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Finite Difference Schemes and Convergence Analysis
Hard
·
25 min read
Numerical Methods
Finite Differences
Black-Scholes PDE
Convergence Analysis
1
Article
2
Notebook
3
Quiz
Quick Quiz
1.
Why apply the log-spot transformation
x
=
ln
S
x=\ln S
x
=
ln
S
before discretising the Black-Scholes PDE?
It removes the variable coefficient
1
2
σ
2
S
2
\tfrac12\sigma^2 S^2
2
1
σ
2
S
2
, giving constant coefficients
It linearises a nonlinear terminal condition
It converts the PDE from parabolic to elliptic, which is easier to solve
It maps the semi-infinite domain
(
0
,
∞
)
(0,\infty)
(
0
,
∞
)
onto the bounded interval
[
0
,
1
]
[0,1]
[
0
,
1
]
2.
The explicit (FTCS) scheme needs
λ
=
σ
2
Δ
τ
/
Δ
x
2
≤
1
2
\lambda=\sigma^2\Delta\tau/\Delta x^2\le\tfrac12
λ
=
σ
2
Δ
τ
/Δ
x
2
≤
2
1
. If
Δ
x
\Delta x
Δ
x
is halved, by what factor must
Δ
τ
\Delta\tau
Δ
τ
shrink to stay stable?
Factor of 8
No change is needed
Factor of 4
Factor of 2
3.
Crank-Nicolson is
O
(
Δ
τ
2
+
Δ
x
2
)
O(\Delta\tau^2+\Delta x^2)
O
(
Δ
τ
2
+
Δ
x
2
)
but oscillates near a payoff kink/discontinuity. The standard fix is:
Increase the time step to damp the oscillations
Two implicit (BTCS) start-up steps, then revert to CN (Rannacher)
Replace CN with the explicit scheme to remove the oscillations
Refine the spatial grid only near the strike
4.
The Lax equivalence theorem states that, for a consistent finite-difference approximation of a well-posed linear PDE, stability is equivalent to convergence.
True
False
5.
Von Neumann analysis of BTCS gives
ξ
=
1
/
[
1
+
2
λ
(
1
−
cos
α
Δ
x
)
]
\xi=1/[1+2\lambda(1-\cos\alpha\Delta x)]
ξ
=
1/
[
1
+
2
λ
(
1
−
cos
α
Δ
x
)]
. What is
max
α
∣
ξ
∣
\max_\alpha|\xi|
max
α
∣
ξ
∣
?
Exactly
1
/
(
1
+
4
λ
)
1/(1+4\lambda)
1/
(
1
+
4
λ
)
, attained at
α
=
π
/
Δ
x
\alpha=\pi/\Delta x
α
=
π
/Δ
x
Unbounded for large
λ
\lambda
λ
Exactly 1, attained at
α
=
0
\alpha=0
α
=
0
It depends on the boundary conditions
6.
For a 2-D model (e.g. Heston), plain Crank-Nicolson is inefficient because at each step the implicit system:
Involves an
O
(
M
2
)
×
O
(
M
2
)
O(M^2)\times O(M^2)
O
(
M
2
)
×
O
(
M
2
)
matrix that is not tridiagonal
Crank-Nicolson becomes only conditionally stable in two dimensions
The 2-D PDE is hyperbolic and has no finite-difference approximation
The cross-derivative term makes the PDE nonlinear
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