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

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:

Value of a caplet

Where

Value of a caplet 1is 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 caplet 2

Value of a floorlet

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

Value of a floorlet

Where

Value of a floorlet 1is 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

Value of a binary call option

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

Value of a binary call option 1

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

Value of a binary call option 2