SLIDE 1
The Tempered Multistable Approach and Asset Return Modeling Olivier - - PowerPoint PPT Presentation
The Tempered Multistable Approach and Asset Return Modeling Olivier - - PowerPoint PPT Presentation
The Tempered Multistable Approach and Asset Return Modeling Olivier Le Courtois Professor of Finance and Insurance EM Lyon Business School 1 Outline of the Talk 1. Bibliography 2. Beyond Lvy Processes 3. Series Representations 4.
SLIDE 2
SLIDE 3
Bibliography
➠ Madan and Milne, MF [1991] ➠ Barndorff-Nielsen, FS [1998] ➠ Eberlein, Keller and Prause, JoB [1998] ➠ Carr, Geman, Madan and Yor, JoB [2002] ➠ Carr, Geman, Madan and Yor, MF [2003] ➠ Falconer and Lévy-Véhel, JTP [2009] ➠ Lévy-Véhel and Liu, WP [2013]
3
SLIDE 4
Beyond Lévy Processes
stable process -> multistable process : tail parameter α -> α(t) so that : Lévy measure
1 x1+α -> 1 x1+α(t)
4
SLIDE 5
Beyond Lévy Processes
Multivariate characteristic function of the independent increments multistable process :
E
e
i
d
- j=1
θjLII(tj)
= e
−
- d
j=1 θj1[0,tj](s)
- α(s)
ds.
This is an additive process.
5
SLIDE 6
Beyond Lévy Processes
Multivariate characteristic function of the field-based multistable process :
E
e
i
m
- j=1
θjLFB(tj)
= e
−2
[0,T]
+∞
sin2
m
- j=1
θj
C 1/α(tj) α(tj) 2y1/α(tj)1[0,tj](x)
dy dx
This process has dependent non-stationary increments... ... but still Pareto-like
6
SLIDE 7
Beyond Lévy Processes
Univariate characteristic function of the independent increments tempered multistable process : ϕZII(t)(θ) = e
C t
0 Γ(−Y (v))
- (M−iθ)Y (v)−MY (v)+(G+iθ)Y (v)−GY (v)
dv
7
SLIDE 8
Beyond Lévy Processes
Univariate characteristic function of the field-based tempered multistable process : ϕZFB(t)(θ) = e
tCΓ(−Y (t))
- (M−iθ)Y (t)−MY (t)+(G+iθ)Y (t)−GY (t)
8
SLIDE 9
Beyond Lévy Processes
First goal : obtain the multivariate characteristic functions
- f these processes
Second goal : study their properties and applications in finance
9
SLIDE 10
Series Representations
Using Rosiński [2007]’s results, we have the following series representation for the CGMY process when Y ∈ (0, 1) : X(t) =
∞
- j=1
γj
ΓjY
2CT
−1/Y
∧ ejV 1/Y
j M+G 2
+ γj M−G
2
1(Uj≤t), 0 < t ≤ T,
where Γj is an arrival time of a Poisson process with unit arrival rate, Uj is a uniform random variable on [0, T], Vj is a uniform random variable on [0, 1], ej is a standard exponential random variable, and γj is a random variable with distribution P(γj = 1) = P(γj = −1) = 1/2. All these random variables are independent.
10
SLIDE 11
Series Representations
Then, when Y ∈ [1, 2) : X(t) =
∞
- j=1
γj
ΓjY
2CT
−1/Y
∧ ejV 1/Y
j M+G 2
+ γj M−G
2
1(Uj≤t)+t η, 0 < t ≤ T,
where, for Y ∈ (1, 2) : η = − Γ(1 − Y ) C
- MY −1 − GY −1
and for Y = 1 : η = (2κ+ln(2T)) C
- MY −1 − GY −1
+C
- MY −1 ln(M) − GY −1 ln(G)
- and where κ is the Euler constant
and x ∧ y stands for min(x, y).
