Frequently Asked Questions In Quantitative Finance

304 Frequently Asked Questions In Quantitative Finance

Table 6.4:Formulæ for European binary put.

Binary Put

Payoff 1 ifS<Kotherwise 0

ValueVe−r(T−t)(1−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)(1−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 )− r−D
σ
√
T−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)∂∂Vr −(T−t)e−r(T−t)(1−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