Manajemen | Fakultas Ekonomi Universitas Maritim Raja Ali Haji jbes%2E2009%2E08105
Journal of Business & Economic Statistics
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Derivative Pricing With Wishart Multivariate
Stochastic Volatility
Christian Gourieroux & Razvan Sufana
To cite this article: Christian Gourieroux & Razvan Sufana (2010) Derivative Pricing With
Wishart Multivariate Stochastic Volatility, Journal of Business & Economic Statistics, 28:3,
438-451, DOI: 10.1198/jbes.2009.08105
To link to this article: http://dx.doi.org/10.1198/jbes.2009.08105
Published online: 01 Jan 2012.
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Date: 12 January 2016, At: 00:29
Derivative Pricing With Wishart Multivariate
Stochastic Volatility
Christian G OURIEROUX
CREST, 92245 Malakoff Cedex, France, CEPREMAP, 75013 Paris, France, and Department of Economics,
University of Toronto, Toronto, Ontario M5S 3G7, Canada (c.gourieroux@utoronto.ca)
Razvan S UFANA
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Department of Economics, York University, Toronto, Ontario M3J 1P3, Canada (rsufana@yorku.ca)
This paper deals with the pricing of derivatives written on several underlying assets or factors satisfying a
multivariate model with Wishart stochastic volatility matrix. This multivariate stochastic volatility model
leads to a closed-form solution for the conditional Laplace transform, and quasi-explicit solutions for
derivative prices written on more than one asset or underlying factor. Two examples are presented: (i) a
multiasset extension of the stochastic volatility model introduced by Heston (1993), and (ii) a model for
credit risk analysis that extends the model of Merton (1974) to a framework with stochastic firm liability,
stochastic volatility, and several firms. A bivariate version of the stochastic volatility model is estimated
using stock prices and moment conditions derived from the joint unconditional Laplace transform of the
stock returns.
KEY WORDS: Credit default swap; Credit risk; Derivative pricing; Stochastic volatility; Wishart process.
1. INTRODUCTION
The standard Black–Scholes model (Black and Scholes
1973) is not flexible enough to reproduce some stylized facts
observed on derivative prices such as the smile effect, that is
a U-shaped relationship between the implied Black–Scholes
volatility and the strike price (for any given residual maturity).
It is well known that a smile can be created by introducing stochastic volatility in the Black–Scholes model. This approach
was introduced by Hull and White (1987) and improved by
Heston (1993), and Ball and Roma (1994), who changed the
volatility dynamics to ensure a positive volatility.
Another stylized fact widely documented in the empirical literature and not captured in the Black–Scholes framework is the
skewness of the univariate implied volatilities as a function of
the stock price, called the leverage effect. The leverage effect
can be recovered at least in two different ways:
(i) It can be created by considering a stochastic volatility
model with an instantaneous correlation between the stock return and volatility shock (see, e.g., Heston 1993). In this extension the instantaneous correlation is conditional on a rather
small information set including the past values of the asset return and volatility shock only.
(ii) An alternative way for creating a leverage effect consists
in increasing the information set. The conditional correlation
depends on the selected information set, and the asset return
and stochastic volatility can be conditionally uncorrelated for a
large information set and conditionally correlated for a smaller
information set as a consequence of the covariance analysis
equation. More precisely, let us consider two information sets at
time t: It and Jt , with It ⊂ Jt . The covariance analysis equation
written for the change in return r and volatility σ is
Thus, we can have simultaneously Cov(drt , dσt |Jt ) = 0 and
Cov(drt , dσt |It ) = 0. This case arises when a common unobservable factor is introduced in the drifts of drt and dσt [i.e., in
E(drt |Jt ) and E(dσt |Jt )] and then integrated out. Loosely speaking, a leverage effect observed on a single asset can be recovered by considering an appropriate multifactor framework and
integrating out the effect of some of the factors.
The present paper introduces a multivariate model in which
the stochastic volatility matrix follows the Wishart autoregressive (WAR) process introduced by Bru (1991) and studied in
Gourieroux, Jasiak, and Sufana (2009) (see also Gourieroux
2006 for a survey). The Wishart process is a multivariate extension of the CIR process (Cox, Ingersoll, and Ross 1985), and
allows us to jointly model not only the dynamics of volatilities,
but also the evolution of covolatilities. The WAR specification
ensures by construction the positive definiteness of the stochastic volatility matrix.
