Online Finance – Interest Rate Options – Pricing Caps & Floors

Caps and Floors

The most commonly used options in the swaps market are caps and floors. A cap is a call on the rates where the payoff depends on Max (LIBOR – Strike, 0). A floor is a put on the rates where the payoff depends on Max (Strike-LIBOR, 0).

Cap

A cap may be considered as a portfolio of caplets on the underlying asset which is the LIBOR. The value of the caplet may be derived using Black’s Formula. The value of a caplet which resets at time ti and payoffs at time ti+1 is:

Where  is known as the forward premium, X is the Strike, Fi is the forward rate.

This is illustrated below for a 4 year cap on a 1 year Inter-bank offer rate with a strike of 12.5%. The annualized constant volatility is 3.13%. The valuation date is 31/5/2010 and the payment frequency is annual. The notional amount is 100,000.

 Period Start (i) Period End (i+1) ti ti+1 ZCti+1 Fi d1 d2 Forward Premium Value of Caplet 01/01/10 01/01/11 n/a 0.59 12.150% 12.150% n/a n/a – – 01/01/11 01/01/12 0.59 1.59 12.225% 12.269% -0.76 -0.79 37.26 31.02 01/01/12 01/01/13 1.59 2.59 12.349% 12.583% 0.19 0.15 241.75 178.77 01/01/13 01/01/14 2.59 3.59 12.418% 12.595% 0.18 0.12 302.29 198.53 Price of Cap 408.33

The calculation of d1 and d2 are based on the duration at the start of the period, i.e. ti. Therefore for the first row we see that these values cannot be calculated. The forward premium in this case is simply equal to the payoff i.e. Max(Fi-X,0)*Notional*?t/365=max(12.15%-12.5%,0)*100000*0.59=0

For the other rows as d1 and d2 can be calculated the Forward Premium from Black’s formula will be used.

The price of the cap is the sum of the values of the caplets, which are the present values of the forward premiums.

Floor

A floor may be considered as a portfolio of floorlets on the underlying asset which is the LIBOR. The value of the floorlet may be derived using Black’s Formula. The value of a caplet which resets at time ti and payoffs at time ti+1 is:

Where  is the forward premium. d1 and d2 are as given above. The value of a 4 year floor with strike, volatility, notional, maturity and payment frequency the same as for the cap above is illustrated below:

 Period Start (i) Period End (i+1) ti ti+1 ZCti+1 Fi 01/01/2010 01/01/2011 n/a 0.59 12.150% 12.150% 01/01/2011 01/01/2012 0.59 1.59 12.225% 12.269% 01/01/2012 01/01/2013 1.59 2.59 12.349% 12.583% 01/01/2013 01/01/2014 2.59 3.59 12.418% 12.595%
 d1 d2 Forward Premium Value of Floorlet n/a n/a 206.16 192.70 -0.76 – 0.79 268.02 223.14 0.19 0.15 158.39 117.12 0.18 0.12 207.47 136.26 Price of Floor 669.22

Again it may be noted that the calculation of d1 and d2 are based on the duration at the start of the period, i.e. ti. Therefore for the first row we see that these values cannot be calculated. The forward premium in this case is simply equal to the payoff i.e. Max(X-Fi,0) * Notional * ?t/365 = max(12.5%-12.15%,0)*100000*0.59=206.16

For the other rows as d1 and d2 can be calculated the Forward Premium from Black’s formula will be used.

The price of the floor is the sum of the values of the floorlets, which are the present values of the forward premiums.