On moment non-explosions for Wishart-based stochastic volatility models

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Stochastics and Statistics

On moment non-explosions for Wishart-based stochastic volatility models José Da Fonseca∗ Auckland University of Technology, Business School, Department of Finance, Private Bag 92006, Auckland 1142, New Zealand

a r t i c l e

i n f o

Article history: Received 23 January 2016 Accepted 23 April 2016 Available online xxx Keywords: Pricing Moment non-explosions Wishart multidimensional stochastic volatility model Wishart affine stochastic correlation model

a b s t r a c t This paper provides a result on moment non-explosions for a stock following a Wishart multidimensional stochastic volatility dynamics or a Wishart affine stochastic correlation dynamics when the parameter values satisfy certain constraints. By reformulating the stock dynamics in terms of the volatility path along with standard results on matrix Lyapunov and Riccati equations, a non-explosion result of the moment of order greater than one can be obtained. It extends to these frameworks a property well known for the Heston model.

1. Introduction In (Andersen & Piterbarg, 2007), the authors consider the problem of moment explosions for different stochastic volatility models. They show that under certain constraints on the parameters the moment of order higher than one of the stock can explode in finite time. By complementarity, for certain parameter values there is no explosion, a result that is also useful for option pricing using a Fourier transform approach. This kind of result is quite easy to obtain for the model proposed by Heston (1993) as the momentgenerating function is known in closed form. The Wishart Multidimensional Stochastic Volatility model (WMSV), proposed in (Da Fonseca, Grasselli, & Tebaldi, 2008), and the Wishart Affine Stochastic Correlation model (WASC), proposed in (Da Fonseca, Grasselli, & Tebaldi, 2007), are extensions of the Heston model whose main feature is that they involve for the volatility dynamics a Wishart process proposed by Bru (1991). The WMSV model is a single-stock multidimensional stochastic volatility model while the WASC model is a multi-asset stochastic volatilityn-explosions directly from the moment-generating function of the stock is not feasible (or at least we did not manage to obtain such kind of results). However, by rewriting the stock price dynamics as a function of the volatility path and by performing a convenient conditioning, similar to the one used in (Da Fonseca, Gnoatto, & Grasselli, 2015), the problem can be reformulated in terms of the

∗

Tel.: +64 9 9219999x5063; fax: +64 9 9219940. E-mail address: [email protected]

© 2016 Elsevier B.V. All rights reserved.

Wishart moment-generating function for which standard results allow us to draw conclusions on moment non-explosions in these two models. The structure of the paper is as follows: in Section 2 we present the models and the moment-generating functions; in Section 3 we provide the results on moment non-explosions; in Section 4 numerical examples based on real parameter values are given; the last section concludes. 2. The models 2.1. The WMSV model In this section we briefly review the main results regarding the WMSV proposed in (Da Fonseca et al., 2008). In this model the volatility is described by the Wishart process, a matrix-valued stochastic process defined in (Bru, 1991) and introduced in finance in (Gouriéroux & Sufana, 2010), and the dynamics for the stock price which is given by the following stochastic differential equation (SDE):

dst = st Tr



   t dWt R + dBt In − RR ,

(1)

with s0 > 0, Tr is the trace operator, In is the identity matrix of size n, Wt , Bt ∈ Mn (the set of square matrices) are composed by n2 independent Brownian motions under the risk-neutral measure (Bt and Wt are independent), R ∈ Mn represents the correlation matrix and  t belongs to Sn+ , the set of n × n symmetric positive semidefinite matrices (we suppose that the risk free rate is zero). In this specification the volatility is multi-dimensional and depends

http://dx.doi.org/10.1016/j.ejor.2016.04.042 0377-2217/© 2016 Elsevier B.V. All rights reserved.

