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 Lévy Processes 3. Series Representations 4. Dependence 5. Moments and Risk Indicators 6. General Properties 7. Illustration 2

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

Beyond Lévy Processes stable process -> multistable process : tail parameter α -> α ( t ) so that : 1 1 Lévy measure x 1+ α -> x 1+ α ( t ) 4

Beyond Lévy Processes Multivariate characteristic function of the independent increments multistable process :   d α ( s ) � θ j L II ( t j ) i − � � � � d j =1 θ j 1 [0 ,tj ] ( s ) ds . � �   j =1  e  = e E � �     This is an additive process. 5

Beyond Lévy Processes Multivariate characteristic function of the field-based multistable process :   1 /α ( tj ) C m � + ∞ α ( tj ) sin 2   − 2 � � θ j 2 y 1 /α ( tj ) 1 [0 ,tj ] ( x )  dy dx m     [0 ,T ] 0 � i θ j L FB ( t j )  j =1  j =1   = e E  e   This process has dependent non-stationary increments... ... but still Pareto-like 6

Beyond Lévy Processes Univariate characteristic function of the independent increments tempered multistable process : C � t � ( M − iθ ) Y ( v ) − M Y ( v ) +( G + iθ ) Y ( v ) − G Y ( v ) � 0 Γ( − Y ( v )) dv ϕ Z II ( t ) ( θ ) = e 7

Beyond Lévy Processes Univariate characteristic function of the field-based tempered multistable process : � ( M − iθ ) Y ( t ) − M Y ( t ) +( G + iθ ) Y ( t ) − G Y ( t ) � tC Γ( − Y ( t )) ϕ Z FB ( t ) ( θ ) = e 8

Beyond Lévy Processes First goal : obtain the multivariate characteristic functions of these processes Second goal : study their properties and applications in finance 9

Series Representations Using Rosiński [2007]’s results, we have the following series representation for the CGMY process when Y ∈ (0 , 1) : � Γ j Y  e j V 1 /Y  � − 1 /Y ∞ j � X ( t ) =  1 ( U j ≤ t ) , 0 < t ≤ T, γ j ∧   M + G + γ j M − G 2 CT  j =1 2 2 where Γ j is an arrival time of a Poisson process with unit arrival rate, U j is a uniform random variable on [0 , T ] , V j is a uniform random variable on [0 , 1] , e j is a standard exponential random variable, and γ j is a random variable with distribution P ( γ j = 1) = P ( γ j = − 1) = 1 / 2 . All these random variables are independent. 10

Series Representations Then, when Y ∈ [1 , 2) : � Γ j Y  e j V 1 /Y  � − 1 /Y ∞ j � X ( t ) = γ j  1 ( U j ≤ t ) + t η, 0 < t ≤ T, ∧   M + G + γ j M − G  2 CT j =1 2 2 where, for Y ∈ (1 , 2) : M Y − 1 − G Y − 1 � � η = − Γ(1 − Y ) C and for Y = 1 : M Y − 1 ln( M ) − G Y − 1 ln( G ) M Y − 1 − G Y − 1 � � � � η = (2 κ +ln(2 T )) C + C and where κ is the Euler constant and x ∧ y stands for min( x, y ) . 11

Series Representations For the independent increments tempered multistable process : e j V 1 /Y ( U j )   � − 1 /Y ( U j ) ∞ � Γ j Y ( U j ) j � Z II ( t ) = γ j  1 ( U j ≤ t ) , 0 < t ≤ T, ∧   M + G + γ j M − G  2 CT j =1 2 2 12

Series Representations For the field-based tempered multistable process :  e j V 1 /Y ( t )  � − 1 /Y ( t ) ∞ � Γ j Y ( t ) j � Z FB ( t ) = γ j  1 ( U j ≤ t ) , 0 < t ≤ T, ∧   M + G + γ j M − G  2 CT j =1 2 2 13

Series Representations FB-CGMY simulation experiment with T = 20 , C = 1 , G = 30 , M = 30 and Y ( T ) = 0 . 5 : 10 3 10 4 10 5 10 6 10 7 j max Z FB ( T ) 0.83353 0.90376 0.89712 0.89714 0.89714 14

Dependence Multivariate char. function of the FB-CGMY process : K   � i θkhM ( tk ) 1 u ≤ tk + ∞ + ∞ T 1 1 e − g   � � � � k =1  1 − e  dgdvdxdu  K    2 T � θ k Z FB ( t k ) i u =0 x =0 v =0 g =0   E  e k =1  = e   K   � − i θkhG ( tk ) 1 u ≤ tk T + ∞ 1 + ∞ 1 e − g � � � �   k =1  1 − e  dgdvdxdu   2 T u =0 x =0 v =0 g =0 × e �� xY ( t ) � � − 1 /Y ( t ) ∧ gv 1 /Y ( t ) where h M ( t ) = . 2 CT M 15

