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Pricing Algorithms for financial derivatives onbaskets modeled by Lévy copulas
Christoph Winter, ETH Zurich, Seminar for AppliedMathematics
École Polytechnique, Paris, September 6–8, 2006
Introduction Option pricing Discretization Numerical results Conclusion
Introduction
Option pricingPartial integrodifferential equationVariational formulation
DiscretizationSparse tensor product finite element spaceGalerkin discretizationNumerical quadrature of the Lévy copula kernel
Numerical results
Conclusion
C. Winter École Polytechnique, Paris, September 6–8, 2006 p. 2
Introduction Option pricing Discretization Numerical results Conclusion
LiteratureW. Farkas, N. Reich, C. Schwab, “Anisotropic stable Lévycopula processes – Analytical and numerical aspects.”,Research report No. 2006-08, SAM, ETH Zurich , 2006.
J. Kallsen and P. Tankov, “Characterization of dependence ofmultidimensional Lévy processes using Lévy copulas.”, Journalof Multivariate Analysis, Vol. 97, pp. 1551–1572, (2006).
A.-M. Matache, T. von Petersdorff, C. Schwab, “Fastdeterministic pricing of options on Lévy driven assets.”, M2ANMath. Model. Numer. Anal., Vol. 38, pp. 37–71, (2004).
T. von Petersdorff and C. Schwab, “Numerical solution ofparabolic equations in high dimensions.”, M2AN Math. Model.Numer. Anal., Vol. 38, pp. 93–127, (2004).
C. Winter École Polytechnique, Paris, September 6–8, 2006 p. 3
Introduction Option pricing Discretization Numerical results Conclusion
Lévy copula and tail integralA function F : R
d→ R is called Lévy copula if
F (u1, . . . ,ud) 6= ∞ for (u1, . . . ,ud) 6= (∞, . . . ,∞),
F (u1, . . . ,ud) = 0 if ui = 0 for at least one i ∈ 1, . . . ,d,
F is d -increasing,
F i(u) = u for any i ∈ 1, . . . ,d, u ∈ R.
The tail integral U : Rd\0 → R
U(x1, . . . , x2) =
d∏
i=1
sgn(xj)ν
d∏
j=1
I(xj)
.
C. Winter École Polytechnique, Paris, September 6–8, 2006 p. 4
Introduction Option pricing Discretization Numerical results Conclusion
Theorem (Sklar’s theorem for Lévy copulas)For any Lévy process X ∈ R
d exists a Lévy copula F such thatthe tail integrals of X satisfy
U I ((xi )i∈I) = F I ((Ui(xi))i∈I) ,
for any nonempty I ⊂ 1, . . . ,d and any (xi )i∈I ∈ R|I|\0. The
Lévy copula F is unique on∏d
i=1 RanUi .
Lévy density k with marginal Lévy densities k1, . . . , kd ,
k(x1, . . . , xd ) = ∂1 . . . ∂d F |ξ1=U1(x1),...,ξd=Ud (xd )k1(x) . . . kd (x) .
C. Winter École Polytechnique, Paris, September 6–8, 2006 p. 5
Introduction Option pricing Discretization Numerical results Conclusion
Clayton Lévy copulaIn two dimensions (d=2)
F (u, v) =(|u|−θ + |v |−θ
)− 1θ (η1uv≥0 − (1 − η)1uv≤0
),
(α1, α2)-stable marginal densities
k(x1, x2) = (1 + θ)αθ+11 αθ+1
2 |x1|α1θ−1 |x2|
α2θ−1
·(αθ
1 |x1|α1θ + αθ
2 |x2|α2θ)− 1
θ−2 (
η1x1x2≥0 + (1 − η)1x1x2<0).
C. Winter École Polytechnique, Paris, September 6–8, 2006 p. 6
Introduction Option pricing Discretization Numerical results Conclusion
Clayton Lévy copula with marginal of CGMY typeCi = 1, Gi = Mi = 4, Yi = 1 for i = 1,2 and η = 1
2Independent θ = 0.5 (left) and dependent θ = 10 (right) tails.
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Introduction Option pricing Discretization Numerical results Conclusion
Tempered stable Lévy copula processesLet densities k1, . . . , kd be tempered stable.
With Sklar’s theorem for Lévy copulas there exist a Lévyprocess Xt ∈ R
d with marginal densities k1, . . . , kd .
Log prices are solution of the generalized BS equation
∂u∂t
+ Au = 0 , u|t=T = g ,
where A is the infinitesimal generator of the process Xt withdomain D(A).