Please cite this article as: J. Da Fonseca, On moment non-explosions for Wishart-based stochastic volatility models, European Journal of Operational Research (2016), http://dx.doi.org/10.1016/j.ejor.2016.04.042

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on the elements of the matrix process  t , which is assumed to satisfy the following dynamics:



dt = 2 + Mt + t M

 



with



dt



A12 (t )

A21 (t )

A22 (t )

+ t dWt Q + QdWt t , (2) √ with · denoting the matrix square root, initial condition  0 a strictly positive definite matrix, M ∈ Mn ,  ∈ Sn++ and Q ∈ Sn++ , the set of n × n symmetric definite positive matrices. Eq. (2) characterizes the Wishart process and represents the matrix analogue of the square root mean-reverting process. In order to grant the typical mean reverting feature of the volatility, the matrix M is assumed to be such that λm < 0; i = 1 . . . n with i λm ∈ Spec(M ) (i.e. the spectra of M). The constant drift part sati isfies 2 = β Q 2 with the real parameter β ≥ n − 1 and ensures that  t is a (symmetric) positive semi-definite matrix. If β satisfies the stronger assumption β ≥ n + 1 then the unique strong solution to the SDE (2) evolves in Sn++ , see Mayerhofer, Pfaffel, and Stelzer (2011). The constraint on  can be relaxed as explained in (Cuchiero, Filipovic, Mayerhofer, & Teichmann, 2011) but we keep this more parsimonious choice in view of numerical applications as to the best of our knowledge it is the only specification for which a calibration on market option prices is available. This model and the other models presented in this work belong to the class of affine processes, see for example Duffie, Filipovic, and Schachermayer (2003) and Keller-Ressel and Mayerhofer (2015).

and

Remark 2.1. In (Da Fonseca et al., 2015) the noise of (2) is given by   t dWt Q + Q  dWt t with Q ∈ GL(n), the set of invertible n × n matrices. The polar decomposition of Q = U Q˜ with U a unitary matrix and Q˜ ∈ Sn++ along with the invariance of the law of the Brownian motion Wt with respect to unitary transformations imply that the dynamics proposed here is similar to the one considered in (Da Fonseca et al., 2015). Lastly, dWt R + dBt In − RR =  dWt U (RU ) + dBt In − (RU )(RU ) and R˜ = RU is the correlation

A(t ) =

matrix. In this model the instantaneous variance of the asset returns is associated to the trace of the Wishart matrix, that is: dln st  = Tr[t ]dt, which alone is not Markovian and constitutes also a multivariate extension of the Heston model. Lemma 2.1. (see Da Fonseca et al. (2008)) Given a scalar z and two square (symmetric) matrices  , I , the joint moment gener t ating function of (ln st , t , 0 u du ) is denoted Gwmsv (t, z,  , I ) and given by

E[ez ln st +Tr[ t ]+Tr[I

t 0

u du]

] = ez ln s0 +Tr[A(t )0 ]+b(t ) ,

(3)

where the deterministic matrix function A(t) and the scalar function b(t) satisfy the following ODE (ordinary differential equations) where we omit the time variable t:

    dA = A M + zQR + M + zQR A dt z (z − 1 ) + 2AQ 2 A + In + I , 2

(4)

(5)

A(t ) = ( A12 (t ) + A22 (t ))−1 ( A11 (t ) + A21 (t )), 2

⎛

M + zQR

A = ⎝ z (z − 1 ) 2

(8)

⎞

−2Q 2



In + I

− M + zQR

 ⎠.

When n = 1 then the model corresponds to the Heston model presented in (Heston, 1993). In that case the dynamics can be written as

dst = st

√

vt (ρ dw1,t +



dvt = κ (θ − vt )dt + σ

1 − ρ 2 dw2,t ),

(9)

√

vt dw1,t ,

(10)

with s0 > 0, v0 > 0, wt = (w1,t , w2,t )t≥0 a two-dimensional Brownian motion, κ ∈ R, κθ ∈ R+ , σ > 0 and ρ ∈ [−1, 1]. The moment-generating function for the (log) stock st is known in closed form as we have the following lemma. Lemma 2.2. In the model proposed by Heston (1993), the

moment generating function of (ln st ), defined by Ghes (t, z ) = E ez ln st , is equal to exp(z ln s0 + A(t )v0 + b(t )) with the deterministic functions A(t), b(t) defined as: √

z 2 − z 1 − e−  t √ , 2 λ+ − λ− e− t





(11) √

− b(t ) = 2σκθ2 t λ− − log λ+ λ−+λ−− λe −

with 2λ± = (κ − zρσ ) ±

√

t



,

(12)

 ,  = (κ − zρσ )2 − σ 2 (z2 − z ).

