Frequently Asked Questions In Quantitative Finance

Chapter 6: Black–Scholes and Greeks 301

Table 6.1:Formulæ for European call.

Call

Payoff max(S−K,0)

ValueVSe−D(T−t)N(d 1 )−Ke−r(T−t)N(d 2 )
Black–Scholes value

Delta∂∂VS e−D(T−t)N(d 1 )
Sensitivity to underlying

Gamma∂^2 V
∂S^2

e−D(T−t)N′(d 1 )
σS
√
T−t
Sensitivity of delta to
underlying

Theta∂∂Vt
Sensitivity to time

−

σSe−D(T−t)N′(d 1 )

2

√

T−t +DSN(d^1 )e

−D(T−t)

−rKe−r(T−t)N(d 2 )

Speed∂

(^3) V
∂S^3
−e
−D(T−t)N′(d 1 )
σ^2 S^2 (T−t)
×
(
d 1 +σ
√
T−t
)
Sensitivity of gamma to
underlying
Charm∂
(^2) V
∂S∂t De
−D(T−t)N(d
1 )+e
−D(T−t)N′(d
1 )
Sensitivity of delta to time ×
(
d 2
2(T−t)−
r−D
σ
√
T−t
)
Colour ∂
(^3) V
∂S^2 ∂t
e−D(T−t)N′(d 1 )
σS
√
T−t
Sensitivity of gamma to time ×
(
D+^1 2(−Td^1 −dt)^2 −dσ^1 √(rT−−Dt)
)
Vega∂∂σV S
√
T−te−D(T−t)N′(d 1 )
Sensitivity to volatility
Rho(r)∂∂Vr K(T−t)e−r(T−t)N(d 2 )
Sensitivity to interest rate
Rho(D)∂∂DV −(T−t)Se−D(T−t)N(d 1 )
Sensitivity to dividend yield
Vanna∂∂S^2 ∂σV −e−D(T−t)N′(d 1 )dσ^2
Sensitivity of delta to
volatility
Volga/Vomma∂
(^2) V
∂σ^2
S
√
T−te−D(T−t)N′(d 1 )d^1 σd^2
Sensitivity of vega to volatility