A swaption is an over-the-counter contract that gives the buyer the right to enter into a fixed-for-floating interest rate swap at a predetermined swap rate. There are two types: receiver swaptions give the buyer the right to receive fixed rates, while payer swaptions give the buyer the right to pay fixed rates. The Black-Scholes model can be applied to price swaptions, using inputs like the current swap rate, time to expiration, strike rate, and volatility. However, in reality swaption volatility varies with the strike rate, known as swaption skew.

1. Fixed Income Options- Swaptions
Sapna Desai

2. Swaptions
• A swaption is an OTC contract that gives the
buyer the right, at expiration, to enter into
fixed for floating interest rate swap at maturity
and strike rate agreed to in the contract.
• Receiver Swaption gives the buyer the right to
receive fixed and pay floating
• Payer Swaption gives the buyer the right to
pay fixed and receive floating.

3. Example
• $100mm 5.28% 5y5y receiver swaption traded on
Feb 9, 2011.
• This option gives the buyer the right in five years
on Feb9,2016 to receive 5.85% and pay LIBOR on
$100mm for five years until Feb 9, 2021
• Value = $100mm(5.28-C5(5,10)) *A5(5,10)
• C5(5,10) = 5 yr par swap rate five years from
now.
• A5(5,10)= value of the annuity of $1 five years
from now for 5 years paid on fixed rate payment
dates from 5 to 10 years.

4. Applying BS to the example
• Calculating receiver swaption price per 100
notional amount of swaps
Quantity Value
S0 5.28%
T 5
t 5
K 5.28
sigma 1.145%
A0(t,T+t) 3.8285
Pi^N(S0,T,K,sigma) .010213

5. Value of swaption
• A0(t,t+T) times pi^N(S0,T,K,sigma)
• For Payer swaption =
• A0(t,t+T) times epsilon^N(S0,T,K,sigma)

6. Swaption Skew
• If the BS model is true for all swaptions of
given tenor and expiry, a single bp volatility
would price al swaptions of all strikes.
• However that is not the case. The
phenomenon that volatility is not constant at
all strikes is called skew