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01 — Article
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Quiz
Quiz: Black-Scholes: Derivation, Greeks, Limitations
Module 1 of 4 · Medium
Quick Quiz
1.
Applying Itô's lemma to
C
(
t
,
S
t
)
C(t,S_t)
C
(
t
,
S
t
)
and forming
Π
=
C
−
Δ
S
\Pi = C - \Delta S
Π
=
C
−
Δ
S
, which term is cancelled by the choice
Δ
t
=
∂
C
/
∂
S
\Delta_t = \partial C/\partial S
Δ
t
=
∂
C
/
∂
S
?
The
d
t
dt
d
t
term containing
1
2
σ
2
S
2
∂
S
S
C
\tfrac12\sigma^2 S^2\,\partial_{SS}C
2
1
σ
2
S
2
∂
S
S
C
(the gamma term)
The
d
W
t
dW_t
d
W
t
diffusion term, leaving a locally riskless portfolio
The drift term containing
μ
S
∂
S
C
\mu S\,\partial_S C
μ
S
∂
S
C
(removing
μ
\mu
μ
directly)
The
d
t
dt
d
t
term containing
∂
t
C
\partial_t C
∂
t
C
(the theta term)
2.
Why does the physical drift
μ
\mu
μ
of the stock not appear in the Black-Scholes price?
Because Black-Scholes is valid only in the special case where
μ
=
r
\mu = r
μ
=
r
holds
Because the stock is quoted under
P
\mathbb{P}
P
with
μ
\mu
μ
calibrated to market option prices
Because
μ
\mu
μ
is unobservable, so it is conventionally set to zero in the formula
Because hedging removes all
d
W
t
dW_t
d
W
t
risk, and no-arbitrage then fixes the return at
r
r
r
3.
How do the Gammas of a European call and a European put with identical
K
K
K
and
T
T
T
compare?
Equal only at the money, i.e. when
S
=
K
S=K
S
=
K
Equal in magnitude, opposite in sign:
Γ
P
=
−
n
(
d
1
)
/
(
S
σ
τ
)
\Gamma_P=-n(d_1)/(S\sigma\sqrt{\tau})
Γ
P
=
−
n
(
d
1
)
/
(
S
σ
τ
)
Equal and positive:
Γ
C
=
Γ
P
=
n
(
d
1
)
/
(
S
σ
τ
)
\Gamma_C=\Gamma_P=n(d_1)/(S\sigma\sqrt{\tau})
Γ
C
=
Γ
P
=
n
(
d
1
)
/
(
S
σ
τ
)
Different by a financing term:
Γ
P
=
Γ
C
−
e
−
r
τ
\Gamma_P=\Gamma_C-e^{-r\tau}
Γ
P
=
Γ
C
−
e
−
r
τ
4.
Over a small interval
d
t
dt
d
t
, the P&L of a delta-hedged long call when realised vol
σ
R
\sigma_R
σ
R
differs from implied vol
σ
^
\hat{\sigma}
σ
^
is approximately:
1
2
S
2
Γ
(
σ
R
−
σ
^
)
d
t
\tfrac12 S^2\Gamma\,(\sigma_R-\hat{\sigma})\,dt
2
1
S
2
Γ
(
σ
R
−
σ
^
)
d
t
S
2
Γ
(
σ
R
2
−
σ
^
2
)
d
t
S^2\Gamma\,(\sigma_R^2-\hat{\sigma}^2)\,dt
S
2
Γ
(
σ
R
2
−
σ
^
2
)
d
t
1
2
S
Δ
(
σ
R
2
−
σ
^
2
)
d
t
\tfrac12 S\,\Delta\,(\sigma_R^2-\hat{\sigma}^2)\,dt
2
1
S
Δ
(
σ
R
2
−
σ
^
2
)
d
t
1
2
S
2
Γ
(
σ
R
2
−
σ
^
2
)
d
t
\tfrac12 S^2\Gamma\,(\sigma_R^2-\hat{\sigma}^2)\,dt
2
1
S
2
Γ
(
σ
R
2
−
σ
^
2
)
d
t
5.
Breeden-Litzenberger states
∂
2
C
/
∂
K
2
=
e
−
r
T
p
S
T
Q
(
K
)
\partial^2 C/\partial K^2 = e^{-rT} p^{\mathbb{Q}}_{S_T}(K)
∂
2
C
/
∂
K
2
=
e
−
r
T
p
S
T
Q
(
K
)
, so a full call-price surface determines the risk-neutral marginal density of
S
T
S_T
S
T
at each maturity.
True
False
6.
Which observation is the single most direct empirical contradiction of the Black-Scholes assumptions?
Implied volatility varies with strike and maturity (the volatility smile/skew)
Black-Scholes cannot be applied to American-style options at all
Black-Scholes needs the drift
μ
\mu
μ
, which cannot be observed in the market
Black-Scholes can return negative prices for deep out-of-the-money calls
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