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Risk & Greeks
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01 — Article
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Quiz
Quiz: Analytical Greeks for European Options
Module 1 of 4 · Medium
Quick Quiz
1.
Why is
Δ
call
=
N
(
d
1
)
\Delta_{\text{call}}=N(d_1)
Δ
call
=
N
(
d
1
)
rather than
N
(
d
2
)
=
Q
(
S
T
>
K
)
N(d_2)=\mathbb{Q}(S_T>K)
N
(
d
2
)
=
Q
(
S
T
>
K
)
, the risk-neutral probability of finishing in the money?
Because delta is computed under
P
\mathbb{P}
P
, while
N
(
d
2
)
N(d_2)
N
(
d
2
)
uses
Q
\mathbb{Q}
Q
Because delta is the hedge ratio
∂
C
/
∂
S
\partial C/\partial S
∂
C
/
∂
S
, a price sensitivity not a likelihood
Because
d
1
d_1
d
1
and
d
2
d_2
d
2
differ only by a sign, so
N
(
d
1
)
=
N
(
d
2
)
N(d_1)=N(d_2)
N
(
d
1
)
=
N
(
d
2
)
Because
N
(
d
1
)
>
N
(
d
2
)
N(d_1)>N(d_2)
N
(
d
1
)
>
N
(
d
2
)
always, giving a more conservative hedge
2.
By put-call parity, which pair of Greeks is identical for European calls and puts (same
K
K
K
,
T
T
T
, underlying)?
Theta and Rho
Gamma and Vega
Delta and Theta
Delta and Rho
3.
From
Θ
+
1
2
σ
2
S
2
Γ
+
r
S
Δ
−
r
C
=
0
\Theta+\tfrac12\sigma^2 S^2\Gamma+rS\Delta-rC=0
Θ
+
2
1
σ
2
S
2
Γ
+
r
S
Δ
−
r
C
=
0
, the daily P&L of a delta-hedged call (zero net delta) from a spot move
d
S
dS
d
S
and time
d
t
dt
d
t
is approximately:
Θ
d
t
\Theta\,dt
Θ
d
t
1
2
Γ
(
d
S
)
2
\tfrac12\Gamma(dS)^2
2
1
Γ
(
d
S
)
2
Δ
d
S
+
Θ
d
t
\Delta\,dS+\Theta\,dt
Δ
d
S
+
Θ
d
t
1
2
Γ
(
d
S
)
2
+
Θ
d
t
\tfrac12\Gamma(dS)^2+\Theta\,dt
2
1
Γ
(
d
S
)
2
+
Θ
d
t
4.
Vanna
=
∂
2
C
/
∂
S
∂
σ
=\partial^2 C/\partial S\,\partial\sigma
=
∂
2
C
/
∂
S
∂
σ
(no dividends). Which expression is correct?
ν
d
1
d
2
σ
\dfrac{\nu\,d_1 d_2}{\sigma}
σ
ν
d
1
d
2
n
(
d
1
)
d
1
σ
\dfrac{n(d_1)\,d_1}{\sigma}
σ
n
(
d
1
)
d
1
−
n
(
d
1
)
S
σ
τ
-\dfrac{n(d_1)}{S\sigma\sqrt{\tau}}
−
S
σ
τ
n
(
d
1
)
−
n
(
d
1
)
d
2
σ
-\dfrac{n(d_1)\,d_2}{\sigma}
−
σ
n
(
d
1
)
d
2
5.
For a deep in-the-money European call near expiry (
τ
→
0
\tau\to 0
τ
→
0
,
S
≫
K
S\gg K
S
≫
K
), Gamma approaches zero while Delta approaches 1.
True
False
6.
A long OTM call has positive vanna (
−
n
(
d
1
)
d
2
/
σ
>
0
-n(d_1)d_2/\sigma>0
−
n
(
d
1
)
d
2
/
σ
>
0
since
d
2
<
0
d_2<0
d
2
<
0
). With the equity leverage effect (spot down ⇒ implied vol up), what is the vanna P&L on a down day?
Zero — vanna is only relevant for barrier-type options
Negative — the spot-vol cross move costs the position
Indeterminate without knowing the magnitudes of both moves
Positive — the negative spot/vol correlation earns vanna P&L
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