)$: Brownian motion under the $T_{i+1}$-forward measure $\mathbb{Q}^{T_{i+1}}$

The lognormal dynamics implied by this equation have profound practical consequences. Because the rate follows geometric Brownian motion under its natural measure, its distribution at any future time is lognormal. This lognormal distribution leads directly to Black's formula for caplet pricing, providing the theoretical justification for the market convention that predated the formal LMM framework.

A caplet pays $\delta \max(L_i(T_i) - K, 0)$ at time $T_{i+1}$, where $K$ is the cap rate. This payoff represents the protection provided by the cap: if the realized LIBOR rate exceeds the cap strike, the holder receives the difference applied to the notional over the accrual period. Since $L_i(t)$ is a martingale under $\mathbb{Q}^{T_{i+1}}$ and follows lognormal dynamics, the caplet price has a Black-Scholes-type closed-form solution known as Black's formula:

where:

• $L_i(0)$: initial forward LIBOR rate
• $K$: cap strike rate
• $P(0, T_{i+1})$: discount factor for the payment date
• $N(\cdot)$: cumulative standard normal distribution function
• $d_1 = \frac{\ln(L_i(0)/K) + \frac{1}{2}v_i^2}{v_i}$
• $d_2 = d_1 - v_i$
• $v_i = \sqrt{\int_0^{T_i} \sigma_i(t)^2 \, dt}$: integrated volatility (approximated as $\sigma_i \sqrt{T_i}$ for constant volatility)

The formula terms provide the following intuition:

• $N(d_2)$: probability that the option expires in-the-money under the forward measure
• $L_i(0) N(d_1)$: expected value of the floating rate at maturity, conditional on exercise
• $K N(d_2)$: expected cost of the fixed rate strike payment, conditional on exercise

The structure mirrors the Black-Scholes formula for equity options, which is no coincidence. Both arise from the same mathematical setting: lognormal dynamics under the relevant pricing measure. The discount factor $P(0, T_{i+1})$ converts the expected payoff under the forward measure to a present value.

A cap is simply a portfolio of caplets:

where:

• Cap: total value of the interest rate cap
• Caplet_i: value of the $i$-th individual caplet
• $n$: number of caplets in the contract

Each caplet in the cap protects against high rates in a different future period, and since the caplets have non-overlapping payoff dates, they can be valued independently and summed.

The cap price of approximately 1.77% of notional represents the premium required to hedge against floating rates exceeding 5% over the next 3 years. This value reflects the sum of individual caplet prices, each weighted by the discount factor and the probability of the forward rate exceeding the strike. For an at-the-money cap where the strike equals the forward rate, roughly half of the value comes from intrinsic value considerations and half from time value.

The plot illustrates the inverse relationship between cap prices and strike rates. As the strike increases, the cap becomes more out-of-the-money, reducing its value. A higher strike means the cap provides protection only against more extreme rate increases, which are less likely to occur. The curvature reflects the non-linear nature of option pricing with respect to the strike, with the steepest gradient occurring near the at-the-money rate where the option's moneyness is most sensitive to small changes in the strike.

While Black's formula works beautifully for individual caplets, pricing more complex derivatives like swaptions requires simulating multiple forward rates simultaneously. The challenge is that each forward rate $L_i(t)$ is a martingale under a different measure $\mathbb{Q}^{T_{i+1}}$. We cannot simply simulate each rate as a driftless process because they all need to evolve consistently under the same probability measure.

To simulate all rates consistently, we typically work under a single measure, commonly the terminal measure $\mathbb{Q}^{T_n}$ (using $P(t, T_n)$ as numeraire). Under this measure, only $L_{n-1}(t)$ is a driftless martingale. Other forward rates acquire drift terms that arise from the change of measure between their natural measures and the terminal measure.

