This document discusses fixed income options like caps and floors. It defines a caplet as an individual component of a cap contract and provides an example of pricing a caplet. It explains that the value of a caplet is determined using the Black-Scholes formula, with inputs like the forward rate, strike price, and volatility. The document also discusses how a cap is the sum of the values of its underlying caplets and how the implied volatility of a cap is calibrated. Finally, it notes that floors and floorlets are analogous to caps and caplets, with floorlet payments determined by the difference between the strike and interest rate.

1. Fixed Income Options
Sapna Desai

2. Caps and Floors
• Example :
• A caplet with three month Libor reset date of
Nov 28th, 2012 a payment date of February 28,
2013 and a strike of .97%. There are 92 days in
the period
• Caplet pays on payment date
• (L-0.97%) times 92/360

3. Caplet pricing
• Value of a caplet with reset at time T and
payment at time T +t is given by
• tdf(T+t)BS(S0,T,K,sigma) where t is the term of
reference rate, Df is the discount factor to
payment date, S0 is todays forward rate from
T to T+t, K is the strike, sigma is the volatility
of the forward rate in bps.
• Applying BS, we get the value of the caplet

4. Pricing a caplet with LIBOR Reset on
Nov 28, 2012 and Pmt date Feb
28,2013, as of Feb 28,2011
Quantity Value
S0 1.9077%
T 639/365 = 1.7507
t 92/360 = 0.2556
K 0.97%
Sigma .7722%
Df(t+T) 0.981801
Value of Caplet = 100*tdf(t+T) *BS .2602

5. Cap
• Cap is portfolio of caplets with the value of
the cap being the sum of the value of its
component caplets. The implied volatility of a
cap is the volatility when used to value every
component caplet results in the market price
of the cap.

6. Structure and pricing of a 2 year cap as
of feb 28, 2011
• Cap strike = .97%
• Cap vol = .7722%
Reset date Pay date Forward rate (%) Caplet premium
2/28/11 5/31/11 .310
5/31/11 8/31/11 .377 .0027
8/30/11 11/30/11 .452 .0128
11/28/11 2/28/12 .599 .0300
2/28/12 5/29/12 .830 .0611
5/29/12 8/29/12 1.176 .1157
8/28/12 11/28/12 1.555 .1864
11/28/12 2/28/13 1.908 0.2602
Sum 0.6688

7. • The cap is ATM because strike is equal to the
corresponding swap rate which is a 2 year swap
rate in this example.
• This is a spot cap example
• One can have 5x5 cap which means a 5 year cap 5
year forward.
• So 1st reset will happen in 5 years and 1st
payment in 5 years + length of accrual period ( 5y
+3mths)and last payment would be in 10 years.

8. Floorlet and Floor
• Analogous to cap and caplet. Payment of a
floorlet at time T+t deteremined by the Libor rate
set at time T =
• (K-L)times t
• Assuming normal forward rates, the value of the
floorlet is given by
• tdf(T+t)BS(S0,T,K,sigma)
• As per put-call parity, prices of ATM caplet and
floorlet with same expiration are equal as are the
matched date ATM caps and floor with same
expiration are equal