On the application of spectral filters in a Fourier
option pricing technique
M. J. Ruijter1
M. Versteegh2
1 CWI,
C. W. Oosterlee1,2
Centrum Wiskunde & Informatica
2 Delft
University of Technology
CWI Scientific Meeting
21 June 2013
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Introduction
PhD student since October 2010 (Prof.dr.ir. C.W. Oosterlee)
Group Scientific Computing
Computational finance
2 / 14
Introduction
PhD student since October 2010 (Prof.dr.ir. C.W. Oosterlee)
Group Scientific Computing
Computational finance
Fourier option pricing techniques
Stochastic process for asset price
Expected value
Non-smooth functions
Spectral filters
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Asset price St
Real world market
Figure 1: Asset price Imtech.
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Asset price St
Model:
Real world market
Geometric Brownian motion
dSt = µSt dt + σSt dWt
240
220
Asset price St
200
180
160
140
120
100
Figure 1: Asset price Imtech.
80
60
0
0.2
0.4
0.6
Time t
0.8
1
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Asset price St
Model:
Real world market
Geometric Brownian motion
dSt = µSt dt + σSt dWt
Merton’s jump diffusion
240
220
Asset price St
200
180
160
140
120
100
Figure 1: Asset price Imtech.
80
60
0
0.2
0.4
0.6
Time t
0.8
1
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Asset price St
Model:
Real world market
Geometric Brownian motion
dSt = µSt dt + σSt dWt
Merton’s jump diffusion
Variance Gamma process
140
135
130
Asset price St
125
120
115
110
105
100
Figure 1: Asset price Imtech.
95
90
0
0.2
0.4
0.6
Time t
0.8
1
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Option valuation techniques
100
Payoff g(ST)
80
Option value
60
40
v (t0 , S0 ) = E [g (ST )]
20
0
0
50
100
Asset price S
150
200
T
Figure 2: Put payoff function g .
Option valuation techniques
100
Payoff g(ST)
80
Option value
60
40
20
0
0
50
100
Asset price S
150
200
T
v (t0 , S0 ) = E [g (ST )]
Z
g (y )p(y )dy
=
R
Figure 2: Put payoff function g .
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Option valuation techniques
100
Payoff g(ST)
80
Option value
60
40
20
0
0
50
100
Asset price S
150
200
T
v (t0 , S0 ) = E [g (ST )]
Z
g (y )p(y )dy
=
R
Figure 2: Put payoff function g .
Monte Carlo simulation
Numerical integration
Fourier methods (Fourier transform density p is known)
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Fourier-cosine series
The Fourier-cosine series of f (y ) is defined as
f (y ) =
∞
X
′
fˆn cos (ny ) , y ∈ [0, π]
n=0
with Fourier-cosine coefficients
Z
2 π
fˆn =
f (y ) cos (ny ) dy .
π 0
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Fourier-cosine series
The Fourier-cosine series of f (y ) is defined as
f (y ) =
∞
X
′
fˆn cos (ny ) , y ∈ [0, π]
n=0
with Fourier-cosine coefficients
Z
2 π
fˆn =
f (y ) cos (ny ) dy .
π 0
The partial sum is given by
fN (y ) =
N
X
′
fˆn cos (ny ) .
n=0
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Put option payoff g (y )
30
0
g
gN
log10 error
20
g(y)
2
−1
log10 error
25
−2
15
−3
10
−4
5
−5
70
80
90
100
y
110
120
130
−6
70
−4
−6
−8
−5
0
0
−2
−10
80
90
100
y
110
120
130
−12
0
1
2
3
log10 N
4
5
Figure 3: Put payoff and error (N = 16).
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Put option payoff g (y )
30
0
g
gN
log10 error
20
g(y)
2
−1
log10 error
25
−2
15
−3
10
−4
5
−5
70
80
90
100
y
110
120
130
−6
70
−4
−6
−8
−5
0
0
−2
−10
80
90
100
y
110
120
130
−12
0
1
2
3
log10 N
4
5
Figure 3: Put payoff and error (N = 16).
Option value
v (t0 , S0 ) = E [g (ST )]
Z
g (y )p(y )dy .