Under $\mathbb{Q}^{T_n}$, the dynamics of $L_i(t)$ for $i < n-1$ become:

where:

• $L_i(t)$: forward LIBOR rate
• $\mu_i(t)$: drift term under the terminal measure
• $\sigma_i(t)$: volatility of the forward rate
• $W_i^{T_n}(t)$: Brownian motion under the terminal measure $\mathbb{Q}^{T_n}$

The drift term is defined as:

where:

• $L_i(t), L_j(t)$: forward LIBOR rates
• $\delta$: accrual period
• $\rho_{ij}$: correlation between forward rates $L_i$ and $L_j$
• $\sigma_i(t), \sigma_j(t)$: volatilities of the respective forward rates
• $\mu_i(t)$: state-dependent drift ensuring no arbitrage under the terminal measure

This drift term ensures consistency when modeling all rates under the single terminal measure $\mathbb{Q}^{T_n}$. To understand each component, consider the following interpretation. The fraction $\frac{\delta L_j(t)}{1 + \delta L_j(t)}$ measures the sensitivity of the bond price ratio $P(t, T_j)/P(t, T_{j+1})$ to the rate $L_j$. This is essentially the duration of a single-period bond with respect to its yield. The product $\rho_{ij} \sigma_i(t) \sigma_j(t)$ captures the covariance between the rate being modeled ($L_i$) and the rates acting as numeraires ($L_j$). The summation accumulates adjustments for all intermediate rates between the payment date and the terminal horizon.

This drift term arises from the change of measure from each forward rate's natural measure to the terminal measure. The negative sign indicates that rates with earlier payment dates tend to drift downward under the terminal measure, which compensates for the fact that they are martingales under different measures.

The correlation structure $\rho_{ij}$ between forward rates is crucial for pricing products that depend on multiple rates (like swaptions). A swap rate is essentially a weighted average of forward rates, so the volatility of the swap rate depends not just on the individual forward rate volatilities but also on how the forward rates co-move. Common parameterizations include:

Constant correlation:

where:

• $\rho_{ij}$: correlation between forward rates $L_i$ and $L_j$
• $\rho$: constant correlation parameter

This simple specification assumes that all pairs of forward rates have the same correlation. While convenient for calibration, it lacks economic nuance because it treats the correlation between 1-year and 2-year rates the same as the correlation between 1-year and 10-year rates.

Exponential decay:

where:

• $\beta$: correlation decay parameter
• $|i - j|$: distance between tenor indices

This captures the intuition that forward rates with similar maturities are more correlated than rates far apart. Economic factors that drive short-term rates (monetary policy, near-term economic outlook) differ from those driving long-term rates (inflation expectations, long-run growth potential). The exponential form provides a smooth transition from high correlation for adjacent rates to lower correlation for distant rates.

To price complex derivatives under the LMM, we simulate forward rate paths using the discretized dynamics under the terminal measure. The simulation requires careful handling of the state-dependent drift: at each time step, we must first evaluate the current drift based on the current forward rates before updating the rates for the next step. This creates a natural ordering where we update rates from longest maturity to shortest, since shorter rates depend on longer rates through the drift formula.

The simulation captures the stochastic evolution of forward rates, including their correlation structure. Notice how the rate fixes at its fixing time and remains constant thereafter. Before the fixing time, the rate fluctuates randomly around its initial value, with the distribution widening as time progresses. After fixing, the rate becomes a realized constant, which reflects the actual market mechanics where LIBOR rates are set at the beginning of each accrual period.

The key parameters for the LIBOR Market Model are:

• L_i(0): Initial forward LIBOR rates. Observable directly from the current yield curve, these provide the starting point for all simulations and ensure the model fits today's market prices exactly.
• σ_i: Volatility of the i-th forward rate. Determines the width of the distribution for each rate and is typically calibrated to cap volatility data.
• ρ_ij: Correlation between forward rates L_i and L_j. Crucial for pricing multi-rate products like swaptions, as it determines how the rates co-move and thus affects the volatility of rate combinations.
• δ: Accrual period (tenor spacing). Typically 0.25 (3 months) or 0.5 (6 months), matching the conventions of the underlying interest rate contracts.
• β: Correlation decay parameter. Controls how quickly correlation decreases as the distance between rate maturities increases, with higher values producing faster decay.

