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Multivariate option pricing models: some extensions of the VG model Florence Guillaume Actuarial and Financial Statistics workshop, Eindhoven August 29th-30th, 2011 1 / 41 Time change in finance time-changed Brownian motion W ( T ( t )) in

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SLIDE 1

Multivariate option pricing models: some extensions of the αVG model

Florence Guillaume

Actuarial and Financial Statistics workshop, Eindhoven

August 29th-30th, 2011

1 / 41

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SLIDE 2

Time change in finance

time-changed Brownian motion W (T(t)) in finance: first proposed by Clark (1973) ⇒ Business clock T(t) (positive stochastic process) quantifies the information arrival rate Motivation: information flow directly affects evolution of the price: when low amount of available information, slow trading and price process evolves slowly time change T: characteristics

no loss of information over time ⇒ non decreasing process amount of new information independent on the amount of information previously released ⇒ independent increments amount of released information during [t, t + dt] only depends

  • n the length of that interval dt ⇒ stationary increments

⇒ T ≡ subordinator (non-decreasing L´ evy process)

2 / 41

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SLIDE 3

L´ evy processes

Definition

X = {Xt, t ≥ 0} defined on (Ω, F, P) is a L´ evy process if

X starts at 0: X0 = 0 a.s.; X has independent increments; X has stationary increments. ⇒ Xt+s − Xs ∼ infinitely divisible distribution and has (φ(u))t as CF, where φ(u) is the CF of X1.

Definition

L´ evy-Khintchine representation of an infinitely divisible distribution CF:

log(φX(u)) = iγu − σ2 2 u2 + +∞

−∞

  • exp(iux) − 1 − iux1|x|<1
  • ν(dx)

where γ ∈ R, σ2 ≥ 0 and ν is a measure on R\ {0} satisfying +∞

−∞ inf

  • 1, x2

ν(dx) < ∞

3 / 41

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SLIDE 4

Example: the VG process as time-changed Brownian motion (BM)

The Gamma process CF of Gamma(a, b) with a > 0, b > 0: φGamma(u; a, b) =

  • 1 − iu

b −a Scaling property: if X ∼ Gamma(a, b) then cX ∼ Gamma(a, b/c), c > 0 (1) Xt ∼ Gamma(at, b) (2) If Xi ∼ Gamma(ai, b) are N independent r.v. then

N

  • i=1

Xi ∼ Gamma N

  • i=1

ai, b

  • (3)

4 / 41

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SLIDE 5

Example: the VG process as time-changed Brownian motion (Cont)

The VG process VG(σ, ν, θ) process ≡ Gamma time-changed BM with drift X VG

t

(σ, ν, θ) = θGt + σWGt where G = {Gt, t ≥ 0} ∼ Gamma(t/ν, 1/ν) process and W = {Wt, t ≥ 0} is a standard Brownian motion CF of VG(σ, ν, θ) with σ > 0, ν > 0, θ ∈ R:

φVG(u; σ, ν, θ) =

  • 1 − iuθν + u2σ2ν

2 −1

ν

VG(σ, ν, θ) VG(σ, ν, 0) mean θ variance σ2 + νθ2 σ2 skewness

θν

  • 3σ2+2νθ2
  • σ2+νθ2 3

2

kurtosis 3

  • 1 + 2ν −

νσ4 (σ2+νθ2)2

  • 3(1 + ν)

5 / 41

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SLIDE 6

Multivariate L´ evy models as time changed BM

1 multivariate BM subordinated by an univariate time change

(Madan and Senata, 1990). Unique business clock ⇒ independence can not be captured

2 αVG model (Semeraro, 2008):

extend multivariate time changed BM by considering a subordinator = sum of idiosyncratic and common Gamma subordinators: G (i)

t

= X (i)

t

+ αiZt Original model: constraints on the Gamma subordinator parameters such that G (i)

1

∼ Gamma. CFs: independent of common subordinator settings ⇒ calibration requires multivariate derivatives

3 multivariate L´

evy two factors models (Luciano and Semeraro, 2010): extend the αVG model to other subordinator distributions

6 / 41

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SLIDE 7

Some extensions of the αVG model

1 relax the constraints imposed on the subordinator

parameters in the original model

2 consider Sato processes instead of L´

evy processes

7 / 41

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SLIDE 8

The αVG model

St =      S(1)

t

S(2)

t

. . . S(N)

t

     =           

S(1) exp

  • (r−q1)t+Y (1)

t

  • E
  • exp
  • Y (1)

t

  • S(2)

exp

  • (r−q2)t+Y (2)

t

  • E
  • exp
  • Y (2)

t

  • . . .