11
SLIDE 12
Series Representations
For the independent increments tempered multistable process : ZII(t) =
∞
- j=1
γj
ΓjY (Uj)
2CT
−1/Y (Uj)
∧ ejV 1/Y (Uj)
j M+G 2
+ γj M−G
2
1(Uj≤t), 0 < t ≤ T,
12
SLIDE 13
Series Representations
For the field-based tempered multistable process : ZFB(t) =
∞
- j=1
γj
ΓjY (t)
2CT
−1/Y (t)
∧ ejV 1/Y (t)
j M+G 2
+ γj M−G
2
1(Uj≤t), 0 < t ≤ T,
13
SLIDE 14
Series Representations
FB-CGMY simulation experiment with T = 20, C = 1, G = 30, M = 30 and Y (T) = 0.5 : jmax 103 104 105 106 107 ZFB(T) 0.83353 0.90376 0.89712 0.89714 0.89714
14
SLIDE 15
Dependence
Multivariate char. function of the FB-CGMY process : E
e
i
K
- k=1
θkZFB(tk)
= e
1 2T T
- u=0
+∞
- x=0
1
- v=0
+∞
- g=0
e−g
1−e
i K
- k=1
θkhM(tk)1u≤tk
dgdvdxdu
× e
1 2T T
- u=0
+∞
- x=0
1
- v=0
+∞
- g=0
e−g
1−e
−i K
- k=1
θkhG(tk)1u≤tk
dgdvdxdu
where hM(t) =
xY (t)
2CT
−1/Y (t) ∧ gv1/Y (t)
M
- .
15
SLIDE 16
Dependence
Let s < t. The correlation between the increments ZFB(t) − ZFB(s) and ZFB(t + δ) − ZFB(s + δ) satisfies : ρs,t(δ) =
∂Ψ ∂θ (0, 0)∂Ψ ∂η (0, 0) − ∂2Ψ ∂θ∂η(0, 0)
∂Ψ
∂θ (0, 0)
2 − ∂2Ψ
∂θ2 (0, 0)
∂Ψ
∂η (0, 0)
2 − ∂2Ψ
∂η2 (0, 0)
where θ = θ1 and η = θ2.
16
SLIDE 17
Dependence
Let us assume : s t M G C T 1 60 40 1 20 and : Y (t, a) = 0.1 + 0.8 (1 − e−at)
17
SLIDE 18
Dependence
1 1.5 2 2.5 3 3.5 4 4.5 5 0.16 0.18 0.2 0.22 0.24 0.26 0.28
Lag Correlation a=0.1 a=0.2 a=0.5
18
SLIDE 19
Dependence
Let us assume : s t M G C T 1 50 45 1 20 and : Y (t, a) = 0.1 + 0.8 e−at
19
SLIDE 20
Dependence
1 1.5 2 2.5 3 3.5 4 4.5 5 −0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0.05
Lag Correlation a=1.4 a=1.7 a=2
20
SLIDE 21
Moments and Risk Indicators
The first four moments of the field-based tempered multistable process are given by : Mean (ZFB(t)) = CtΓ(1 − Y (t))
- 1
M1−Y (t) − 1 G1−Y (t)
- ,
Variance (ZFB(t)) = CtΓ(2 − Y (t))
- 1
M2−Y (t) + 1 G2−Y (t)
- ,
21
SLIDE 22
Moments and Risk Indicators
and : Skewness (ZFB(t)) = CtΓ(3 − Y (t))
- 1
M3−Y (t) − 1 G3−Y (t)
- CtΓ(2 − Y (t))
- 1
M2−Y (t) + 1 G2−Y (t)
3/2,
Kurtosis (ZFB(t)) = 3 + CtΓ(4 − Y (t))
- 1
M4−Y (t) + 1 G4−Y (t)
- CtΓ(2 − Y (t))
- 1
M2−Y (t) + 1 G2−Y (t)
2,
and so on at higher orders.
22
SLIDE 23
Moments and Risk Indicators
1 2 3 4 5 5 10 15 20 25 30
Time Kurtosis a=0.1 a=10
23
SLIDE 24
Moments and Risk Indicators
VaR can be computed using for instance : F(x) = eαx 2π
+∞
−∞ eiux Φ(iα − u)
α + iu du where α can take any positive value.
