Chapter 5: Models and Equations 287
the bond is significantly greater than the expiration of
the option. The relevant formulæ are, for a call option
e−r(T−t)(FN(d 1 )−KN(d 2 )),and for a put
e−r(T−t)(−FN(−d 1 )+KN(d− 2 )),where
d 1 =ln(F/K)+^12 σ^2 (Ti−t)
σ√
Ti−t,d 2 =ln(F/K)−^12 σ^2 (Ti−t)
σ√
Ti−t.HereFis the forward price of the underlying bond at
the option maturity dateT. The volatility of this forward
price isσ. The interest rateris the rate applicable to
the option’s expiration andKis the strike.
Caps and floors A cap is made up of a string of caplets
with a regular time interval between them. The payoff
for theith caplet is max(ri−K,0) at timeTi+ 1 whereri
is the interest rate applicable fromtitoti+ 1 andKis the
strike.
Each caplet is valued under Black ’76 as
e−r(Ti+^1 −t)(FN(d 1 )−KN(d 2 )),whereris the continuously compounded interest rate
applicable fromttoTi+ 1 ,Fis the forward rate from time
Tito timeTi+ 1 ,Kthe strike and
d 1 =ln(F/K)+^12 σ^2 (Ti−t)
σ√
Ti−t,d 2 =ln(F/K)−^12 σ^2 (Ti−t)
σ√
Ti−t,whereσis the volatility of the forward rate.