Numerical methods for option pricing in Feller
Lévy models
O.Reichmann (Seminar for Applied Mathematics, ETH Zürich)
(joint w. Ch.Schwab (SAM, ETH) and R.Schneider (TU-Berlin))
May 20/21 2010, EFEF Warwick
Motivation Theoretical background Implementation Numerical examples Outlook
Motivation
Theoretical background
Implementation
Numerical examples
Outlook
O. Reichmann
May 20/21 2010, EFEF Warwick
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Motivation Theoretical background Implementation Numerical examples Outlook
Pricing problem
Numerical approximation of (weak!) solution
u(x, t) ∈ Domain(AX ) in G ⊆ Rq of fwd Kolmogoroff PIDE
ut + (AX (x; D)u)(x) = f ∈ (0, T ] × G,
u|t=0 = u0 .
(AX (x; D)u)(t, x) :=
1
c(x)u(t, x) + γ(x)⊤ ∇x u(t, x) + σ(x)σ(x)⊤ D 2 u(t, x)
2
+
Z
1
N (x, dy),
u(x + y) − u(x) − y · ∇x u(x)
1 + kyk2
y∈Rd
Applications:
Markovian projection of Semimartingales
Modelling of bounded (jump) processes
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Motivation Theoretical background Implementation Numerical examples Outlook
Feller-processes I
Definition
Assume q = 1. Let X be a strong R-valued Markov process
and let
(Tt g)(x) = E[g(Xt )|X0 = x].
X is called Feller iff
1. Tt : C0 (R) → C0 (R),
2. limt→0+ ku − Tt ukL∞ (R) = 0 for all u ∈ C0 (R).
Theorem
Feller generators admit the positive maximum principle, i.e., if
u ∈ D(AX ) and supx∈R u(x) = u(x0 ) > 0, then (AX u)(x0 ) ≤ 0.
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Motivation Theoretical background Implementation Numerical examples Outlook
Feller-processes II
Theorem
Let AX be the generator of a Feller-process with
C0∞ (R) ⊂ D(AX ), then A|C0∞ (R) is a pseudodifferential operator
(PDO):
(AX u)(x) = −a(x, D)u(x)
Z
1
= −(2π)− 2
a(x, ξ)û(ξ)eixξ dξ, u ∈ C0∞ (R),
R
with symbol a(x, ξ) : R × R → C which is measurable and
locally bounded in (x, ξ) and which admits the Lévy-Khintchine
representation.
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Motivation Theoretical background Implementation Numerical examples Outlook
Feller-processes III
1
a(x, ξ) = c(x) − iγ(x)ξ + (σ(x))2 ξ 2
2
Z iyξ
iyξ
+
1−e +
N (x, dy),
1 + y2
R
R
where supx∈R R min(1, y 2 )N (x, dy) < ∞.
Examples:
1. Brownian motion (local vol) a(x, ξ) = 21 σ(x)2 ξ 2 .
2. Lévy process
R a(x, ξ) = c − iγξ + 12 (σ)2 ξ 2 + R 1 − eiyξ +
O. Reichmann
May 20/21 2010, EFEF Warwick
iyξ
1+y 2
ν(dy).
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Motivation Theoretical background Implementation Numerical examples Outlook
Feller-processes IV
Definition
m(x)
(Symbol class Sρ,δ )
Let 0 ≤ δ ≤ ρ ≤ 1 and let m(x) ∈ C ∞ (R). A symbol a(x, ξ)
m(x)
belongs to Sρ,δ (R) iff
1. a(x, ξ) ∈ C ∞ (R × R),
2. m(x) = s + m(x)
e
with m(x)
e
∈ S(R), s ∈ R,
3. for α, β ∈ N0 there are constants cα,β such that
β α
∀x, ξ ∈ R : Dx Dξ a(x, ξ) ≤ cα,β hξim(x)−ρα+δβ ,
1
where hξi := (1 + ξ 2 ) 2 , ξ ∈ R.
m(x)
The corresponding set of PDOs is denoted by Ψρ,δ (R).