An advantage of the multivariate stochastic volatility model
presented in this paper is the closed-form expression of the
conditional Laplace transform (moment generating function).
Since the conditional Laplace transform is the basis for pricing derivatives (Duffie, Pan, and Singleton 2000), the model
implies quasi-explicit solutions for derivative prices. The multivariate stochastic volatility model for asset returns provides
a more accurate analysis of the joint risks captured in the covolatilities, a focus on the risk premia corresponding to these
covolatilities, and is needed for pricing derivatives written on
several underlying assets or factors. The multivariate stochastic volatility model can also be applied with some unobservable
Cov(drt , dσt |It )
= Cov[E(drt |Jt ), E(dσt |Jt )|It ] + E[Cov(drt , dσ t |Jt )|It ].
438
© 2010 American Statistical Association
Journal of Business & Economic Statistics
July 2010, Vol. 28, No. 3
DOI: 10.1198/jbes.2009.08105
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Gourieroux and Sufana: Derivative Pricing With Wishart Multivariate Stochastic Volatility
factors in order to capture the observed leverage effect in the alternative way mentioned above, or to model default dependence
and price basket derivatives in credit risk. The introduction of
unobservable common factors is required by the Basel 2 regulation concerning internal risk models (Basel Committee on Bank
Supervision 2001). The cost of using the multivariate model is
the large number of parameters, which increases the complexity of the model and represents the usual curse of dimensionality encountered in multivariate stochastic volatility models.
It can be solved by considering WAR factor models or by introducing appropriate parameter restrictions (Bonato, Caporin,
and Ranaldo 2008; Gourieroux, Jasiak, and Sufana 2009).
Two examples of multivariate stochastic volatility models
illustrate the flexibility and tractability of the general framework. The first model is a multiasset extension of the stochastic
volatility model introduced by Heston (1993), that specifies a
price equation with a volatility-in-mean effect to capture the
risk premia on each asset. By introducing interactions between
covolatilities and expected returns, it will also capture the tendency for the volatility and stock price to move together (i.e.,
the leverage effect) without assuming an instantaneous correlation between the stock return and the volatility-covolatility
innovations. The second example of the general multivariate
model is a model for credit risk analysis that extends the model
of Merton (1974) to a framework with stochastic firm liability, stochastic volatility, and more than one firm. The extended
model allows for default dependence and a joint pricing of risky
bonds and credit default swaps.
The paper is organized as follows. Section 2 introduces the
factor model with multivariate stochastic volatility. Then we derive a closed-form solution for the conditional Laplace transform of the joint process of integrated factors, volatilities, and
covolatilities. This derivation requires the analysis of the Riccati differential system associated with the WAR process. We
prove that the structure of the Riccati equations allows for an
explicit solution of this differential system. Section 3 studies
two examples of multivariate stochastic volatility models, a
multiasset model and a credit risk model, and explains how to
obtain solutions for derivative prices. In Section 4, a bivariate
version of the model is estimated by the generalized method
of moments, using daily stock prices and moment conditions
derived from the joint unconditional Laplace transform of the
stock returns, and illustrates the differences relative to traditional multivariate models. Section 5 concludes. Proofs and
other technical results are gathered in appendices.
2. FACTOR MODEL WITH MULTIVARIATE
STOCHASTIC VOLATILITY
In this section we introduce and discuss a model for the joint
evolution of a vector of stochastic factors, and of their associated stochastic volatility matrix.
2.1 The Factor Model
Let us consider n positive stochastic factors whose values at
time t are represented in an n-dimensional vector yt . The (n, n)
439
infinitesimal stochastic volatility matrix of the factors is denoted by t . The joint dynamics of log yt and t is given by
the stochastic differential system:
d log yt = μ + (Tr(D1 t ), . . . , Tr(Dn t ))′ dt
1/2
+ t
dt =
dWtS ,
(1)
1/2
(KQ′ Q + Mt + t M ′ ) dt + t dWtσ (Q′ Q)1/2
1/2
+ (Q′ Q)1/2 (dWtσ )′ t
(2)
,
where WtS and Wtσ are a n-dimensional vector and a (n, n)
matrix, respectively, whose elements are independent unidimensional standard Brownian motions, μ is a constant ndimensional vector, K is a scalar such that K > n − 1, and Di ,
i = 1, . . . , n, M, Q are (n, n) matrices with Q′ Q invertible. Tr
1/2
denotes the trace operator and t is the positive symmetric
square root of the volatility matrix t .