We remind the reader of the main result on moment nonexplosion of Andersen and Piterbarg √ (2007). If z > 1 but still such that  > 0 we deduce that  < |κ − ρσ z|. If ρ < 0 (that will be the case in practice) then λ− > 0 (and still λ+ > 0) so t∗ √ ∗ such that λ+ − λ− e− t = 0 is equivalent to t ∗ = − √1 ln( λλ+ ). As −  √ λ+ − λ− =  > 0 we conclude that t∗ < 0, hence there isn’t a moment explosion. The fact that ρ < 0 is the favourable case is intuitive as an increase of the stock implies a decrease of the volatility and therefore a thinner upper tail for the stock. Remark 2.2. The moment explosions problem has attracted quite a lot of attention among academics during the past few years. In addition to the work aforementioned let us also mention the works of Lions and Musiela (2007) and Keller-Ressel (2011). As discussed in (Da Fonseca et al., 2015), in the WMSV model the stock’s variance, the variance of stock’s variance and the correlation between the log-stock and its (instantaneous) variance are given by (in the particular case of n = 2) 2 2 dvar(st ) = Q11 t11 + Q22 t22 dt

with initial conditions A(0 ) =  , b(0 ) = 0. The solution is explicitly given by:

β

= exp tA

dln st  = Tr[t ]dt

db = Tr[2 A], dt

b(t ) = −

A11 (t )

(6)



Tr log( A12 (t ) + A22 (t )) + t (M + zQ  R ) ,

(7)

dCorr(ln st , var(st )) =



Tr[RQ t ]



Tr[t ] Tr[Q  Q t ]

dt.

(13)

The correlation between the log-stock and its (instantaneous) variance, given by (13), is found to be negative in practice (see Da Fonseca & Grasselli (2011)); that is why it was conjectured in (Da Fonseca et al., 2015) that there should be z

 > 1 such that E stz = Gwmsv (t, z, 0n , 0n ) < ∞ ∀t > 0. Hence, a moment greater than one exits for which no explosion occurs. This aspect is also important from an implementation point of view as to perform the pricing of a call option using the Fourier transform, as proposed by Carr and Madan (1999),

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we need to integrate the function z → |Gwmsv (t, iz, 0n , 0n )| ≤ for z∈C with

(z ) < −1, Gwmsv (t, − (z ), 0n , 0n ) thus the finiteness for the moment of order greater than one is needed. It was implicitly used in (Da Fonseca et al., 2008), (Da Fonseca & Grasselli, 2011), (Leung, Wong, & Ng, 2013) or in a WMSV-like foreign exchange option pricing model in (Gnoatto & Grasselli, 2014a). Needless to say that this aspect applies to any option pricing model and is therefore not related to the models presented here, see for example Bao, Li, and Gond (2012) where the parameter α in Eq. (3.6) is used as a damping factor or in (Wong & Lo, 2009) where the parameter also named α in Eq. (3) plays the same role or in (Petrella, 2004). Remark 2.3. From this point we will make the hypothesis that (λri ) < 0; i = 1 . . . n with λri ∈ Spec(R ) (i.e. the spectra of R). 2.2. The WASC model The WASC model of Da Fonseca, Grasselli, and Tebaldi (2007) consists in a n-dimensional vector of risky assets st = (st1 , . . . , stn ) whose dynamics is given by:



dst = diag[st ] t dZt ,

(14)