S(N) exp

  • (r−qN)t+Y (N)

t

  • E
  • exp
  • Y (N)

t

          Yt =      Y (1)

t

Y (2)

t

. . . Y (N)

t

     =        θ1G (1)

t

+ σ1W (1)

G (1)

t

θ2G (2)

t

+ σ2W (2)

G (2)

t

. . . θNG (N)

t

+ σNW (N)

G (N)

t

       Gt =      G (1)

t

G (2)

t

. . . G (N)

t

     =      X (1)

t

+ α1Zt X (2)

t

+ α2Zt . . . X (N)

t

+ αNZt      where W (i)’s are independent standard BM, αi > 0, Z1 ∼ Gamma(c1, c2), c1, c2 > 0 and X (i)

1

∼ Gamma(ai, bi), ai, bi > 0 are independent r.v. and are independent of the W (i)’s.

8 / 41

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The αVG model: CF’s

CF of Yt: φY(u, t) = E

  • exp(iu′Yt)
  • =

N

  • i=1

φX (i)

1

  • uiθi + i1

2σ2

i u2 i , t

  • φZ1

N

  • i=1

αi

  • uiθi + i1

2σ2

i u2 i

  • , t
  • marginal CF’s

φY (i)(u, t) =

  • 1 − iuθi + i 1

2σ2 i u2

bi −ait 1 − iαi c2

  • uθi + i1

2σ2

i u2

−c1t

c2 = 1 (by space scaling property (1)) On average, business time grows as the real time: E

  • G (i)

t

  • = t ⇒ ai

bi = 1 − αi c1 c2

(4) ⇒ bi

  • 1 − αi

c1 c2

  • > 0

(5)

9 / 41

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SLIDE 10

The αVG model: correlation

ρij = Cov

  • Y (i)

t , Y (j) t

  • Var
  • Y (i)

t

  • Var
  • Y (j)

t

  • where

   Cov

  • Y (i)

t , Y (j) t

  • = θiθjαiαj c1

c2

2 t

Var

  • Y (i)

t

  • =
  • θ2

i

  • ai

b2

i + α2

i c1 c2

2

  • + σ2

i

  • ai

bi + αi c1 c2

  • t

⇒ time independent correlations

10 / 41

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The original αVG model

Original αVG model: impose bi = c2

αi

∀i such that G (i)

1 (3)

∼ Gamma(ai + c1, c2

αi ) (4)

≡ Gamma( c2

αi , c2 αi ) ⇒

φY (i)(u, t) =

  • 1 − iαi

c2

  • uθi + i1

2σ2

i u2

− c2

αi t

⇒ Y (i)

1

∼ VG

  • σi, αi

c2 , θi

  • ⇒ CFs independent of the common subordinator setting

ρij = θiθjαiαj θ2

i

bi + σ2 i θ2

j

bj + σ2 j

c1 ∝ c1.

11 / 41

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SLIDE 12

The generalized αVG model

Generalized αVG model: relax the constraints on the bi’s ⇒ φY (i)(u, t) =

  • 1 − i

uθi+i 1

2 σ2 i u2

bi

−ait 1 − i αi

c2

  • uθi + i 1

2σ2 i u2−c1t

= (φY (i)(u, 1))t ⇒ CFs depend on the whole parameter set Y (i)

t

∼ L´ evy process (but not necessarily VG)

12 / 41

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SLIDE 13

Towards the Sato two factors models

Extend the two factors L´ evy model to the class of Sato processes: Sato processes typically lead to a significantly better fit of

  • ption prices in both the strike and time to maturity

dimensions in the univariate case (Carr, Geman, Madan and Yor, 2007)

13 / 41

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Sato processes

Definition

X is self-decomposable if X d = cX + Xc, ∀0 < c < 1, where Xc is independent of X. Self-decomposable distributions are infinitely divisible distributions with a L´ evy-Khintchine representation log ΦX(u) = iγu− σ2 2 u2+ +∞

−∞

  • exp(iux) − 1 − iux1|x|<1

h(x) |x| dx where h(x) ≥ 0 is decreasing for x > 0 and increasing for x < 0. The probability law of the Sato process at time t is obtained by scaling the self-decomposable law of X at unit time: Xt

d

= tγX, where γ = self-similar exponent. Sato processes have independent but time inhomogeneous increments.