The dynamics of the volatility matrix t in Equation (2)
represents a continuous-time process of stochastic, symmetric, positive definite matrices. It corresponds to the continuoustime WAR introduced by Bru (1991) and studied in Gourieroux,
Jasiak, and Sufana (2009) and Gourieroux (2006) as a tool for
modeling the dynamics of volatility matrices. By construction,
the Wishart specification of the volatility matrix ensures that,
at any point in time, the factor volatilities and covolatilities satisfy the constraints imposed by the positive definiteness of the
volatility matrix (see Section A.1). An alternative specification
of the volatility matrix assumes that the inverse of t follows
a Wishart process. This extends the inverted gamma distribution assumed in one-dimensional stochastic volatility models to
get a closed form expression for the return density and to study
its tail magnitude (see, e.g., Praetz 1972; Clark 1973; Blattberg
and Gonedes 1974). The direct Wishart specification used in
our framework is more appropriate for deriving closed form expressions for the moment generating functions and for pricing
derivatives.
The drift in Equation (1) is an affine function of factor volatilities and covolatilities:
Et (d log yi,t ) = [μi + Tr(Di t )] dt,
i = 1, . . . , n,
(3)
where Et denotes the expectation conditional on the information
available at time t. Without loss of generality we can restrict Di
to be symmetric, since Tr(Di t ) = Tr( 21 (Di + D′i )t ).
Even for a rather small dimension n, the model involves a
large number of parameters equal to n2 (n + 5)/2 + n + 1. For
instance, there are 40 and 77 parameters for n = 3 and n = 4,
respectively. This is the curse of dimensionality encountered in
multivariate volatility models. It explains the flexibility of the
model, but can create difficulties for estimation. A reduction
in the number of parameters can be achieved by considering
WAR-in-mean models with factors as discussed in Gourieroux,
Jasiak, and Sufana (2009), or by introducing appropriate restrictions on the parameters as in Bonato, Caporin, and Ranaldo
(2008).
440
Journal of Business & Economic Statistics, July 2010
2.2 Affine Property and Conditional Laplace Transform
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The joint process (log yt , t ) is an affine process, that admits
drift and volatility functions which are affine functions of log yt
and t (see Duffie and Kan 1996 and Duffie, Filipovic, and
Schachermayer 2003 for the definitions and analysis of standard
affine processes). The results in Section A.2 show that the affine
property is satisfied for the drift and volatility of (log yt , t ).
The general theory of affine processes implies that the conditional Laplace transform (moment generating function) of the
joint process (log yt , t ) is exponential affine. The conditional
Laplace transform of the joint process (log yt , t ) and of its integrated values is defined by
γ , C, c0 ,
C)
t,h (γ , γ0 ,
t+h
= Et exp
(γ ′ log yu + γ0 ) du +
γ ′ log yt+h
+
t
t+h
Tr(Cu + c0 ) du + Tr(Ct+h ) ,
(4)
where the coefficients γ , γ0 ,
γ , c0 , and the symmetric matrices
C,
C can be real or complex whenever the expectation exists.
The property below is a direct consequence of general results
on affine processes (see Duffie, Filipovic, and Schachermayer
2003).
Proposition 1. The conditional Laplace transform of the
affine process (1)–(2) is
γ , C, c0 ,
C)
t,h (γ , γ0 ,
= exp a(h)′ log yt + Tr(B(h)t ) + b(h) ,
(5)
where a, b, and the symmetric matrix B satisfy the system of
Riccati equations:
da(h)
= γ,
dh
dB(h)
= B(h)M + M ′ B(h) + 2B(h)Q′ QB(h)
dh
n
1
′
ai (h)Di + C,
+ a(h)a(h) +
2
(6)
(7)
i=1
db(h)
= a(h)′ μ + K Tr[B(h)Q′ Q] + γ0 + c0 ,
dh
with initial conditions: a(0) =
γ , B(0) =
C, b(0) = 0.
(8)
Proof. See Appendix B.
The differential system (6)–(8) involves parameters γ , γ0 ,
γ ,
C define the initial conditions. The differenC, c0 , whereas
tial equation for a admits the explicit solution
a(h) = γ h +
γ.