where Zt ∈ Rn is a vector Brownian motion, while the returns’ variance-covariance matrix  t evolves stochastically, according to the Wishart dynamics (2) introduced previously. The leverage effects and the asymmetric correlation effects are modeled by introducing the  following correlation structure among Brownian motions: dZt = 1 − ρ  ρ dBt + dWt ρ , where ρ is a vector of size n, with ρ ∈ [−1, 1]n and ρ  ρ ≤ 1 (Bt is a n-dimensional Brownian motion under the risk-neutral measure and is independent of Wt ). Remark 2.4. The Remark 2.1 translates to the WASC model case in very similar terms. Indeed, the main difference appears, if we use the  notations of Remark2.1, for the noise driving the stock dZt = 1 − ρ  ρ dBt + dWt ρ = 1 − ρ˜  ρ˜ dBt + dWt U ρ˜ with ρ˜ = U  ρ . Lemma 2.3. (see Da Fonseca et al. (2007)) The joint t moment-generating function of (ln st , t , 0 u du ) is denoted Gwasc (t, z,  , I ) and is equal to:

E[ez



ln st +Tr[ t ]+Tr[I

= ez



t 0

ln s0 +Tr[A(t )0 ]+b(t )

u du]

]

,

(15)

with z ∈ Rn , two square (symmetric) matrices  , I and the deterministic matrix function A(t) satisfies the following ODE:

    dA = A M + Q ρ z + M + Q ρ z A dt n 1  j jj 1 + 2AQ 2 A + zz − z e + I , 2 2

(16)

A=



−2Q 2

L12

− M + Q ρ z

with L12 =

b(t ) = −

1 2



zz −

β

2



n j=1

2 2 dvar(st1 ) = 4(Q11 + Q21 )t11 dt, ρ1 Q11 + ρ2 Q21 dCorr(ln st1 , var(st1 )) =  dt. 2 + Q2 Q11 21

(19)

In (Da Fonseca & Grasselli, 2011) it was found that (19) was negative for the pairs EuroStoxx50/DAX, FTSE/DAX and FTSE/EuroStoxx50 with even the sign constraints ρ 1 < 0 and ρ 2 < 0. As for the WMSV the same remark applies to the WASC regarding the finiteness of the moment-generating function of the stocks. For each stock the moment of order greater than one has to be finite in order to price a call option. This hypothesis was implicitly made in the works Da Fonseca et al. (2007), Da Fonseca and Grasselli (2011) or in Branger and Muck (2012), Chiu, Wong, and Zhao (2015) or Asai and McAleer (2015) dealing with models having a structure similar to the WASC model. Remark 2.5. Although there are several single-stock models there are very few multi-asset models. Apart from the WASC model let us mention Muhle-Karbe, Pfaffel, and Stelzer (2012), Semeraro (2008) and Yoon, Jang, and Roh (2011). 3. Moment non-explosions 3.1. The WMSV case To study the moment non-explosion for the stock we would need to perform similar computations as those done for the Heston model. Although the WMSV is remarkably tractable it is not to the point so as to allow us to determine the behavior of the moment if we work directly with the expressions (6) and (7). However, rewriting conveniently the moment of the stock in terms of the volatility path enables us to bypass this difficulty as the following lemma shows. Lemma 3.1. Let z ∈ R and suppose that R Q = QR then we have





t E[stz ] = c1 E eTr[0 t ]+ 0 Tr[1 u ]du

with



c1 = exp z ln s0 −



(20)

z zt Tr[R 0 Q −1 ] − Tr[R 2 Q −1 ] 2 2



z −1  Q R 2 z z z2 1 = (In − RR ) − In − (Q −1 R M + M RQ −1 ) 2 2 2 Proof. We have

the scalar function b(t) solves the ODE (5), the initial conditions are A(0 ) =  , b(0 ) = 0 and (ei j )i, j=1...n is the canonical basis of Mn . The solution is explicitly given by an equation of the form (6) but with the matrices A11 (t), A12 (t), A21 (t), A22 (t) obtained through the exponentiation (8) of the following matrix

M + Q ρ z

the correlation between the log-stock and its variance for the particular case n = 2 (we consider the first stock but similar equations apply for the second stock) are given by:

0 =

j=1



3

 .

