Bermudan Option Pricing & Early Exercise Lab

Market Presets:

Bermudan Option Price

$0.00

Call Option

European Baseline Price

$0.00

Zero early exercise value

Early Exercise Premium

+$0.00

+0.00% vs European

American Upper Bound

$0.00

Capture: 0.0% of full flex

Contract Parameters

Option Type

Current Spot Price ($S)

Strike Price ($K)

Maturity ($T$ yrs)

Lattice Steps ($N$)

Risk-Free Rate ($r$ %)

Dividend Yield ($q$ %)

Annual Volatility ($\sigma$ %)

Bermudan Exercise Windows

Exercise Schedule Structure

Quarterly dates: Exercisable at t = 0.25, 0.50, 0.75, 1.00 yrs.

Bermudan Greeks

Delta ($\Delta$)	0.0000
Gamma ($\Gamma$)	0.0000
Vega ($\mathcal{V}$)	0.0000
Theta ($\Theta$ / day)	0.0000
Rho ($\rho$)	0.0000

Bermudan Value European Baseline American Upper Bound Intrinsic Payoff

Optimal stopping boundary: Asset prices beyond which immediate exercise dominates holding (continuation).

Value convergence from European (1 window at $T$) through discrete Bermudan schedules to American (continuous exercise).

Exercise Flexibility Decomposition

Exercise Style	Allowed Exercise Dates	Contract Value	Premium vs European	American Capture %
European	Maturity $T$ Only	$0.00	Baseline ($0.00)	0.0%
Bermudan (Selected)	Discrete Windows	$0.00	+$0.00	0.0%
American	Every Trading Day ($N$ steps)	$0.00	+$0.00	100.0%

Spot ($S$) vs. Volatility ($\sigma$) Grid

Exercise Windows vs. Rate ($r$) Grid

Bermudan Option Pricing Theory & Optimal Stopping Dynamics

1. Mathematical Lattice Formulation

On a recombining Cox-Ross-Rubinstein (CRR) binomial tree with $N$ steps and step size $\Delta t = T/N$, the up and down factors and risk-neutral probability $p$ are:

$$u = e^{\sigma \sqrt{\Delta t}}, \quad d = \frac{1}{u}, \quad p = \frac{e^{(r-q)\Delta t} - d}{u - d}$$

At step $i$ and node $j$, the underlying price is $S_{i,j} = S_0 u^j d^{i-j}$. At maturity $N$, terminal node values are set to intrinsic value: $V_{N,j} = \max(\phi (S_{N,j} - K), 0)$, where $\phi = 1$ for calls and $-1$ for puts.

2. Backward Induction Dynamic Programming

Working backwards from step $i = N-1$ down to $0$, the continuation value $C_{i,j}$ is given by the discounted risk-neutral expectation:

$$C_{i,j} = e^{-r \Delta t} \left[ p V_{i+1, j+1} + (1-p) V_{i+1, j} \right]$$

At an allowed Bermudan exercise date ($t_i \in \mathcal{T}_{\text{Berm}}$), the option value equals: $$V_{i,j} = \max\left( \text{Intrinsic}_{i,j}, \, C_{i,j} \right)$$ On unpermitted dates ($t_i \notin \mathcal{T}_{\text{Berm}}$), exercise is forbidden, so $V_{i,j} = C_{i,j}$.

Bermudan Swaptions

In fixed income markets, Bermudan swaptions give the holder the right to enter an interest rate swap on reset dates (e.g. annual or semi-annual coupon dates). They are essential for hedging mortgage prepayment pipelines and structured liabilities.

Optimal Early Exercise Boundary

For a put option, as the risk-free rate rises, the opportunity cost of delaying strike collection increases, widening the early exercise premium. The optimal exercise threshold $S^*(t)$ delineates the region where immediate cash exercise dominates holding.

Dividend Squeeze on Calls

While vanilla European calls on non-dividend assets are never exercised early, discrete dividend payments ($q > 0$) cause the asset price to drop by the dividend on the ex-date, making early Bermudan exercise optimal immediately prior to ex-dividend dates.

Bermudan Option Mastery Quiz

1. If a Bermudan option's allowed exercise dates are increased until exercise is permitted at every time step, its price converges to:

Frequently Asked Questions

A Bermudan option is an exotic derivative that can be exercised only on specified discrete dates prior to and including its expiration date. It represents an intermediate contract between a European option (exercisable only at expiration) and an American option (exercisable on any trading day prior to expiration). Because its exercise flexibility is constrained to discrete windows, its value satisfies European ≤ Bermudan ≤ American.

Bermudan options are priced on recombining binomial or trinomial lattices by working backward from maturity T to time 0. At terminal nodes, the payoff is max(S - K, 0) for calls or max(K - S, 0) for puts. At preceding time steps, if the step is an allowed exercise date, the option value is max(Intrinsic Payoff, Continuation Value). If the step is not an allowed exercise date, the value simply equals the discounted expectation (Continuation Value).

Early exercise is optimal at an allowed exercise date whenever the immediate intrinsic value exceeds the discounted continuation value. For put options, early exercise becomes optimal at sufficiently low stock prices when the interest earned on the strike K outweighs the remaining time value. For call options on non-dividend-paying stocks, early exercise is never optimal prior to maturity; however, discrete dividend yields create optimal early exercise conditions immediately prior to ex-dividend dates.

Bermudan-style exercise is the standard industry structure for Bermudan swaptions (the right to enter an interest rate swap on discrete coupon payment dates), callable corporate bonds (issuer call windows), real options on capital investment tranches, employee stock option vesting windows, and structured equity-linked notes.