C. Winter École Polytechnique, Paris, September 6–8, 2006 p. 8
Introduction Option pricing Discretization Numerical results Conclusion
Partial integrodifferential equationAssume S i
t = Si0ert+X i
t , 1 ≤ i ≤ d . The price
V (t ,S) = E
(e−r(T−t)g(ST )|St = S
),
is the solution of
∂V∂t
(t ,S) +12
d∑
i ,j=1
SiSjAij∂2V∂Si∂Sj
+ rd∑
i=1
Si∂V∂Si
(t ,S) − rV (t ,S)
+
∫
Rd
(V (t ,Sez) − V (t ,S) −
d∑
i=1
Si (ezi − 1)
∂V∂Si
(t ,S)
)ν(dz) = 0 .
Terminal condition V (T ,S) = g(S).
C. Winter École Polytechnique, Paris, September 6–8, 2006 p. 9
Introduction Option pricing Discretization Numerical results Conclusion
Transformation to log priceLet xi = log Si , τ = T − t .
∂u∂τ
+ ABS[u] + AJ[u] = 0 ,
with
ABS[ϕ] = −12
d∑
i ,j=1
Aij∂2ϕ
∂xi∂xj+
d∑
i=1
(12
Aii − r)∂ϕ
∂xi+ rϕ ,
AJ[ϕ] = −
∫
Rd
(ϕ(x + z) − ϕ(x) −
d∑
i=1
(ezi − 1)∂ϕ
∂xi(x)
)ν(dz) .
Initial condition
u(0, x) := u0 = g(ex1 , . . . ,exd ) .
C. Winter École Polytechnique, Paris, September 6–8, 2006 p. 10
Introduction Option pricing Discretization Numerical results Conclusion
Variational formulation
Basket option g(ex1 ,ex2 , . . . ,exd ) =(
1 −∑d
i=1 exi
)+.
Weighted Sobolev space
H1η (Rd) :=
ϕ ∈ L1
loc(Rd) | eη(x)ϕ,eη(x) ∂ϕ
∂xi∈ L2(Rd ), i = 1, . . . ,d
,
Payoff g ∈ H1−η(R
d ) where
η(x) =
d∑
i=1
(µ+
i 1xi >0 + µ−i 1xi<0)|xi | ,
with µ+i > 1, µ−i > 0.
C. Winter École Polytechnique, Paris, September 6–8, 2006 p. 11
Introduction Option pricing Discretization Numerical results Conclusion
Bilinear formsWe associate with ABS the bilinear form
aηBS(u, v) =
∫
RdABS[u](x)v(x)e2η(x)
dx ,
and with AJ
aηJ(u, v) =
∫
RdAJ[u](x)v(x)e2η(x)
dx ,
and setaη(u, v) = aη
BS(u, v) + aηJ(u, v) .
C. Winter École Polytechnique, Paris, September 6–8, 2006 p. 12
Introduction Option pricing Discretization Numerical results Conclusion
Continuity and Gårding inequalityAssume A > 0 and η ∈ L1
loc(Rd) satisfies
(i)∂η
∂xi∈ L∞(Rd ) 1 ≤ i ≤ d ,
(ii) η(x + θz) − η(x) ≤ η(z) ∀x , z ∈ Rd , ∀θ ∈ [0,1] ,
(iii)∫
Rdeη(z) |z| 1|z|>1ν(dz) <∞ .
Then, there exist constants C1, C2, C3 > 0 such that∣∣a−η(u, v)
∣∣ ≤ C1 ‖u‖H1−η(Rd ) ‖v‖H1
−η(Rd ) ,∣∣a−η(u,u)
∣∣ ≥ C2 ‖u‖2H1−η(Rd ) − C3 ‖u‖2
L2−η(Rd ) .
C. Winter École Polytechnique, Paris, September 6–8, 2006 p. 13
Introduction Option pricing Discretization Numerical results Conclusion
Sparse tensor product finite element spaced=1: Wavelet basis on [−R,R]
VL = spanψ`
j |0 ≤ ` ≤ L, 1 ≤ j ≤ M`
= W0 ⊕ · · · ⊕W` ,
with increment spaces
W0 := V0 , W` := spanψ`
j : 1 ≤ j ≤ M`.
In [−R,R]d
V L := VL ⊗ · · · ⊗ VL =⊕
0≤`i≤L
W`1 ⊗ · · · ⊗W`d .
Sparse tensor product space
V L :=⊕
0≤`1+···+`d≤L
W`1 ⊗ · · · ⊗W`d .
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Introduction Option pricing Discretization Numerical results Conclusion
Sparse tensor product space (d = 2)
Difference between V L and V L for level L = 3.