(17)



z

E[stz ] = E ez ln s0 − 2

t 0

√ Tr[u ]du+z 0t Tr[ u dWu R ]

 √ √ t  × E[ez 0 Tr[ u dBu In −RR ] |FW ]

(21)

where FW is the filtration generated by (Wu )0 ≤ u ≤ t . If we denote

 (σti j )i, j=1...n = ( t )i, j=1...n (with σti j σtjk = tik ) and (γ i j )i, j=1...n =  ( In − RR )i, j=1...n (with (γ il γ l j )i, j=1...n = (In − RR )i, j=1...n ), we

deduce that



√ √ t t i1 i2 i2 i3 i i  E[ez 0 Tr[ u dBu In −RR ] |FW ] = E[ez 0 σu dBu γ 3 1 |FW ]

z j e j j + I while b(t) is given by



Tr log( A12 (t ) + A22 (t )) + t (M + Q ρ z ) .

(18)

As mentioned in (Da Fonseca et al., 2015) the stock’s variance given by dln st1  = t11 dt, the variance of the stock’s variance and

z2

=e2

z2

=e2

z2

=e2

t 0

t 0

t 0

i i

i i

σu1 3 σu2 3 γ i4 i1 γ i4 i2 du i i

u1 2 γ i1 i4 γ i4 i2 du

Tr[u (In −RR )]du .

(22)

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Starting from (2) and multiplying by R on the left and Q −1 on the right, takingthe trace and using the hypothesis R Q = QR we obtain 2Tr[ t dWt R ] = Tr[R dt Q −1 ] − Tr[R (2 + Mt + t M )Q −1 ]dt. Combining this equality with (21) and (22) we obtain the result.  Remark 3.1. Notice that R Q = QR is equivalent to RQ −1 = Q −1 R = (RQ −1 ) so RQ −1 is symmetric.

1 =

Proof. According to Lemma 2.1 the expectation in (20) is given by eTr[A(t )0 ]+b(t ) with A(t) satisfying

dA = AM + M A + 2AQ 2 A + 1 dt

(23)

while b solves db/dt = Tr[2 A] and the initial conditions are A(0 ) = 0 and b(0 ) = 0. 1 comprises two parts, the first one 2 is 11 (z ) = z2 (In − RR ) − 2z In while the second one is 21 =  X M + M X with X = −RQ −1 (X is symmetric by hypothesis of Lemma 3.1). Following the hypothesis made on the eigenvalues of R it is clear that 11 (1 ) ∈ Sn−− and as this set is open we deduce that ∃ z > 1 such that 11 (z ) ∈ Sn−− . For 21 we use standard results for Lyapunov equations, see for example Proposition 4.2 in Dullerud and Paganini (20 0 0), and conclude that 21 ∈ Sn−− (the hypothesis made on the sign of the eigenvalues for M is often called the Hurwitz condition). As a result, there exists z > 1 such that 1 ∈ Sn−− and the matrix Riccati equation satisfies the condition of Proposition 1.1 of Dieci and Eirola (1994), which basically requires the coefficient of the quadratic term of (23) to be in Sn++ (Q2 in our case) while the constant term has to be in Sn−− (and the initial condition has to be in Sn−− which is satisfied by 0 by hypothesis), and allows us to conclude that A(t) is well defined for any t and as b is obtained by integrating A, through relation (5), we conclude that there is no explosion. Notice that we could have also used Theorem 2.1 of Wonham (1968).  Remark 3.2. It is important to notice that Dieci and Eirola (1994) allows for weaker conditions on the coefficients. Indeed, the coefficient of the quadratic term of (23) has to be in Sn+ while the constant term has to be in Sn− . Thus, conditions are much weaker than what appears naturally in the WMSV model but these weak conditions will suit nicely for the WASC model as we shall see. 3.2. The WASC case Similarly to the WMSV, we can develop results for the WASC model and focus on the first asset st1 just for illustrative purpose. We have the following lemma giving the expression of the moment of order z1 . Lemma 3.3. Let z1 ∈ R and suppose that ρ e Q = Qe1 ρ  (with 1 (ei )i=1...n the canonical basis of Rn ) then we have





t E[(st1 )z1 ] = c1 E eTr[0 t ]+ 0 Tr[1 u ]du



with z = z1 e1 and



c1 = exp z1 ln s10 −

0 =

1 −1  Q ρz 2

1 t Tr[ρ z 0 Q −1 ] − Tr[ρ z 2 Q −1 ] 2 2

(24)



