Black Formula’s and valuing Interest Rate Caps and Floors

Value of a caplet

The value of a caplet which resets at time t_i and payoffs at time t_i+1 is:

Where

is known as the forward premium

X is the Strike

F_i is the forward rate at time 0 for the period between and t_i+1

?_ti is the volatility of this forward interest rate

ZC_t is the t- period spot rate / zero coupon rate and

N(.) is the cumulative probability distribution function (pdf) for a standardized normal distribution

Value of a floorlet

The value of a floorlet which resets at time t_i and payoffs at time t_i+1 is:

Where

is the forward premium.

Value of a binary call option

The binary call option pays the Fixed rate * Notional if the interbank rate exceeds the cutoff rate. Its value is

Where N(d₂) is the probability that the interbank rate will exceed the cutoff rate and

Where F_i is the forward value of the interbank rate, X is the cut off rate, ? is the volatility of F_i , zc_t is the t- period spot rate / zero coupon rate and t_i is the time from the valuation date to time i.

Value of a binary call option

The binary put option pays the Fixed rate * Notional if the interbank rate is below the cutoff rate. Its value is