14 / 41

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SLIDE 15

The Sato VG process

From the L´ evy density of the VG process, νVG(dx) = C exp(Gx)

|x|

dx x < 0,

C exp(−Mx) |x|

dx x > 0, the VG probability law at unit time is self-decomposable for all acceptable VG parameter sets {σ, ν, θ}. CF of VG Sato process at time t: X VG Sato

t

(σ, ν, θ, γ) = tγX VG

1

(σ, ν, θ) φVG Sato(u, t; σ, ν, θ, γ) = φVG(u, 1; tγσ, ν, tγθ) =

  • 1 − iuνθtγ + σ2νt2γu2

2

−1

ν . 15 / 41

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SLIDE 16

The Sato αVG model

Yt =      Y (1)

t

Y (2)

t

. . . Y (N)

t

     =      θ1tγ1G (1) + σ1tγ1W (1)

G (1)

θ2tγ2G (2) + σ2tγ2W (2)

G (2)

. . . θNtγNG (N) + σNtγNW (N)

G (N)

     G =     G (1) G (2) . . . G (N)     =     X (1) + α1Z X (2) + α2Z . . . X (N) + αNZ     where W (i)’s are independent standard BM, αi > 0, Z ∼ Gamma(c1, c2), c1, c2 > 0 and X (i) ∼ Gamma(ai, bi), ai, bi > 0 are independent random variables and are independent of the W (i)’s.

16 / 41

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The Sato αVG model (Cont)

CF of Yt:

φY(u, t) = E [exp(iu′Yt)] =

N

  • i=1

φX (i)

  • uiθitγi + i1

2σ2

i t2γiu2 i

  • φZ1

N

  • i=1

αi

  • uiθitγi + i1

2σ2

i t2γiu2 i

  • .

marginal CF’s

φY (i)(u, t) =

  • 1 − i uθitγi + i 1

2 σ2 i t2γi u2

bi −ai 1 − i αi c2

  • uθitγi + i 1

2 σ2

i t2γi u2

−c1

c2 = 1 E

  • G (i)

= 1 ⇒ ai

bi = 1 − αi c1 c2 ⇒ bi

  • 1 − αi c1

c2

  • > 0

17 / 41

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The Sato αVG model: correlations

ρij = Cov

  • Y (i)

t , Y (j) t

  • Var
  • Y (i)

t

  • Var
  • Y (j)

t

  • where

Cov

  • Y (i)

t , Y (j) t

  • = θiθjαiαj

c1 c2

2

tγi+γj and Var

  • Y (i)

t

  • =
  • θ2

i

ai b2

i

+ α2

i

c1 c2

2

  • + σ2

i

ai bi + αi c1 c2

  • t2γi

⇒ ρSato

ij

= ρL´

evy ij

18 / 41

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SLIDE 19

The original Sato αVG model

Original Sato αVG model: impose bi = c2

αi

∀i such that G (i) ∼ Gamma(bi, bi) ⇒ φY (i)(u, t) =

  • 1 − iuθitγi + i 1

2σ2 i t2γiu2

bi −bi ⇒ Y (i)

t

∼ VG Sato

  • σi, 1

bi , θi, γi

  • ⇒ CFs independent of the common subordinator setting

ρij = θiθjαiαj θ2

i

bi + σ2 i θ2

j

bj + σ2 j

c1 ∝ c1.

19 / 41

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SLIDE 20

The generalized Sato αVG model

Generalized Sato αVG model: relax the constraints on the bi’s ⇒ φY (i)(u, t) =

  • 1 − i

uθitγi +i 1

2 σ2 i t2γi u2

bi

−ai

  • 1 − i αi

c2

  • uθitγi + i 1

2σ2 i t2γiu2−c1

⇒ CFs depend on the whole parameter set

20 / 41

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SLIDE 21

The generalized Sato αVG model

Under the generalized Sato model, Y (i)

t

∼ Sato

Proof.

φY (i)(u, 1) =

  • 1 − i

uθi+i 1

2 σ2 i u2

bi

−ai 1 − i αi

c2

  • uθi + i 1

2σ2 i u2−c1

= φVG

  • u;
  • ai

bi σi, 1 ai , ai bi θi

  • φVG
  • u;
  • c1

c2 αiσi, 1 c1 , c1 c2 αiθi

  • ⇒ Y (i)

1 d

= U(i)

1

+ U(i)

2 , where U(i) 1

∼ VG

  • ai

bi σi, 1 ai , ai bi θi

  • and

U(i)

2

∼ VG

  • c1

c2 αiσi, 1 c1 , c1 c2 αiθi

  • are independent VG r.v ⇒ Y (i)

1 ’s

are self decomposable since they are the sum of 2 self decomposable r.v.