2

zz −

diag(z ) z1   −1 − (Q −1 ρ e ), 1 M + M e1 ρ Q 2 2

where diag(z) is a n × n matrix with the vector z in its diagonal. Proof.

E[(st1 )z1 ] = E[ez =e

From this lemma we conclude that the moment non-explosion can be determined using the expectation appearing on the right hand side of (20). The following lemma clarifies that point. Lemma 3.2. Under the hypothesis of Lemma 3.1 and λm < i rq rq 0; i = 1 . . . n with λm ∈ Spec ( M ) and λ < 0 ; i = 1 . . . n with λ ∈ i i i Spec(RQ −1 ) then there exists z > 1 such that E[stz ] < ∞ ∀t > 0.

(1 − ρ  ρ )



ln st

z ln s0

= ez



ln s0

]



1

E e− 2



1

E e− 2

 √

×E e

t 0

t 0

Tr[diag(z )u ]du+z Tr[diag(z )u ]du+z

1 −ρ  ρ z 

t 0

√

u dBu

|FW



t

√

0

t 0

√

u dZu



u dWu ρ

.

t  t 1 − ρ  ρ z 0 u dBu )|FW ] = exp((1 − ρ  ρ )/2 0 Tr   t t [zz u ]du ) and z 0 u dWu ρ = 0 Tr[ u dWu ρ z ], multiply −1 ing (2) by ρ z on the left and Q on the right, taking the trace  −1 and using the condition of the lemma we deduce Tr[ρ z t Q ] = Tr[ρ z (2 + Mt + t M )Q −1 ]dt + 2Tr[ t dWt ρ z ]. Integrating this equation and collecting the results we deduce (24).  As

E[exp(



Remark 3.3. For example, in the two-asset case the constraints

ρ ei Q = Qei ρ  for i ∈ {1, 2} lead to

ρ1 Q12 = ρ2 Q11 ρ1 Q22 = ρ2 Q12 . We can proceed as for the WMSV model. The following lemma clarifies the conditions ensuring that there is a stock’s moment of order higher than one that does not explode. Lemma 3.4. Under the hypothesis of Lemma 3.3 and λm < i rq rq 0; i = 1 . . . n with λm ∈ Spec ( M ) and λ ≤ 0 ; i = 1 . . . n with λ ∈ i i i z

Spec(e1 ρ  Q −1 ) then there exists z > 1 such that E[st 1 ] < ∞ 0.

∀t >

Proof. According to Lemma 2.3 the expectation in (24) is given by eTr[A(t )0 ]+b(t ) with A(t) satisfying

dA = AM + M A + 2AQ 2 A + 1 dt

(25)

while b solves db/dt = Tr[2 A] and the initial conditions are A(0 ) = 0 and b(0 ) = 0. 1 comprises two parts, the first one is 11 (z ) = zz (1 − ρ  ρ )/2 − diag(z )/2 (with z = z1 e1 ) while the second one is 21 = X M + M X with X = −e1 ρ  Q −1 (X is symmetric by hypothesis of Lemma 3.3). As z = z1 e1 there is z1 > 1 such that 11 (z ) ∈ Sn− (the main difference with the previous case being that the stronger condition Sn−− cannot be satisfied). For 21 we use standard results for Lyapunov equations, see for example Proposition 4.2 in (Dullerud & Paganini, 20 0 0), and conclude that 21 ∈ Sn− . As a result, there exists z1 > 1 such that 1 ∈ Sn− and the matrix Riccati equation satisfies the condition of Proposition 1.1 of Dieci and Eirola (1994), which basically requires the coefficient of the quadratic term of (25) to be in Sn+ (in our case it is Q2 so even in Sn++ by hypothesis on Q) while the constant term has to be in Sn− (and the initial condition has to be in Sn− which is satisfied by 0 by hypothesis), and allows us to conclude that A(t) is well defined for any t and conclude as in the previous case that there is no explosion.  Remark 3.4. The problem of optimal portfolio choice when the assets follow a WASC dynamics was considered in (Buraschi, Porchia, & Trojani, 2010) when the investor trades only the assets, in (Da Fonseca, Grasselli, & Ielpo, 2011) when the investor can trade also on volatility derivative products, see also Richter (2014) and Bauerle and Li (2013) for related results. To justify the existence of an optimal solution one has to ensure that the matrix differential equation does not explode, the argument often used is the one

