Interest Rate Derivatives
Black-76 & Bachelier
Annuity PVBP Engine
European Swaption Pricing Lab (Black-76 & Bachelier)
Price interest rate payer and receiver swaptions across institutional Expiry × Tenor structures. Model swap annuities (PVBP), compare Black-76 lognormal vs Bachelier normal models, and evaluate Greeks sensitivities.
Payer Swaption: V_{\text{payer}} = A \times [ S \Phi(d_1) - K \Phi(d_2) ] |
Receiver Swaption: V_{\text{rec}} = A \times [ K \Phi(-d_2) - S \Phi(-d_1) ] |
Annuity: A = \sum_{i=1}^m \tau_i P(0, T_i)
Market Presets:
Swaption Parameters
$
%
%
%
bps
%
Black-76 Price
$176,450
176.5 bps of Notional
Bachelier Price
$175,820
Diff: -$630 (-0.6 bps)
Annuity (PVBP)
4.1245
DV01: $4,125 / bp
Swaption Straddle
$352,900
1Y × 5Y PAYER
Swaption Pricing Surface Visualizer
Swaption Risk Sensitivities (Greeks)
Swaption Delta
$2,150 / bp
Swaption Gamma
0.142
Swaption Vega
$7,058 / 1% vol
Swaption Theta
-$241 / day
Swaption Price: Fixed Strike K vs Volatility σ ($)
Swaption Price: Option Expiry vs Underlying Swap Tenor ($)
Swaption Engineering Mastery Quiz
1. What right does a Payer Swaption grant to the holder?
2. Why does the Swaption formula include the Annuity (PVBP) factor?
3. What primary advantage does the Bachelier (Normal) model have over Black-76 in rates markets?
Swaption Pricing Engineering & FAQ
An interest rate swaption (swap option) is an over-the-counter option contract granting the buyer the right, but not the obligation, to enter into an underlying Interest Rate Swap on a specified expiration date at a predetermined fixed rate (the strike rate K). In European swaptions, the option can only be exercised on the precise expiration date.
A Payer Swaption gives the holder the right to PAY fixed interest and receive floating in the underlying swap. It acts like a call option on interest rates, profiting when swap rates rise above the strike. A Receiver Swaption gives the holder the right to RECEIVE fixed interest and pay floating, acting like a put option on interest rates and gaining value when market swap rates fall.
The annuity factor (also known as the Present Value of a Basis Point, or PVBP) represents the present value of receiving 1 basis point or 1% across all fixed-leg coupon payment dates of the underlying swap:
A = Notional × ∑ τ_i P(0, T_i). Because swaption payoffs are settled across the full life of the underlying swap, the Black-76 formula multiplies the standard option spread by this annuity factor.
The classic Black-76 model assumes forward swap rates follow a lognormal distribution, which implies rates cannot become negative and mathematically breaks when rates approach zero (as volatility approaches infinity). The Bachelier (Normal) model assumes swap rates follow an arithmetic Brownian motion with normal distribution, allowing interest rates to be zero or negative, which became the global market standard across EUR, JPY, and USD post-2008.
In swaption market notation, 'Expiry x Tenor' specifies the option duration and the underlying swap length. For example, a '1Yx5Y' swaption is an option that expires in 1 year, giving the holder the right to enter into a 5-year interest rate swap running from year 1 to year 6. A '5Yx10Y' swaption expires in 5 years into a 10-year swap.