21 / 41

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SLIDE 22

Calibration: original αVG model I

Original αVG model: dissociate the calibration of the univariate option surfaces and the calibration of the correlations (Leoni and Schoutens 2008, Luciano and Semeraro 2010):

1 simultaneous calibration of each option surface:

MRMSE =

N

  • i=1

RMSE(i) N =

N

  • i=1

1 N

  • M(i)

j=1

  • P(i)

j

− ˆ P(i)

j

2 M(i) where

N ≡ number of underlying stocks M(i) ≡ number of quoted options for the ith stock P(i)

j

≡ jth market option price of the ith stock ˆ P(i)

j

≡ jth model option price of the ith stock

⇒≡ N univariate option surface calibrations since CF’s do not share any common parameter

22 / 41

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SLIDE 23

Calibration: original αVG model II

2 calibrate parameters which do not influence any marginal CF,

i.e. c1 on market implied or historical stock correlations (Moving Window or EWMA technique): RMSEρ =

  • 1

N2−N 2 N

  • i,j=i

(ρij − ˆ ρij)2 BUT: (5) ⇒ c1 <

1 maxi αi AND only 1 parameter to fit N2−N 2

correlations

23 / 41

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SLIDE 24

Calibration: generalized αVG model

Generalized αVG model Univariate option surfaces only: minimize

  • i

RMSE(i) N =

  • i

RMSE(i)(θi, σi, αi, bi|c1) N and repeat for different values of c1 ⇒ MRMSE∗. Univariate option surfaces + correlation:

MRMSEJ =

N

  • i=1

RMSE(i) N + αρMRMSE∗

  • 1

N2−N 2 N

  • j,k=j

(ρjk − ˆ ρjk)2

where MRMSE∗ = optimal value of MRMSE given by the calibration of the option surfaces and αρ ≥ 0 specifies the relative importance of the correlation matching

24 / 41

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SLIDE 25

Numerical study: data I

basket = 4 of the 10 largest (market cap) S&P500 components which satisfy: Di ≤ K (1 − exp(−r(ti+1 − ti))) , i = 1, 2, . . . , n − 1 and Dn ≤ K (1 − exp(−r(T − tn))) where Di = dividend at ith ex-dividend dates ti with t0 < ti < . . . < tn < T implied correlations: inferred from the S&P500 implied correlation index (= measure of the expected average correlation of price return): ρCBOE ∼ ρ

  • S(i)

Tρ

S(i)

t0

,

S(j)

Tρ

S(j)

t0

  • infer ρ
  • Y (i)

Tρ , Y (j) Tρ

  • by Taylor series expansions

25 / 41

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SLIDE 26

Numerical study: data II

02/06/08 11/08/08 21/10/08 02/01/09 17/03/09 28/05/09 07/08/09 19/10/09 40 50 60 70 80 90 100 Trading day S&P 500 implied correlation index S&P 500 implied correlation index T = January 2009 T = January 2010 T = January 2011 26 / 41

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SLIDE 27

The αVG models: calibration performance (univariate option surfaces) I

04/06/08 10/09/08 24/12/08 08/04/09 22/07/09 28/10/09 0.5 1 1.5

Trading day

MRMSE Option surface calibration performance − Lévy models (αρ = 1) multivariate BS

  • riginal αVG

generalized αVG generalized αVG (step2) 04/06/08 10/09/08 24/12/08 08/04/09 22/07/09 28/10/09 0.5 1 1.5 Trading day MRMSE Option surface calibration performance − Sato models (αρ = 1) multivariate BS

  • riginal αVG Sato

generalized αVG Sato generalized αVG Sato (step2) 04/06/08 10/09/08 24/12/08 08/04/09 22/07/09 28/10/09 20 40 60 80 Trading day VIX VIX index

27 / 41

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SLIDE 28

The αVG models: calibration performance (univariate option surfaces) II

04/06/08 10/09/08 24/12/08 08/04/09 22/07/09 28/10/09 0.1 0.2 0.3 0.4

Trading day

MRMSE Option surface calibration performance − Lévy models (αρ = 1) multivariate BS

