258 CHAPTER 7. THE BLACK-SCHOLES MODEL
Figure 7.7: Value of the call option at various timesMathematical Ideas
To start the examination of each of the sensitivities, restate the Black-Scholes
formula for the value of a European call option:
d 1 =log(S/K) + (r+σ^2 /2)(T−t)
σ√
T−td 2 =log(S/K) + (r−σ^2 /2)(T−t)
σ√
T−tand then
VC(S,t) =SΦ (d 1 )−Ke−r(T−t)Φ (d 2 ).Note thatd 2 =d 1 −σ^2√
T−t.