7.6. SENSITIVITY, HEDGING AND THE “GREEKS” 259
Delta
TheDeltaof a European call option is the rate of change of its value with
respect to the underlying security price:
∆ =
∂VC
∂S
= Φ(d 1 ) +SΦ′(d 1 )∂d 1
∂S
−Kexp(−r(T−t))Φ′(d 2 )∂d 2
∂S
= Φ(d 1 ) +S1
√
2 πexp(−d^21 /2)1
Sσ√
T−t−Kexp(−r(T−t))1
√
2 πexp(−d^22 /2)1
Sσ√
T−t= Φ(d 1 ) +S1
√
2 πexp(−d^21 /2)1
Sσ√
T−t−Kexp(−r(T−t))1
√
2 πexp(
−
(
d 1 −σ√
T−t) 2
/ 2
)
1
Sσ√
T−t= Φ(d 1 ) +exp(−d^21 /2)
√
2 πσ√
T−t×
[
1 −
Kexp(−r(T−t))
Sexp(
d 1 σ√
T−t−σ^2 (T−t)/ 2)]
= Φ(d 1 ) +exp(−d^21 /2)
√
2 πσ√
T−t×
[
1 −
Kexp(−r(T−t))
Sexp(
log(S/K) + (r+σ^2 /2)(T−t)−σ^2 (T−t)/ 2)
]
= Φ(d 1 ) +exp(−d^21 /2)
√
2 πσ√
T−t×
[
1 −
Kexp(−r(T−t))
Sexp (log(S/K) +r(T−t))]
= Φ(d 1 )Note that since 0<Φ(d 1 )<1 (for all reasonable values ofd 1 ), ∆>0, and
so the value of a European call option is always increasing as the underlying
security value increases. This is precisely as we intuitively predicted when