  • riginal αVG

generalized αVG generalized αVG (step2) 04/06/08 10/09/08 24/12/08 08/04/09 22/07/09 28/10/09 0.1 0.2 0.3 0.4 Trading day MRMSE Option surface calibration performance − Sato models (αρ = 1) multivariate BS

  • riginal αVG Sato

generalized αVG Sato generalized αVG Sato (step2)

28 / 41

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SLIDE 29

The αVG models: calibration performance (correlations) I

04/06/08 10/09/08 24/12/08 08/04/09 22/07/09 28/10/09 0.2 0.4 0.6 0.8 1 Trading day RMSEρ Implied correlation calibration performance (αρ = 1)

  • riginal αVG

generalized αVG generalized αVG (step2) 04/06/08 10/09/08 24/12/08 08/04/09 22/07/09 28/10/09 0.2 0.4 0.6 0.8 1 Trading day RMSEρ Implied correlation calibration performance (αρ = 1)

  • riginal αVG Sato

generalized αVG Sato generalized αVG Sato (step2)

29 / 41

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SLIDE 30

The αVG models: calibration performance (correlations) II

04/06/08 10/09/08 24/12/08 08/04/09 22/07/09 28/10/09 20 40 60 80 100 Trading day c1 c1 − Lévy models (αρ = 1)

  • riginal αVG

generalized αVG generalized αVG (step2) 1/max(αi) 04/06/08 10/09/08 24/12/08 08/04/09 22/07/09 28/10/09 5 10 15 20 25 30 Trading day c1 c1 − Sato models(αρ = 1)

  • riginal αVG Sato

generalized αVG Sato generalized αVG Sato (step2) 1/max(αi)

30 / 41

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SLIDE 31

The αVG models: calibration performance (correlations) III

04/06/08 10/09/08 24/12/08 08/04/09 22/07/09 28/10/09 0.2 0.4 0.6 0.8 1 Trading day Maximal attainable ρ Maximal attainable correlation − Original Lévy model 04/06/08 10/09/08 24/12/08 08/04/09 22/07/09 28/10/09 0.2 0.4 0.6 0.8 Trading day Maximal attainable ρ Maximal attainable correlation − Original Sato model ρ (Apple, Exxon) ρ (Apple, Microsoft) ρ (Apple, Intl) ρ (Exxon, Microsoft) ρ (Exxon, Intl) ρ (Microsoft, Intl) 31 / 41

slide-32
SLIDE 32

Alternative calibrations of the original model

04/06/08 10/09/08 24/12/08 08/04/09 22/07/09 28/10/09 0.2 0.4 0.6 0.8 1 1.2 1.4 Trading day MRMSE Option surface calibration performance (αρ = 1) multivariate BS

  • riginal αVG (decoupling)
  • riginal αVG (decoupling step 2)
  • riginal αVG (joint calibration)

generalized αVG generalized αVG (step2) 04/06/08 10/09/08 24/12/08 08/04/09 22/07/09 28/10/09 0.2 0.4 0.6 0.8 1 1.2 Trading day RMSEρ Implied correlation calibration performance (αρ = 1)

  • riginal αVG (decoupling)
  • riginal αVG (decoupling step 2)
  • riginal αVG (joint calibration)

generalized αVG generalized αVG (step2)

32 / 41

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SLIDE 33

Choice of αρ

1 1.5 2 2.5 3 0.145 0.15 0.155 0.16 0.165 0.17 0.175 αρ MRMSE MRMSE(αρ) − generalized αVG model 26/08/2009 1 1.5 2 2.5 3 0.05 0.1 0.15 0.2 αρ RMSEρ RMSEρ(αρ) − generalized αVG model 26/08/2009

Figure: Influence of αρ for the generalized αVG model.

