## Black’s Formula: Pricing Interest Rate Caps and Floors – Calculation reference

*in*Derivatives

# 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}

_{σti} 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(d_{2}) 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, σ 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