December 2015
Intermediate to advanced
522 pages
20h
English
Content preview from Stochastic Volatility Modeling
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458 Stochastic volatility modeling
ϕ (t ≥ 0, S, X) is obtained by solving the forward Kolmogorov equation:
dϕ
dt
= Lϕ (12.12a)
L = L
S
+ L
X
+ L
SX
(12.12b)
with the initial condition:
ϕ (t = 0, S, X) = δ (S − S
0
) δ(X − X
0
)
Linear operators L
S
, L
X
, L
SX
are dened by their action on a function ψ:
L
S
ψ = −(r − q)
d
dS
(S ψ) +
1
2
d
2
dS
2
f(t, X)σ (t, S)
2
S
2
ψ
(12.13a)
L
X
ψ = k
d
dX
(X ψ) +
1
2
d
2
ψ
dX
2
(12.13b)
L
SX
ψ =
d
2
dSdX
ρ
p
f(t, X)σ (t, S) S ψ
(12.13c)
L
S
, L
SX
involve the local volatility function
σ(t, S)
, thus (12.12) has to be solved
self-consistently with (12.11). The idea of calibrating
σ (t, S)
via a forward PDE-based
algorithm was rst proposed by Alex Lipton in [70].
Finite-dierence ...
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