O. Reichmann
May 20/21 2010, EFEF Warwick
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Motivation Theoretical background Implementation Numerical examples Outlook
Feller-processes V
Theorem
(Komatsu, Strook, Jacod 1976, Hoh 1998)
m(x)
For every symbol a(x, ξ) ∈ Sρ,δ there exists a unique Feller
process X with generator AX .
Domain of AX ? Answer: Sobolev spaces of variable order.
Definition
m(x)
The PDO Λm(x) with symbol a(x, ξ) = hξim(x) ∈ S1,δ ,
δ ∈ (0, 1), is called (variable order) Riesz potential.
Corollary
m(x)
(Λm(x) )⊤ ∈ Ψ1,δ
O. Reichmann
2m(x)
and (Λm(x) )⊤ (Λm(x) ) ∈ Ψ1,δ
May 20/21 2010, EFEF Warwick
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Motivation Theoretical background Implementation Numerical examples Outlook
Alternative characterization of PDOs
A PDO in distributional sense can be written as:
Z
KA (x, y)u(y) dy,
Au(x) =
R Z
1
ei(x−y)ξ aA (x, ξ, y) dξ,
KA (x, y) =
2π R
where KA (x, y) is an oscillatory integral, i.e.,
Z
1
KA (x, y) = lim
ei(x−y)ξ aǫA (x, ξ, y) dξ,
ǫ→0 2π R
aǫA (x, ξ, y) = aA (x, ξ, y)µ(ǫy, ǫξ), µ ∈ C0∞ (R × R), µ(0, 0) = 1.
Kikuchi & Negoro 1997, Bass 2002:
a(Λm(x) )⊤ (Λm(x) ) (x, ξ, y) = hξim(x)+m(y) .
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May 20/21 2010, EFEF Warwick
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Motivation Theoretical background Implementation Numerical examples Outlook
Sobolev spaces of variable order
In what follows we always assume m(x) ∈ (0, 1).
Definition
The Sobolev space of variable order is
H m(x) (R) := {u ∈ L2 (R)| kukH m(x) (R) < ∞} where
2
kuk2H m(x) (R) := Λm(x) u
L2 (R)
+ kuk2L2 (R) .
On a bounded domain I we define the space
e m(x) (I) = {u|I u ∈ H m(x) (R), u|
H
R\I
= 0}.
e m(x) (I) is given as
The norm on H
ukH m(x) (R) ,
kukHe m(x) (I) = ke
where u
e is the zero extension
of EFEF
u toWarwick
R.
May 20/21 2010,
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Motivation Theoretical background Implementation Numerical examples Outlook
Wavelets I
e m(x) (I) and obtain a
Aim: Prove a norm equivalence on H
preconditioner for the wavelet matrix of N (x, dz). We require
the following properties of the wavelets:
1. Biorthogonality, i.e., ψl,k , ψel′ ,k′ satisfy
hψl,k , ψel′ ,k′ i = δl,l′ δk,k′ .
2. Local support:
diam suppψl,k ≤ C2−l ,
3. Conformity:
diam suppψel,k ≤ C2−l .
e 1 (I), W
fl ⊂ H
e δ (I) for some δ > 0, l ≥ −1.
Wl ⊂ H
L
L∞ fl
l
4. Density: ∞
l=−1 W ,
l=−1 W dense in L2 (I).
5. Vanishing moments for primal and dual wavelets.
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May 20/21 2010, EFEF Warwick
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Motivation Theoretical background Implementation Numerical examples Outlook
Estimates for extended symbols
Theorem
For any δ ∈ (0, 1) the Schwartz kernel K(Λm(x) )⊤ (Λm(x) ) satisfies
the Caldéron-Zygmund type estimate
α β
−(1+m(x)+m(y)+(1−δ)(α+β))
Dx Dy K(Λm(x) )⊤ (Λm(x) ) (x, y) ≤ Cα,β,δ |x − y|
where x 6= y and |x − y| is small. For large values of |x − y| the
kernel decays faster than |x − y|−N , for any N ∈ N.
