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Feynman-Kac and the Connection to PDEs
Hard
·
20 min read
Stochastic Calculus
Feynman-Kac
PDEs
Black-Scholes
1
Article
2
Quiz
Quick Quiz
1.
The Feynman-Kac formula states that
u
(
t
,
x
)
=
E
[
e
−
∫
t
T
r
d
s
φ
(
X
T
)
∣
X
t
=
x
]
u(t,x) = \mathbb{E}\!\left[e^{-\int_t^T r \, ds} \varphi(X_T) \mid X_t = x\right]
u
(
t
,
x
)
=
E
[
e
−
∫
t
T
r
d
s
φ
(
X
T
)
∣
X
t
=
x
]
satisfies which PDE?
u
t
+
L
u
=
0
u_t + \mathcal{L}u = 0
u
t
+
L
u
=
0
u
t
+
L
u
+
r
u
=
0
u_t + \mathcal{L}u + r u = 0
u
t
+
L
u
+
r
u
=
0
u
t
+
L
u
−
r
u
=
0
u_t + \mathcal{L}u - r u = 0
u
t
+
L
u
−
r
u
=
0
−
u
t
+
L
u
−
r
u
=
0
-u_t + \mathcal{L}u - r u = 0
−
u
t
+
L
u
−
r
u
=
0
2.
In the Feynman-Kac derivation, we define
Y
s
=
e
−
∫
t
s
r
d
u
u
(
s
,
X
s
)
Y_s = e^{-\int_t^s r \, du}\, u(s, X_s)
Y
s
=
e
−
∫
t
s
r
d
u
u
(
s
,
X
s
)
. For
Y
s
Y_s
Y
s
to be a local martingale, the drift of
Y
s
Y_s
Y
s
must be:
Equal to
L
u
\mathcal{L}u
L
u
Equal to
−
r
u
-r u
−
r
u
Equal to
r
u
r u
r
u
Zero
3.
The Black-Scholes PDE for a European option on
S
t
S_t
S
t
with risk-free rate
r
r
r
and constant volatility
σ
\sigma
σ
is:
V
t
+
r
s
V
s
+
1
2
σ
2
s
2
V
s
s
−
r
V
=
0
V_t + r s V_s + \frac{1}{2}\sigma^2 s^2 V_{ss} - rV = 0
V
t
+
r
s
V
s
+
2
1
σ
2
s
2
V
ss
−
r
V
=
0
V
t
+
r
s
V
s
+
σ
2
s
2
V
s
s
−
r
V
=
0
V_t + r s V_s + \sigma^2 s^2 V_{ss} - rV = 0
V
t
+
r
s
V
s
+
σ
2
s
2
V
ss
−
r
V
=
0
V
t
+
μ
s
V
s
+
1
2
σ
2
s
2
V
s
s
−
r
V
=
0
V_t + \mu s V_s + \frac{1}{2}\sigma^2 s^2 V_{ss} - rV = 0
V
t
+
μ
s
V
s
+
2
1
σ
2
s
2
V
ss
−
r
V
=
0
V
t
+
r
s
V
s
+
1
2
σ
2
V
s
s
−
r
V
=
0
V_t + r s V_s + \frac{1}{2}\sigma^2 V_{ss} - rV = 0
V
t
+
r
s
V
s
+
2
1
σ
2
V
ss
−
r
V
=
0
4.
For a European call with payoff
(
S
T
−
K
)
+
(S_T - K)^+
(
S
T
−
K
)
+
, what is the boundary condition as
s
→
0
s \to 0
s
→
0
?
V
(
t
,
0
)
=
0
V(t, 0) = 0
V
(
t
,
0
)
=
0
V
(
t
,
0
)
=
K
V(t, 0) = K
V
(
t
,
0
)
=
K
V
(
t
,
0
)
=
−
K
e
−
r
(
T
−
t
)
V(t, 0) = -K e^{-r(T-t)}
V
(
t
,
0
)
=
−
K
e
−
r
(
T
−
t
)
V
(
t
,
0
)
=
K
e
−
r
(
T
−
t
)
V(t, 0) = K e^{-r(T-t)}
V
(
t
,
0
)
=
K
e
−
r
(
T
−
t
)
5.
The classical Feynman-Kac theorem applies to pricing under rough volatility models where instantaneous variance is driven by fractional Brownian motion with
H
<
1
/
2
H < 1/2
H
<
1/2
.
True
False
6.
For American options, in the continuation region (where early exercise is suboptimal), which statement is correct?
The inequality
V
t
+
r
s
V
s
+
1
2
σ
2
s
2
V
s
s
−
r
V
≥
0
V_t + rsV_s + \frac{1}{2}\sigma^2 s^2 V_{ss} - rV \geq 0
V
t
+
r
s
V
s
+
2
1
σ
2
s
2
V
ss
−
r
V
≥
0
holds
The BS PDE holds with equality, and
V
>
(
s
−
K
)
+
V > (s-K)^+
V
>
(
s
−
K
)
+
The BS PDE holds, and
V
=
(
s
−
K
)
+
V = (s-K)^+
V
=
(
s
−
K
)
+
Early exercise is always optimal
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