Dependence Let s < t . The correlation between the increments Z FB ( t ) − Z FB ( s ) and Z FB ( t + δ ) − Z FB ( s + δ ) satisfies : ∂η (0 , 0) − ∂ 2 Ψ ∂ Ψ ∂θ (0 , 0) ∂ Ψ ∂θ∂η (0 , 0) ρ s,t ( δ ) = �� ∂ Ψ � 2 − ∂ 2 Ψ �� ∂ Ψ � 2 − ∂ 2 Ψ ∂θ (0 , 0) ∂θ 2 (0 , 0) ∂η (0 , 0) ∂η 2 (0 , 0) where θ = θ 1 and η = θ 2 . 16

Dependence Let us assume : s t M G C T 0 1 60 40 1 20 and : Y ( t, a ) = 0 . 1 + 0 . 8 (1 − e − at ) 17

Dependence 0.28 a=0.1 0.26 a=0.2 a=0.5 0.24 Correlation 0.22 0.2 0.18 0.16 1 1.5 2 2.5 3 3.5 4 4.5 5 Lag 18

Dependence Let us assume : s t M G C T 0 1 50 45 1 20 and : Y ( t, a ) = 0 . 1 + 0 . 8 e − at 19

Dependence 0.05 0 −0.05 a=1.4 a=1.7 −0.1 a=2 Correlation −0.15 −0.2 −0.25 −0.3 −0.35 −0.4 1 1.5 2 2.5 3 3.5 4 4.5 5 Lag 20

Moments and Risk Indicators The first four moments of the field-based tempered multistable process are given by : 1 1 � � Mean ( Z FB ( t )) = Ct Γ(1 − Y ( t )) , M 1 − Y ( t ) − G 1 − Y ( t ) 1 1 � � Variance ( Z FB ( t )) = Ct Γ(2 − Y ( t )) M 2 − Y ( t ) + , G 2 − Y ( t ) 21

Moments and Risk Indicators and : � 1 1 � Ct Γ(3 − Y ( t )) M 3 − Y ( t ) − G 3 − Y ( t ) Skewness ( Z FB ( t )) = �� 3 / 2 , 1 1 � � Ct Γ(2 − Y ( t )) M 2 − Y ( t ) + G 2 − Y ( t ) � 1 1 � Ct Γ(4 − Y ( t )) M 4 − Y ( t ) + G 4 − Y ( t ) Kurtosis ( Z FB ( t )) = 3 + �� 2 , � � 1 1 Ct Γ(2 − Y ( t )) M 2 − Y ( t ) + G 2 − Y ( t ) and so on at higher orders. 22

Moments and Risk Indicators 30 a=0.1 a=10 25 20 Kurtosis 15 10 5 0 0 1 2 3 4 5 Time 23

Moments and Risk Indicators VaR can be computed using for instance : � + ∞ F ( x ) = e αx −∞ e iux Φ( iα − u ) du 2 π α + iu where α can take any positive value. 24

Moments and Risk Indicators Let us assume until the end of this presentation : Y ( t, a, b ) = ae − bt 25

Moments and Risk Indicators 50 Y=0.7 45 a=0.7, b=0.001 a=0.7, b=0.002 40 35 30 VaR 25 20 15 10 5 0 0 50 100 150 200 250 t 26

General Properties The independent increments and the field-based tempered multistable processes are semimartingales. 27

General Properties Let us consider an Esscher transform : e θX t � dQ � = � � dP e θX t E t The characteristic triplet of an II-CGMY process X under P :  0 ,    0 , x 1+ Y ( s ) 1 R − + C e − Mx e Gx  C x 1+ Y ( s ) 1 R +   28

General Properties becomes under Q : � 0 � 1 � t � t e ( G + θ ) x − e Gx e − ( M − θ ) x − e − Mx  dxds + C C dxds,  0 − 1 0 0  x Y ( s ) x Y ( s )   0 , C e ( G + θ ) x x 1+ Y ( s ) 1 R − + C e − ( M − θ ) x   x 1+ Y ( s ) 1 R +   29

Illustration We model the logarithmic return of the SP500 Index by the process X defined as follows : X ( t ) = ( µ − q ) t + Z FB ( t ) where Z FB is a field-based tempered multistable process. The calibration of the model is carried out as below : 2 � � N b � � e iθ j x k � � � N � � � e iθ j X ( t ) � k =1 � � � min E − � � � N b � µ,C,G,M,Y j =1 � � � � � � � � 30

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

Illustration Real(CF)−Data Real(CF)−Data 1 1 Real(CF)−Model Real(CF)−Model Imag(CF)−Data Imag(CF)−Data Imag(CF)−Model Imag(CF)−Model 0.8 0.8 Characteristic function Characteristic function 0.6 0.6 0.4 0.4 0.2 0.2 0 0 −0.2 −0.2 −2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 2500 −2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 2500 Theta Theta 32

Illustration The calibrated parameters are : µ (bp/day) Res1 C G M Y 1 2.92 0.2261 344.12 310.26 0.2422 0.1061 Y 10 Res10 a b 0.1481 2.3275 0.2558 0.0547 33

Illustration The autocorrelations are : ρ data ρ model -4.41% -3.73% 34

Illustration For pricing derivatives, we directly model the stock dynamics in the risk-neutral world as follows : S t = S 0 e ( r − q + ω ) t + Z FB ( t ) where ω is defined by : e − ωt = ϕ Z FB ( t ) ( − i ) = E Q � e Z FB ( t ) � 35