C. Winter École Polytechnique, Paris, September 6–8, 2006 p. 15
Introduction Option pricing Discretization Numerical results Conclusion
Galerkin discretizationAnsatz
uL(τ, x) =∑
`,j
u`j (τ)ψ
`j (x) .
Linear system
MdU ′(τ) + AdU(τ) = 0 , ∀τ ∈ (0,T ) ,
U(0) = U0 .
Backward Euler time stepping(
Md + ∆tAd)
U(τm) = MdU(τm−1) , m = 1, . . . ,M ,
U(0) = U0 .
C. Winter École Polytechnique, Paris, September 6–8, 2006 p. 16
Introduction Option pricing Discretization Numerical results Conclusion
Discretized operator (d = 2)We need to compute
aBS(ψ`j , ψ
`′
j ′ ) =d∑
i ,k=1
12
Aik
∫
ΩR
∂ψ`j
∂xi
∂ψ`′
j ′
∂xkdx +
d∑
i=1
12
Aii
∫
ΩR
∂ψ`j
∂xiψ`′
j ′ dx .
In matrix form
A2BS :=
12
A11S ⊗M +12
A22M ⊗S + A12C ⊗(−C)
+12
A11C ⊗M +12
A22M ⊗C .
C. Winter École Polytechnique, Paris, September 6–8, 2006 p. 17
Introduction Option pricing Discretization Numerical results Conclusion
Discretized operator (d = 2)For the jump part
aJ(ψ`j , ψ
`′
j ′ ) = −
∫
Rd
∫
ΩR
(ψ`
j (x + z)ψ`′
j ′ (x) − ψ`j (x)ψ`′
j ′ (x)
−
d∑
i=1
(ezi − 1)∂ψ`
j
∂xiψ`′
j ′ dx
)ν(dz) .
In matrix form
A2J := −
∫
Rd
(M0,z1 ⊗M0,z2 − M ⊗M
− (ez1 − 1)C ⊗M − (ez2 − 1)M ⊗C
)ν(dz) .
C. Winter École Polytechnique, Paris, September 6–8, 2006 p. 18
Introduction Option pricing Discretization Numerical results Conclusion
Numerical quadrature of the Lévy Copula kernelQuadrature points for N = 6 and θ = 0.5.Computation of ∫
ΩR
|z|2 k(z)dz
with Ci = 1, Gi = Mi = 8, Yi = 1 for i = 1,2 and η = 1.
−10 −5 0 5 10−10
−8
−6
−4
−2
0
2
4
6
8
10
6 8 10 12 14 16 18 2010
−12
10−10
10−8
10−6
10−4
10−2
100
Number of Refinments
Err
or
theta =0.5theta =10
C. Winter École Polytechnique, Paris, September 6–8, 2006 p. 19
Introduction Option pricing Discretization Numerical results Conclusion
Operator Matrix (in wavelet basis)For d = 2, L = 5 and R = 5C = (1,1), Y = (1,1), G = (8,8), M = (8,8) and η = 1
2 , θ = 10.
Matrix on full grid (left) and on sparse grid (right).
50 100 150 200 250 300
50
100
150
200
250
300
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Introduction Option pricing Discretization Numerical results Conclusion
Comparison stable vs tempered stable processesFor level L = 6.
Matrix for stable processes (left) and tempered stableprocesses (right).
100 200 300 400 500 600 700
100
200
300
400
500
600
700
100 200 300 400 500 600 700
100
200
300
400
500
600
700
C. Winter École Polytechnique, Paris, September 6–8, 2006 p. 21
Introduction Option pricing Discretization Numerical results Conclusion
Multi-asset optionsLet T = 0.5 and r = 0.
Maximum put options (left) g = (1 − max(S1,S2, . . . ,Sd))+
Basket options (right) g =(
1 −∑d
i=1 Si
)+
0
1
2
00.5
11.5
2
0
0.2
0.4
0.6
0.8
1
SxSy
0
1
2
00.5
11.5
2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
SxSy
C. Winter École Polytechnique, Paris, September 6–8, 2006 p. 22
Introduction Option pricing Discretization Numerical results Conclusion
Influence of the dependence structureDifference between strong and weak dependence for maximumput option (left) and basket option (right)
0
1
2
00.5
11.5
2
−1
−0.5
0
0.5
1
1.5
2
x 10−3
SxSy
0
1
2
00.5
11.5
2
−10
−5
0
5
x 10−4
SxSy
C. Winter École Polytechnique, Paris, September 6–8, 2006 p. 23