Bermudan Swaptions and Callable Bonds

A Bermudan swaption allows exercise on several specified dates, making it an optimal-stopping problem rather than a portfolio of European swaptions. Its value depends on the joint evolution of rates and future exercise opportunities, which is why callable bonds and mortgages carry embedded option risk that cannot be captured by a static duration measure.

Exercise economics

At each exercise date t_e, the holder compares the immediate value of entering the swap with continuation value:

exercise if intrinsic_value(t_e) > continuation_value(t_e)
V(t_e) = max(intrinsic_value, E_Q[discounted V(t_next) | state])

A payer Bermudan grants the right to enter a pay-fixed swap, while a receiver Bermudan grants the right to receive fixed. The exercise value comes from a full swap valuation using the then-current curve. A sequence of European swaptions ignores the fact that exercising today destroys later rights, so its sum is only an upper bound.

Product	Embedded choice	Typically benefits when
payer Bermudan	pay fixed at selected dates	rates rise enough to make fixed cheap
receiver Bermudan	receive fixed at selected dates	rates fall enough to make fixed attractive
callable bond	issuer redeems early	rates fall / refinancing is attractive
mortgage-backed security	borrower prepays	mortgage rates fall or turnover rises

Trees and one-factor models

For a one-factor short-rate model, a recombining lattice is transparent. Calibrate its deterministic shift to the initial curve, roll backward through nodes, and take the maximum of exercise and continuation at each exercise date. Hull–White trees are widely used because a time-dependent drift fits the curve exactly; Vasicek and Hull–White discusses their assumptions.

continuation(node) = discount(node) *
                     sum(probability(child) * value(child))
value(node) = max(exercise_value(node), continuation(node))

The exercise boundary should be inspected, not hidden inside the engine. If it jumps erratically across adjacent nodes, likely causes are a schedule error, inadequate grid resolution, or an inconsistent curve/volatility calibration. Test the tree’s European swaption prices against an analytic or high-accuracy benchmark before trusting Bermudan values.

Least-squares Monte Carlo

In a market model or multifactor setting, the state is too large for a simple tree. Longstaff–Schwartz least-squares Monte Carlo (LSMC) estimates conditional continuation value from simulated paths. Work backward: at each date, regress discounted future cash flows on basis functions of the current state, then exercise paths whose intrinsic value exceeds the fitted continuation.

import numpy as np

def lsmc_step(states, intrinsic, discounted_future):
    in_money = intrinsic > 0
    x = states[in_money]
    # Basis selection must be cross-validated in a real implementation.
    X = np.column_stack([np.ones(len(x)), x, x**2])
    beta, *_ = np.linalg.lstsq(X, discounted_future[in_money], rcond=None)
    continuation = np.column_stack(
        [np.ones(len(states)), states, states**2]
    ) @ beta
    return np.where(intrinsic > continuation, intrinsic, discounted_future)

Regression error creates an upward bias when the same paths select and value exercise. Use out-of-sample paths, dual bounds, or a separate policy-evaluation set. Basis functions should represent relevant state variables—swap rate, annuity, curve factors—not merely a convenient short rate. Common random numbers reduce noise for Greeks, but do not eliminate policy instability near the exercise boundary.

Callable bonds as a short option

A callable bond equals a non-callable bond minus the issuer’s call option:

callable_bond = straight_bond - issuer_call

When yields decline, the straight bond appreciates but the issuer is more likely to call and refinance. The investor therefore has negative convexity: duration shortens as rates fall and extends as rates rise. A callable's option-adjusted spread (OAS) is the constant spread added to model discounting that matches its market price; it is model dependent because exercise probabilities are model dependent.

Mortgage securities add borrower behavior. Rate incentive is necessary but insufficient: burnout, seasoning, housing turnover, loan size, and servicing frictions shape prepayment. Treating borrowers as perfect option exercisers systematically overstates prepayments in some regimes and understates them in others.

Calibration, hedging, and stress

Calibrate the rate model to the relevant European swaption surface and ensure the curve fit is exact. A model that fits ATM vol but misses the smile can misvalue cancellable structures because exercise occurs in tail states. Model choice—one-factor versus multifactor, normal versus lognormal, deterministic versus stochastic volatility—is an exposure, not just a technical implementation detail.

Hedge a Bermudan dynamically with swaps and Europeans, but expect residual gamma, vega, and correlation risk. European swaptions are imperfect hedges because their expiry distribution differs from the Bermudan exercise distribution. Stress the exercise boundary under curve twists, volatility moves, and liquidity-driven spread widening; classical options Greeks are useful local diagnostics, not a substitute for this analysis.

Key takeaways

• Bermudans require optimal stopping: exercise is compared with continuation at every allowed date.
• Lattices suit low-dimensional models; LSMC generalizes to market-model states.
• Validate European calibration, numerical convergence, and exercise boundaries independently.
• Callable bonds embed a short call and therefore negative convexity for investors.
• Hedging with Europeans leaves residual path-dependent and model risk.