Interest Rate Derivatives
Hull-White Trinomial Tree
Dynamic Programming
Bermudan Swaption Pricing Lab (Hull-White Tree)
Price callable and Bermudan interest rate swaptions across discrete exercise schedules. Model optimal early exercise stopping times, quantify early exercise premiums over European swaptions, and simulate Hull-White short-rate lattice dynamics.
Optimal Exercise Rule: V_j = \max\left( V_{\text{hold}, j}, \; \text{SwapNPV}_j(K) \right) |
Early Exercise Premium: \Delta_{\text{Berm}} = V_{\text{Bermudan}} - V_{\text{European, min}}
Market Presets:
Bermudan Contract Parameters
$
%
%
%
Bermudan Swaption Valuation Summary
5Ync1Y PayerBermudan Swaption PV
$294,850
294.9 bps
European Bound (Min)
$248,120
1Y × 4Y Swaption
Early Exercise Premium
+$46,730
+18.8% Premium
Expected Exercise Time
2.14 Years
Stopping Time E[T*]
Lattice & Risk Metrics
Trinomial Tree Time Steps:
10 steps (Δt = 0.5Y)
Exercise Opportunities:
4 discrete dates
Rate Step Size Δr:
104.2 bps (√3 σ)
DV01 Interest Rate Risk:
$1,240 / bp
Vega Sensitivity (per 10 bps σ):
$18,450
Lockout Cushion:
1.0 Year Protected
Optimal exercise boundary r*(t): Short rate threshold triggering immediate early exercise versus holding.
Comparison between the Bermudan swaption and constituent European swaptions exercisable at each individual date.
Early exercise flexibility premium growth as short-rate volatility increases.
Bermudan Swaption PV: Strike K vs Rate Volatility σ ($)
Early Exercise Premium: Mean Reversion a vs Lockout (bps)
Bermudan Swaption Engineering Mastery Quiz
1. Why is a Bermudan swaption worth more than a single European swaption with the same strike?
2. What happens to the remaining underlying swap tenor when a Bermudan swaption is exercised early?
3. How does higher mean reversion (a) affect the early exercise premium?
Bermudan Swaption Engineering & FAQ
A Bermudan swaption is an interest rate option granting the holder the right to enter into an underlying interest rate swap on any of several specified discrete dates (such as annually or semi-annually after an initial lockout period), unlike a European swaption which can only be exercised on a single date.
Black-76 is a closed-form formula designed for European-style options where the exercise decision occurs on a single date. Because Bermudan swaptions involve multiple early exercise opportunities, path-dependent continuation values, and variable remaining swap tenors at each exercise node, they require term structure models like the Hull-White trinomial tree or the Libor Market Model (LMM) with dynamic programming.
The early exercise premium is the difference between the Bermudan swaption value and the corresponding European swaption value maturing at the earliest or longest exercise date. It quantifies the financial value of the flexibility to enter the swap early when interest rate movements make immediate exercise optimal.
Higher mean reversion pulls future short rates back toward the long-term mean faster, dampening the dispersion of interest rates over long horizons. This reduces the option value of long-dated exercise dates and decreases the overall early exercise premium relative to lower mean reversion regimes.
In institutional market conventions, '10Ync1Y' denotes a 10-year total swap structure with a 1-year lockout period during which the option cannot be exercised. Starting at year 1 (and typically every year or coupon date thereafter until maturity), the holder has the right to enter into the remaining swap.