=
R
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Density function p(y )
Its Fourier-cosine series expansion reads
p(y ) =
N
X
′
p̂n cos (ny ) ,
n=0
with Fourier-cosine coefficients
Z
2 π
p(y ) cos (ny ) dy
p̂n =
π 0
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Density function p(y )
Its Fourier-cosine series expansion reads
p(y ) =
N
X
′
p̂n cos (ny ) ,
n=0
with Fourier-cosine coefficients
Z
2 π
p(y ) cos (ny ) dy
p̂n =
π 0
Z π
2
= Real
p(y ) exp (iny ) dy
π
0
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Density function p(y )
Its Fourier-cosine series expansion reads
p(y ) =
N
X
′
p̂n cos (ny ) ,
n=0
with Fourier-cosine coefficients
Z
2 π
p(y ) cos (ny ) dy
p̂n =
π 0
Z π
2
= Real
p(y ) exp (iny ) dy
π
0
2
≈ Real {ϕ (n)}
π
ϕ is the Fourier transform of the density function.
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Density function p(y )
Its Fourier-cosine series expansion reads
p(y ) =
N
X
′
p̂n cos (ny ) ,
n=0
with Fourier-cosine coefficients
Z
2 π
p(y ) cos (ny ) dy
p̂n =
π 0
Z π
2
= Real
p(y ) exp (iny ) dy
π
0
2
≈ Real {ϕ (n)}
π
ϕ is the Fourier transform of the density function.
⇒ COS method
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Variance Gamma asset price process
40
2
p
pN
35
1
log10 error
30
p(y)
25
20
0
−1
15
10
−2
5
−3
0
−5
−0.2
−0.1
0
0.1
y
0.2
0.3
0.4
−4
−0.2
−0.1
0
0.1
y
0.2
0.3
0.4
Figure 4: Density function VG process (N = 128).
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Put option
Option value v (t0 , S0 )
∂v (t0 , S0 )
Delta
∂S
∂ 2 v (t0 , S0 )
Gamma
∂S 2
They are approximated by the COS method with N coefficients.
Put value
Put Delta
0
Put Gamma
1
3
log10 error
−2
log10 error
log10 error
0
−1
−4
−2
−3
−6
−4
−8
−5
2
1
0
−1
−6
−10
−2
−7
−12
1
2
3
log10 N
4
−8
1
2
3
log10 N
4
−3
1
2
3
log10 N
4
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Spectral filter
The partial sum is given by
fN (y ) =
N
X
′
fˆn cos (ny ) .
n=0
The filtered partial sum is defined by
fN,σ (y ) =
N
X
′
σ(n/N)fˆn cos (ny ) .
n=0
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Spectral filter
The partial sum is given by
fN (y ) =
N
X
′
fˆn cos (ny ) .
n=0
The filtered partial sum is defined by
fN,σ (y ) =
N
X
′
σ(n/N)fˆn cos (ny ) .
n=0
We can rewrite this as a convolution in the physical space:
Z
2 π
fN,σ (y ) =
S(y − t)f (t)dt.
π 0
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Put option payoff g (y )
2
g(y)
20
log10 error
g
gN
gN,σ
25
15
10
−2
−4
−6
5
80
90
100
y
110
120
130
−10
70
gN
gN,σ
0
−2
−4
−6
−8
−8
0
−5
70
2
gN
gN,σ
0
log10 error
30
−10
80
90
100
y
110
120
130
−12
0
1
2
3
log10 N
4
5
Figure 6: Put payoff and error, exponential filter (q = 4).
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Variance Gamma asset price process
40
2
p
pN
pN,σ
30
25
p(y)
pN
pN,σ
0
log10 error
35
20
15
10
−2
−4
−6
5
−8
0
−5
−0.2
−0.1
0
0.1
y
0.2
0.3
0.4
−10
−0.2
−0.1
0
0.1
y
0.2
0.3
0.4
Figure 7: Density function VG process, exponential filter (q = 4).
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Put option
Exponential filter with order q
Put value
Put Delta
0
2
−2
0
Put Gamma
−4
−6
−8
−10
−12
−14
1
log10 error
log10 error
log10 error
2
−2
−4
−6
3
log10 N
4
−4
−6
−8
−8
−10
−10
−12
2
0
−2
−14
1
−12
2
3
log10 N
4
−14
1
no filter
q=4
q=6
q=8
2
3
log10 N
4
Figure 8: Error option value and the Greeks.
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Summary and conclusion
Stochastic process for asset price
Fourier option pricing technique
Slow convergence (Gibbs phenomenon)
Spectral filters
Contact: [email protected]
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