33 / 41

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SLIDE 34

Multivariate exotic options: Rainbow options

Worst-of call

WC = exp(−rT)EQ max

  • min
  • S(1)

T

− S(1) S(1) , S(2)

T

− S(2) S(2) , S(3)

T

− S(3) S(3) , S(4)

T

− S(4) S(4)

  • , 0
  • Best-of call

BC = exp(−rT)EQ max

  • max
  • S(1)

T

− S(1) S(1) , S(2)

T

− S(2) S(2) , S(3)

T

− S(3) S(3) , S(4)

T

− S(4) S(4)

  • , 0
  • ⇒ when ρ increases ⇒ asset prices tend to be the same ⇒

minimum of asset prices increases and maximum decreases

34 / 41

slide-35
SLIDE 35

Rainbow options: model risk

04/06/08 10/09/08 24/12/08 08/04/09 22/07/09 28/10/09 0.02 0.04 0.06 0.08 0.1 Trading day Worst−of call 6 months ATM worst−of call

  • riginal αVG

generalized αVG

  • riginal αVG Sato

generalized αVG Sato 04/06/08 10/09/08 24/12/08 08/04/09 22/07/09 28/10/09 0.5 1 1.5 Trading day Best−of call 6 months ATM best−of call

  • riginal αVG

generalized αVG

  • riginal αVG Sato

generalized αVG Sato

35 / 41

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SLIDE 36

Multivariate exotic options: Spread options

Spread options Spread = exp(−rT)EQ max

  • S(1)

T

S(1) − S(2)

T

S(2) , 0

  • ⇒ when ρ increases ⇒ asset prices tend to be the same ⇒ spread

decreases

36 / 41

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SLIDE 37

Spread options: model risk

04/06/08 10/09/08 24/12/08 08/04/09 22/07/09 28/10/09 0.1 0.2 0.3 0.4 0.5 Trading day Spread option 6 months Exxon Apple spread option

  • riginal αVG

generalized αVG

  • riginal αVG Sato

generalized αVG Sato 04/06/08 10/09/08 24/12/08 08/04/09 22/07/09 28/10/09 0.1 0.2 0.3 0.4 Trading day Spread option 6 months Exxon Microsoft spread option

  • riginal αVG

generalized αVG

  • riginal αVG Sato

generalized αVG Sato

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SLIDE 38

Alternative calibrations of the original model: impact on rainbow option prices

04/06/08 10/09/08 24/12/08 08/04/09 22/07/09 28/10/09 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Trading day WC 1 year ATM worst−of call option price

  • riginal α VG (decoupling)
  • riginal α VG (decoupling step 2)
  • riginal α VG (joint calibration)

generalized α VG generalized α VG (step2) 04/06/08 10/09/08 24/12/08 08/04/09 22/07/09 28/10/09 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 Trading day BC 1 year ATM best−of call option price

  • riginal α VG (decoupling)
  • riginal α VG (decoupling step 2)
  • riginal α VG (joint calibration)

generalized α VG generalized α VG (step2)

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SLIDE 39

The generalized αVG model: conclusions

generalized αVG model: CFs which remain of L´ evy type but become dependent on the whole parameter set ⇒ market-implied calibration does not require anymore the existence of a liquid market for multivariate derivatives volatility (trading activity) depends on both the idiosyncratic and common subordinators ⇒ in line with empirical evidence

  • f the presence of both individual and common business clocks

generalized model: better fit of univariate option surfaces except during high volatility regime periods + correlation goodness of fit significantly improved by performing a second calibration (penalty term assessing the correlation goodness of fit) shortfall of the decoupling calibration procedure for the original αVG model: condition that the business time grows on average as the calendar time implies an upper bound on the common parameter c1 which is a function of the αi’s ⇒ decoupling calibration limits severely admissible value range of c1

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SLIDE 40

The αVG Sato models: Conclusions

Sato processes ⇒ significantly better fit of univariate option surfaces (especially during high volatility regime) Similar correlation fit than the αVG models Test higher values of αρ to improve the correlation fit Significantly different multivariate exotic option prices under the original and generalized models

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SLIDE 41

[1] Carr, P., Geman, H., Madan, D.B. and Yor, M. (2007). Self-decomposability and Option Pricing. Mathematical Finance, 17:1, 31-57. [2] Chicago Board Options Exchange (2009) CBOE S&P 500 Implied Correlation Index. Working paper, Chicago. [3] Leoni, P. and Schoutens, W. (2008) Multivariate smiling. Wilmott magazine [4] Luciano, E. and Semeraro, P. (2010) Multivariate time changes for L´ evy asset models: Characterization and calibration. Journal of Computational and Applied Mathematics, 233, 1937-1953. [5] Madan, D.B. and Senata, E. (1990) The Variance Gamma (V.G.) Model for Share Market Returns. Journal of Business, 63, 511-524. [6] Semeraro, P. (2008) A multivariate Variance Gamma model for financial applications. International Journal of Theoretical and Applied Finance, 11, 1-18.

Thank you for your attention

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