Proposition 2. Let us assume γ = 0, and the existence of a
symmetric matrix B∗ which satisfies
n
1 ′
M ′ B∗ + B∗ M + 2B∗ Q′ QB∗ + γ
γi Di + C = 0.
γ +
2
(9)
i=1
The solution B(h) is given by
B(h) = B∗ + exp[(M + 2Q′ QB∗ )h]′
h
× (
C − B∗ )−1 − 2
exp[(M + 2Q′ QB∗ )u]Q′ Q
0
t
This is an example of multidimensional Riccati equation with
a fundamental system of solutions (see Walcher 1986, 1991 for
a general definition, and Grasselli and Tebaldi 2008 for a discussion of quadratic models, which do not include the present
WAR process).
Then, system (6)–(8) can be solved recursively. The Riccati
Equation (7) provides the matrix B(h) and finally the form of
b(h) is obtained substituting a(h) and B(h) by their expressions
in Equation (8).
In general, Riccati equations do not admit closed-form solutions and have to be solved numerically. The main result of
the paper is the following proposition which provides a closedform solution for Riccati Equation (7) in some special cases.
× exp[(M + 2Q′ QB∗ )u]′ du
× exp[(M + 2Q′ QB∗ )h].
−1
(10)
The expression of b(h) can be derived by integrating Equation (8):
h
′
′
b(h) = (
γ μ + γ0 + c0 )h + K Tr
B(u) du Q Q . (11)
0
Proof. See Appendix D.
The expression of B(h) in Proposition 2 admits a closed
form. Indeed, up to an appropriate change of basis, the matrix
exp[(M + 2Q′ QB∗ )u] can be written under a diagonal or triangular form, whose elements are simple functions of u (exponential or exponential times polynomial), which can be explicitly
integrated between 0 and h.
The existence of a symmetric solution B∗ of the quadratic
system (9) in Proposition 2 implies joint restrictions on the parameters and on the argument of the conditional Laplace transform.
Proposition 3. Under the conditions of Proposition 2, if
γ, C
are real, there exists an orthonormal basis ei , i = 1, . . . , n, such
that
n
′ 1
2
′ ′
′
′
(ei Mei ) − 2ei Q Qei ei
γi Di + C ei ≥ 0,
γ
γ +
2
i=1
i = 1, . . . , n.
Proof. See Appendix E.
The restriction in Proposition 3 is related to the existence of
the real conditional Laplace transforms t,h in Equation (4).
Since the stochastic volatility matrix is positive definite, the
t+h
function exp t Tr(Cu ) du is integrable for any negative definite matrix C. However, the presence of stochastic volatility
will increase the tails of the factor return distribution and therefore the argument
γ cannot be too large.
Under additional conditions on the parameters, we can derive explicit expressions for the solution B∗ of the system (9),
and for the coefficients B(h) and b(h) in the conditional Laplace
transform (5) without considering a diagonal or triangular decomposition of exp[(M + 2Q′ QB∗ )u].
Gourieroux and Sufana: Derivative Pricing With Wishart Multivariate Stochastic Volatility
γ′ +
Proposition 4. Let Ŵ = 21 γ
that:
n
γi D i
i=1
+ C, and assume
(i) (Q′ Q)−1 M is a symmetric matrix,
(ii) M ′ (2Q′ Q)−1 M − Ŵ is positive definite.
Then, we get
B∗ = −(2Q′ Q)−1 M
+ (2Q′ Q)−1/2 (2Q′ Q)1/2 (M ′ (2Q′ Q)−1 M − Ŵ)
1/2
× (2Q′ Q)1/2
(2Q′ Q)−1/2 .
(12)
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Proof. See Appendix F.
In particular, we get the following corollary, by noting that
(Q′ Q)−1 M = (Q′ Q)−1/2 M(Q′ Q)−1/2 is symmetric when M is
symmetric and M and (Q′ Q)1/2 commute.
γ
γ ′ + ni=1
γi D i +
Corollary 5. Let P = (2Q′ Q)1/2 , Ŵ = 21
C, and assume that:
(i) M ′ = M,
(ii) PM = MP,
(iii) M 2 − PŴP is positive definite.