24
SLIDE 25
Moments and Risk Indicators
Let us assume until the end of this presentation : Y (t, a, b) = ae−bt
25
SLIDE 26
Moments and Risk Indicators
50 100 150 200 250 5 10 15 20 25 30 35 40 45 50
t VaR Y=0.7 a=0.7, b=0.001 a=0.7, b=0.002
26
SLIDE 27
General Properties
The independent increments and the field-based tempered multistable processes are semimartingales.
27
SLIDE 28
General Properties
Let us consider an Esscher transform :
dQ
dP
- t
= eθXt E
- eθXt
- The characteristic triplet of an II-CGMY process X under P :
0, 0, C
eGx x1+Y (s)1R− + C e−Mx x1+Y (s)1R+
28
SLIDE 29
General Properties
becomes under Q :
C
t
−1 e(G+θ)x−eGx xY (s)
dxds + C
t 1
e−(M−θ)x−e−Mx xY (s)
dxds, 0, C e(G+θ)x
x1+Y (s)1R− + C e−(M−θ)x x1+Y (s) 1R+
29
SLIDE 30
Illustration
We model the logarithmic return of the SP500 Index by the process X defined as follows : X(t) = (µ − q)t + ZFB(t) where ZFB is a field-based tempered multistable process. The calibration of the model is carried out as below : min
µ,C,G,M,Y N
- j=1
- E
- eiθjX(t)
−
Nb
- k=1
eiθjxk Nb
- 2
30
SLIDE 31
Illustration
The calibration is performed in two steps. First, we calibrate the model on daily returns and estimate µ, C, G, M and Y (1, a, b). Then, we calibrate the model on ten-day returns and estimate Y (10, a, b). The knowledge of Y (1, a, b) and Y (10, a, b) readily gives a and b.
31
SLIDE 32
Illustration
−2500 −2000 −1500 −1000 −500 500 1000 1500 2000 2500 −0.2 0.2 0.4 0.6 0.8 1
Theta Characteristic function
Real(CF)−Data Real(CF)−Model Imag(CF)−Data Imag(CF)−Model −2500 −2000 −1500 −1000 −500 500 1000 1500 2000 2500 −0.2 0.2 0.4 0.6 0.8 1
Theta Characteristic function
Real(CF)−Data Real(CF)−Model Imag(CF)−Data Imag(CF)−Model
32
SLIDE 33
Illustration
The calibrated parameters are : µ (bp/day) C G M Y1 Res1 2.92 0.2261 344.12 310.26 0.2422 0.1061 Y10 Res10 a b 0.1481 2.3275 0.2558 0.0547
33
SLIDE 34
Illustration
The autocorrelations are : ρdata ρmodel
- 4.41%
- 3.73%
34
SLIDE 35
Illustration
For pricing derivatives, we directly model the stock dynamics in the risk-neutral world as follows : St = S0e(r−q+ω)t+ZFB(t) where ω is defined by : e−ωt = ϕZFB(t)(−i) = EQ
- eZFB(t)
35
SLIDE 36
Illustration
For European call options : C = S0e−qT
1
2 − i 2π
- R
φ(θ − i)e−iθk − 1 θ dθ
− Ke−rT
1
2 − i 2π
- R
φ(θ)e−iθk − 1 θ dθ
where : k = ln
Ke−(r−q)T
S0
36
SLIDE 37
Illustration
and where : φ(θ) = EQ
e
iθ ln
- ST e−(r−q)T
S0
and : φ(θ) = eiθωTEQ
- eiθZFB(T)
= ϕZFB(T)(θ)
- ϕZFB(T)(−i)
iθ
37
SLIDE 38
Illustration
Calibrating on SP500 Index options, we obtain for the CGMY model : C G M Y 0.0208 1861 1.4310 1.4508 and for the field-based extended model : C G M Y 1 Y 2 Y 3 0.2683 24.4760 4.8688 0.9348 0.5270 0.5826
38
SLIDE 39
Illustration
1400 1500 1600 1700 1800 1900 2000 2100 2200 50 100 150 200 250 300
Strike Price
1400 1500 1600 1700 1800 1900 2000 2100 2200 50 100 150 200 250 300
Strike Price
Fit of SPX options by CGMY and FBCGMY models
39
SLIDE 40