Proof:
Littlewood Paley decomposition of unity
Decomposition of the symbol
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Motivation Theoretical background Implementation Numerical examples Outlook
Norm Equivalences I
We consider the infinite matrix (λ = (l, k), λ′ = (l, k′ )):
m(x) ⊤ m(x)
′ , ψλ i
M := hΛm(x) ψλ′ , Λm(x) ψλ i
=
h(Λ
)
Λ
ψ
λ
′
λ,λ ∈I
λ,λ′ ∈I
The following variables will be useful:
mλ = sup{m(x) : x ∈ Ωλ }, mλ = inf{m(x) : x ∈ Ωλ },
[
Ωλ =
{suppψλ′ : suppψλ ∩ suppψλ′ 6= ∅} .
l′ >l
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May 20/21 2010, EFEF Warwick
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Motivation Theoretical background Implementation Numerical examples Outlook
Norm Equivalences II
Theorem
Let D−m(x) := (2−lmλ δλ,λ′ )λ,λ′ and
A := D−m(x) MD−m(x) .
Then A is compressible i.e. there exists s > 0 s.t.
Aλ,λ′ . 2−|l−l′ |(s+ 21 ) (1 + dist(suppψ ′ , suppψλ )−1−2(d−m)(1−δ) .
λ
As compressible matrices have a bounded spectral norm and
D−m(x) Dm(x) also has a bounded spectral norm, we obtain the
norm equivalence:
kukHe m(x)(I) ∼ u⊤ D2m(x) u.
O. Reichmann
May 20/21 2010, EFEF Warwick
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Motivation Theoretical background Implementation Numerical examples Outlook
Implementation of the PIDE
Implementation of FFT methods for the PDO not feasible,
due to nonstationarity of X
Alternative: solve PIDE in “x-space” Rq
Weak solutions: FEM
Compression of jump measure: Wavelets
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May 20/21 2010, EFEF Warwick
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Motivation Theoretical background Implementation Numerical examples Outlook
Assumptions
Let N (x, dz) = k(x, z)dz. Assume that the jump density k(x, z)
satisfies: there exist constants β − > 0 and β + > 1,
0 ≤ δ ≤ ρ ≤ 1 independent of x s.t.
1.
(
−
e−β |z| , z < −1,
k(x, z) ≤ C
+
e−β z , z > 1.
2.
1
2m(x)
|z|
3.
O. Reichmann
∼ k(x, z),
0 < |z| ≤ 1.
β α
−1−2m(x)−αρ−βδ
Dx Dz k(x, z) ≤ cα!β! |z|
May 20/21 2010, EFEF Warwick
∀α, β ∈ N0 , z 6= 0.
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Motivation Theoretical background Implementation Numerical examples Outlook
Derivation of the PIDE I
Let X be a pure jump process without drift. Then
Z
a(x, ξ) = (1 − eizξ + izξ)k(x, z)dz.
R
We can derive for all u(x) ∈ S(R):
Z
1
(AX u)(x) = −
eixξ a(x, ξ)û(ξ) dξ
2π R
Z
= (u(x + z) − u(x) − z∂x u(x))k(x, z) dz.
R
For sufficiently smooth u this can be written as:
Z
u′′ (x + z)k(−2) (x, z) dz,
(AX u)(x) =
R
where
O. Reichmann
k(−i)
is the i-th antiderivative w.r.t. z.
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Motivation Theoretical background Implementation Numerical examples Outlook
Derivation of the PIDE II
The bilinear form for a test function v ∈ C0∞ (R) reads:
Z
(AX u)(x)v(x) dx
b(u, v) =
RZ Z
u′ (x + z)v ′ (x)k(−2) (x, z) dz dx
= −
ZR ZR
u′ (x + z)v(x)kx(−2) (x, z) dz dx.
