Advanced Interest Rate Models (HJM and LMM) › Swaption Pricing Under the LMM

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 their prices are used to calibrate the LMM. These prices reflect market expectations about both the level and volatility of future interest rates.

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:

$$S_{\alpha,\beta}(t) = \frac{P(t, T_\alpha) - P(t, T_\beta)}{A_{\alpha,\beta}(t)}$$

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 $T_\alpha$:

$$\text{Swaption Payoff} = A_{\alpha,\beta}(T_\alpha) \cdot \max(S_{\alpha,\beta}(T_\alpha) - K, 0)$$

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:

$$\text{Swaption} = A_{\alpha,\beta}(0) \cdot [S_{\alpha,\beta}(0) N(d_1) - K N(d_2)]$$

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:

$$v_S^2 \approx \sum_{i=\alpha}^{\beta-1} \sum_{j=\alpha}^{\beta-1} w_i w_j \rho_{ij} \sigma_i \sigma_j T_\alpha$$

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 $i \neq j$).

For more accurate pricing that captures the correlation structure, we can use Monte Carlo simulation under the LMM. This approach simulates the joint evolution of all forward rates, calculates the swap rate at expiry from the simulated rates, and averages the discounted payoffs across all paths. The Monte Carlo method naturally handles the complex dependence of the swap rate on multiple forward rates without requiring the approximations used in the analytical formula.