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presented above (see proof of Proposition 3.3 in (Da Fonseca et al., 2011) or Remark 4.5 of Bauerle and Li (2013)). 4. Numerical examples

5

(Gnoatto & Grasselli, 2014b). It might be possible to combine the results presented here with those obtained in (Deelstra, Grasselli, & Van Weverberg, 2015), which deal with moment explosions for the Wishart process alone, to obtain more precise conclusions on moment explosions for the WMSV and WASC models.

4.1. The WMSV case In (Da Fonseca & Grasselli, 2011) the WMSV model was calibrated on several index option sets. We report hereafter the values found for the matrices when using DAX options and check to what extent the above constraints are satisfied.



0 =

M=

Q =

R=

0.029772169

0.011941184

0.011941184

0.010797031

,

−1.247938145

−0.898452467

−0.081968542

−1.143334085

,

0.341746785

0.349304718

0.184780314

0.309027751

,

−0.224325097

−0.124423682

−0.254535739

−0.723046566

.

We can check that Tr[RQ 0 ] = −0.0103683 and after using the polar decomposition for Q and adjusting R as explained in Remark 2.1 we obtain that Spec (R ) = (−0.783843, −0.166522 ) so the constraint of Remark 2.3 is satisfied (we still denote the correlation matrix R, instead of R˜). Moreover, as Spec (M ) = (−1.47201, −0.919266 ) and Spec(RQ −1 ) = (−4.84579, −0.655948 ) the constraints required in Lemma 3.2 are also satisfied. Lastly, we have||R Q − QR||2 = 0.174976 and ||R Q ||2 = 0.435308 (with ||A||2 = λmax (A A ), i.e. the square root of the largest eigenvalue of A A), so a possibility if the difference between these two norms is not considered large enough would be to calibrate the model parameters with the constraint R Q = QR, which amounts to requiring QR to be symmetric. 4.2. The WASC case In (Da Fonseca & Grasselli, 2011) the WASC model was calibrated on several pairs of index options. We report hereafter the values found for the matrices when using the pair EuroStoxx50/DAX and check to what extent the constraints imposed by the lemmas are satisfied.



0 =

M=



0.044628818

0.036569563

0.036569563

0.042381462

,

−0.377233173

−0.053882017

−1.249657697 0.357283814

0.280896903

0.336204623

ρ  = −0.64070105

−0.110535411



,

0.389785977

Q =

−0.781980114

,



and β = 0.733196469. It is clear that (19) will be negative and after using the polar decomposition for Q and adjusting ρ as explained in Remark 2.4 we obtain that Spec (e1 ρ  Q −1 ) = (−5.73236, 0.0 ) and as Spec (M ) = (−1.28969, −0.741945 ) the constraints required in Lemma 3.3 are also satisfied. Lastly, we have ||ρ e Q − Qe1 ρ  ||2 = 1 0.136886 and ||ρ e Q || = 0 . 312375 . 2 1 Remark 4.1. It might be possible to derive the conclusions presented here using the expressions for the moment-generating functions for the stock, both for the WMSV and WASC models, given in (Gnoatto & Grasselli, 2014b). Interestingly, the symmetry constraints found in Lemmas 3.1 and 3.3 also appear in

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