European, Bermudan and Range Accrual: Which Model for Which Product#

The earlier pages price three different rate options. This page puts them side by side, because the real skill is choosing the right model for the payoff and knowing what each model leaves out. It also shows how the series fits together.

The three products#

From vanilla to structured#
Aspect	European swaption	Bermudan swaption	Range accrual
Payoff	One exercise date. Enter a swap at a fixed rate if it is worth it.	Several exercise dates. Enter the swap, to a common end date, at the best of them.	A coupon for each fixing where a rate is inside a range.
What the payoff depends on	One forward swap rate, at one date.	The path of the whole curve, because the exercise date is chosen.	The rate at many dates, at two barriers.
Standard model	Black or Bachelier with a SABR smile.	A short-rate or market model fitted to the curve and to the co-terminal European swaptions.	A strip of digitals priced from the smile, with a wedge for the barrier.
Main market input	Volatility at the strike. Observable at liquid points.	Volatilities of the co-terminal Europeans, and a mean reversion or correlation assumption.	Smile at the barrier strikes for each fixing date. Often not observable.
Biggest model risk	Smile at strikes away from the quotes.	Mean reversion, which sets the value of the choice of exercise date.	Skew at the barriers, and the treatment of pin risk.
Key Greek issue	Vega by strike and expiry.	Mean reversion is not hedgeable with vanilla options.	Delta spikes at the barrier near each fixing.
Typical level	2 for liquid points, 3 for far strikes and long dates.	2 or 3, depending on how much the mean reversion matters.	Often 3.
Page	Swaptions and the Black-76 Model, SABR: Modelling the Volatility Smile	Bermudan Swaptions and Mean Reversion	Range Accruals, Digitals and the Wedge

How the models build on each other#

The curve (Interest Rate Curve Build, Curve Build, Part 2: Jacobians, Global Solvers and Multi-Curve) gives forward rates and discount factors. Every product needs it.
Black-76 (Swaptions and the Black-76 Model) prices a European option on one forward rate with one volatility, which is the quoting convention of the market.
SABR (SABR: Modelling the Volatility Smile, Hagan’s Approximation: From SABR to a Black Volatility) gives a volatility for every strike at one expiry, which prices Europeans away from the quoted points and the digitals that make up a range accrual.
Hull-White and its relatives (Bermudan Swaptions and Mean Reversion) describe how the whole curve moves over time. They price the choice of exercise date, at the cost of a worse fit to the smile.
Digitals and the wedge (Range Accruals, Digitals and the Wedge) use the smile from SABR and add a rule for the barrier.

Each step adds a feature and gives something up. SABR fits the smile of one rate and cannot price a Bermudan, because it has no description of how rates at different dates move together. Hull-White does that but has a flat smile, so it cannot price a digital far from the money. A callable range accrual needs both, which is why it is usually priced with a multi-factor market model or a model with a local and stochastic volatility, calibrated to the smile and to the correlation.

Consistency checks between the products#

Valuation control can use the products to test each other:

A Bermudan is worth at least the highest co-terminal European. The tree in Bermudan Swaptions and Mean Reversion gives a switch value above zero for every mean reversion, as it should.
The Hull-White model prices the calibration swaption back to the Black price, to within a small tolerance.
A digital is the slope of the call price. A call spread of a small width gives the same value as the digital, which the range accrual script confirms.
Put-call parity holds for swaptions, at every strike on the smile.
The density must not be negative. A smile that gives a negative density gives a negative digital price, which is an arbitrage. See Hagan’s Approximation: From SABR to a Black Volatility.

A valuation control view#

Model fitness: the model has to be fit for the product. A model that fits the vanilla prices can still misprice a product that depends on something else, such as the smile far from the money or the correlation between dates.
A model inventory: each product is mapped to its approved model, its calibration instruments, the range of validity, and the reserves.
A ladder of reserves: the European needs a reserve for illiquid strikes, the Bermudan for mean reversion, and the range accrual for the skew and the wedge. All of them feed the bridge in Financial Statements, and the prudent valuation adjustment in Prudent Valuation.
Independent pricing: a second model for each, for example a tree against a closed form, a different wedge, or a different mean reversion. The difference is evidence for the reserve.