Frequently Asked Questions In Quantitative Finance

302 Frequently Asked Questions In Quantitative Finance

Table 6.2:Formulæ for European put.

Put

Payoff max(K−S,0)

ValueV −Se−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 )−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 −

σSe−D(T−t)N′(−d 1 )

2

√

T−t −DSN(−d^1 )e

−D(T−t)

Sensitivity to time +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 )−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 )−d^1 (r−D)
σ
√
T−t
)
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 ∂
(^2) V
∂S∂σ −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