A swaption is an option to enter into an interest rate swap at a future date. A payer swaption gives you the right to pay fixed and receive floating, while a receiver swaption gives the right to receive fixed and pay floating. Swaptions are among the most liquid interest rate derivatives and are crucial for calibrating the LMM. Their prices reflect market expectations about both the level and volatility of future interest rates, making them valuable sources of information for model calibration.

The forward swap rate $S_{\alpha,\beta}(t)$ for a swap starting at $T_\alpha$ and ending at $T_\beta$ is the fixed rate that makes the swap have zero value at inception. It can be expressed as the ratio of two present values:

where:

• $P(t, T_\alpha)$: price of bond maturing at swap start date
• $P(t, T_\beta)$: price of bond maturing at swap end date
• $A_{\alpha,\beta}(t) = \sum_{i=\alpha}^{\beta-1} \delta P(t, T_{i+1})$: present value of the fixed leg's annuity factor (PVBP)

The numerator $P(t, T_\alpha) - P(t, T_\beta)$ represents the present value of receiving one dollar at the swap start and paying one dollar at the swap end. This is exactly the floating leg's present value for a swap with a notional of one dollar. The denominator, the annuity factor, represents the present value of receiving one dollar at each payment date. The ratio therefore gives the fixed rate that equates the present values of the fixed and floating legs.

A payer swaption with strike $K$ has payoff at expiry $T_\alpha$:

where:

• $A_{\alpha,\beta}(T_\alpha)$: annuity factor at expiry
• $S_{\alpha,\beta}(T_\alpha)$: forward swap rate at expiry
• $K$: swaption strike rate

The payoff structure reflects that exercising a payer swaption is valuable when the market swap rate exceeds the strike: you can immediately enter a swap paying below-market rates. The multiplication by the annuity factor converts the rate difference into a present value, accounting for all future payment dates.

While the swap rate under LMM does not follow exactly lognormal dynamics (it's a function of multiple forward rates), a common approximation assumes lognormal swap rate dynamics. This approximation is remarkably accurate in practice because the swap rate, being a weighted average of forward rates, tends toward normality by the central limit theorem and the log of a sum is approximately the sum of logs when the rates are similar in magnitude. This leads to Black's formula for swaptions:

where:

• $A_{\alpha,\beta}(0)$: initial annuity factor
• $S_{\alpha,\beta}(0)$: initial forward swap rate
• $K$: swaption strike rate
• $d_1 = \frac{\ln(S_{\alpha,\beta}(0)/K) + \frac{1}{2}v_{S}^2}{v_S}$
• $d_2 = d_1 - v_S$
• $v_S$: total swaption volatility ($\sigma_{swap}\sqrt{T_\alpha}$)

As with the caplet formula, this expression decomposes the value into interpretable components:

• $S_{\alpha,\beta}(0) N(d_1)$ is the expected value of the floating leg conditional on exercise
• $K N(d_2)$ is the expected value of the fixed leg conditional on exercise
• The annuity factor $A_{\alpha,\beta}(0)$ discounts the future swap cash flows to a present lump sum

Since the swap rate behaves approximately as a weighted average of the forward rates, its variance can be derived from the covariances of the underlying forward rates. This is a direct application of the formula for the variance of a linear combination of random variables. The total swaption volatility squared $v_S^2$ is approximated by:

where:

• $w_i = \frac{L_i(0)}{S_{\alpha,\beta}(0)} \frac{\partial S_{\alpha,\beta}(0)}{\partial L_i(0)}$: weights representing the elasticity of the swap rate with respect to each forward rate
• $\rho_{ij}$: correlation between forward rates
• $\sigma_i$: volatility of forward rate $L_i$
• $T_\alpha$: time to swaption expiry

The weights $w_i$ measure how sensitive the swap rate is to each underlying forward rate. These weights sum to one (approximately) and tend to be larger for rates near the middle of the swap than for rates at the ends. The double summation captures both the variance contributions from each rate (when $i = j$) and the covariance contributions from pairs of rates (when $i \neq j$).