# Black Formula’s and valuing Interest Rate Caps and Floors

### Value of a caplet

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 at time 0 for the period between and ti+1

σti is the volatility of this forward interest rate

ZCt 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 ti and payoffs at time ti+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 Fi is the forward value of the interbank rate, X is the cut off rate, σ is the volatility of Fi , zct is the t- period spot rate / zero coupon rate and ti 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 