−
R
O. Reichmann
R
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Motivation Theoretical background Implementation Numerical examples Outlook
Numerical quadrature
Density k(x, z) of N (x, dz) satisfies conditions of
Chernov, von Peterdorff & Schwab 2009
Composite Gauss quadrature used to deal with singularity
at x = y.
Idea: geometric quadrature node refinement towards
singularity of k(x, z).
Exponential convergence of the tensorized (composite)
Gauss quadrature
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Motivation Theoretical background Implementation Numerical examples Outlook
Extension to multidimensional framework
Symbol classes can analogously be defined on Rq
H m(x) , for m(x) = (m1 (x1 ), . . . , mq (xq )) can be
characterized as
H m(x) =
q
\
mj (xj )
Hj
.
(1)
j=1
Construction of the jump measure using a Lévy copula
function
Discretization using tensor product Wavelet basis ⇒ Norm
equivalences.
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Motivation Theoretical background Implementation Numerical examples Outlook
Model problem I
We consider CGMY-type processes:
(
−
e−β z z −1−α(x) ,
z>0
k(x, z) = C
+ |z|
−1−α(x)
−β
e
|z|
, z < 0,
2
α(x) = ke−x + 0.5.
Levy jump kernel
Feller jump kernel
5
4
4
3
3
k(y)
k(x,y)
5
2
2
1
1
0
1
0
1
1
0.5
1
0.5
0.5
0
0.5
0
0
−0.5
X
0
−0.5
−0.5
−1
−1
X
Y
(a) α(x) = 1.75
O. Reichmann
−0.5
−1
−1
(b) α(x) = 1.25e
Figure:
May 20/21 2010, EFEF Warwick
Y
−x2
+ 0.5
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Motivation Theoretical background Implementation Numerical examples Outlook
Stiffness matrices
2
Stiffness matrices for Example I with Y (x) = 1.25e−x + 0.5:
100
100
200
200
300
300
400
400
500
500
600
600
700
700
800
800
900
900
1000
1000
200
400
600
800
1000
200
400
(−2)
(a) k(−2) (x, z)
(b) kx
600
800
1000
(x, z)
Figure: Stiffness matrices
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Motivation Theoretical background Implementation Numerical examples Outlook
Preconditioning
2
Parameter study for condition numbers
10
2
Parameter study for condition numbers
10
k=4.00
k=2.00
k=1.50
k=1.25
k=1.00
k=0.75
k=0.50
Condtition number
Condition number
k=0.25
k=0.50
k=0.75
k=1.00
k=1.25
k=1.45
k=1.49
1
10
1
10
0
10 1
10
2
3
10
10
4
10
Degrees of freedom
0
10 1
10
2
3
10
10
4
10
Degrees of freedom
(a) Model problem I (smooth α)
(b) Model problem II (Lipschitz continuous α)
Figure: Condition numbers for different levels and choices of k.
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May 20/21 2010, EFEF Warwick
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Motivation Theoretical background Implementation Numerical examples Outlook
Option prices
50
Payoff
Y=0.9
variable
Y=0.8
Y=0.7
Y=0.5
Y=0.1
45
40
35
Option price
30
25
20
15
10
5
0
0.5
0.6
0.7
0.8
0.9
1
Moneyness
1.1
1.2
1.3
1.4
1.5
Figure: Option prices for several models for a European put option
with T = 1 and K = 100.
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May 20/21 2010, EFEF Warwick
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Motivation Theoretical background Implementation Numerical examples Outlook
Outlook
Quadratic Hedging
Analysis of model risk via hierarchical models, i.e.
BS ⊆ Local Volatility ⊆ Feller-Lévy.
Preconditioning methods for α(x) ≈ 2.
(Piecewise) smooth time dependent coefficients.
Fast Calibration (P. Carr 2009).
References
R. Schneider, O.R., Ch. Schwab, Wavelet solution of
variable order pseudodifferential equations, Calcolo’09.
O.R. and Ch. Schwab, Numerical analysis of additive, Lévy
and Feller processes with applications to option pricing,
SAM-Report 06-2010.
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