Frequently Asked Questions In Quantitative Finance

Chapter 6: Black–Scholes and Greeks 303

Table 6.3:Formulæ for European binary call.

Binary Call

Payoff 1 ifS>Kotherwise 0
ValueVe−r(T−t)N(d 2 )
Black–Scholes value

Delta∂∂VS
e−r(T−t)N′(d 2 )
σS
√
T−t
Sensitivity to underlying

Gamma∂^2 V
∂S^2
−e

−r(T−t)d 1 N′(d 2 )
σ^2 S^2 (T−t)
Sensitivity of delta to
underlying
Theta∂∂Vt re−r(T−t)N(d 2 )+e−r(T−t)N′(d 2 )
Sensitivity to time ×

( d

1

2(T−t)−

r−D

σ

√

T−t

)

Speed∂

(^3) V
∂S^3
−e
−r(T−t)N′(d 2 )
σ^2 S^3 (T−t)
×
(
− 2 d 1 +^1 σ−√dT^1 −d^2 t
)
Sensitivity of gamma to
underlying
Charm∂
(^2) V
∂S∂t
e−r(T−t)N′(d 2 )
σS
√
T−t ×
(
r+
1 −d 1 d 2
2(T−t)+
d 2 (r−D)
σ
√
T−t
)
Sensitivity of delta to time
Colour ∂
(^3) V
∂S^2 ∂t
−e
−r(T−t)N′(d
2 )
σ^2 S^2 (T−t)
×
(
rd 1 +^2 2(d^1 T+−dt^2 )−σr√−TD−t
Sensitivity of gamma to
time
×d 1 d 2
( d
1
2(T−t)−
r−D
σ
√
T−t
))
Vega∂∂σV −e−r(T−t)N′(d 2 )dσ^1
Sensitivity to volatility
Rho(r)∂∂rV −(T−t)e−r(T−t)N(d 2 )
Sensitivity to interest rate+
√
T−t
σ e
−r(T−t)N′(d 2 )
Rho(D)∂∂VD −
√
T−t
σ e
−r(T−t)N′(d
2 )
Sensitivity to
dividend yield
Vanna ∂
(^2) V
∂S∂σ −
e−r(T−t)
σ^2 S
√
T−t
N′(d 2 )
(
1 −d 1 d 2
)
Sensitivity of delta to
volatility
Volga/Vomma∂
(^2) V
∂σ^2
e−r(T−t)
σ^2
N′(d 2 )
(
d^21 d 2 −d 1 −d 2
)
Sensitivity of vega to
volatility