Then, denoting G = −(M 2 − PŴP)1/2 , the matrix B∗ is given
by
B∗ = P−1 [G − M]P−1 ,
and the expressions of B(h) and b(h) are
CP − G + M)−1
B(h) = P−1 G − M + exp(Gh) (P
−1
1
−1
+ (Id − exp(2Gh))G
exp(Gh) P−1 ,
2
K
b(h) =
γ ′ μ + γ0 + c0 + Tr(G − M) h
2
1
K
CP − G + M)
− log det Id + (P
2
2
−1
× (Id − exp(2Gh))G
.
(13)
(14)
(15)
Proof. See Appendix G.
When the parameters do not satisfy the restrictions of Proposition 4, the solution B∗ of Equation (9) has to be found numerically or the Riccati Equation (7) can be solved by applying the
linearization technique considered in Fonseca, Grasselli, and
Tebaldi (2005) (see also Buraschi, Cieslak, and Trojani 2007,
appendix A.5).
3. MULTIVARIATE EXTENSIONS OF HESTON
AND MERTON MODELS
In this section we present two illustrations of models with
multivariate stochastic volatility. The first illustration is an extension to the multiasset framework of the stochastic volatility
model introduced by Heston (1993). The second illustration is
a model for credit risk analysis.
441
3.1 A Stochastic Volatility Model for Asset Prices
3.1.1 The Model. Let us consider a market with one riskfree asset and n risky assets. The risk-free rate r is assumed to
be constant, whereas the infinitesimal geometric returns of the
risky assets are represented in a n-dimensional vector d log st ,
with st being the vector of asset prices at time t. The (infinitesimal) stochastic volatility matrix of the risky returns is denoted
by t .
The model for the n risky assets is obtained from the multivariate stochastic volatility model in Section 2 by replacing yt
with st . Thus, the joint dynamics of log st and t is given by the
stochastic differential system:
d log st = μ + (Tr(D1 t ), . . . , Tr(Dn t ))′ dt
1/2
+ t
dWtS ,
(16)
1/2
dt = (KQ′ Q + Mt + t M ′ ) dt + t
1/2
+ (Q′ Q)1/2 (dWtσ )′ t
dWtσ (Q′ Q)1/2
,
(17)
where the notation is explained in Section 2.
The volatility matrix in the drift of the price Equation (16)
accounts for a risk premium. Section A.3 proves that, if the
matrix Di is symmetric positive semidefinite for any i, the risk
premium is positive: Tr(Di ) ≥ 0, and that the risk premium
is an increasing function of risk: Tr(Di ) ≥ Tr(Di ∗ ), for any
≫ ∗ , and for any i, where ≫ denotes the standard ordering on symmetric matrices. Moreover, despite the assumption
of independence between WtS and Wtσ , this multivariate model
is able to capture the leverage effect observed separately on a
single asset. The reason is that by introducing for instance the
covolatility between assets 1 and 2 in the expressions of the expected return and volatility of asset 1, we can introduce a tendency for the volatility and return of asset 1 to move together
when this unobserved stochastic covolatility is integrated out.
This argument is valid for any number of additional assets introduced in the system. By increasing their number, we get a
larger number of unobserved covolatility factors.
As a consequence of Propositions 1 and 2, a closed-form solution can be derived for the conditional Laplace transform of
γ ′ log st+h )], which fully characterthe log asset prices Et [exp(
izes the conditional distribution of the asset returns:
γ ′ log st+h )] = t,h (0, 0,
γ , 0, 0, 0).
Et [exp(
As usual, the introduction of stochastic volatility increases the
tail magnitude of the stock returns. In the present framework,
it is easily checked that the Laplace transform admits a series
expansion in a neighborhood of zero and the power moments
of stock returns exist at any nonnegative order. Thus, the tail
increase due to Wishart stochastic volatility does not imply the
nonexistence of some moments of stock returns.
The associated transition of the stock returns can be derived
by inverting the Fourier transform, which is the Laplace transform evaluated at pure imaginary arguments, or in a more direct
way. Indeed, for a given volatility path, the return process is
multivariate Gaussian. The conditional distribution of log st+h
442
Journal of Business & Economic Statistics, July 2010
given (log st , t ) is normal with mean
t+h
log st + μh + Tr D1
u du , . . . ,
t
Tr Dn
t+h
u du
t
t+h
′
,
and variance–covariance matrix t u du. Thus the transition of log st+h given (log st , t ) is deduced by integrating
t+h
out the cumulated volatility t u du given t . The distrib
t+h
ution of the integrated volatility t u du is characterized by
t+h
Et [exp t Tr(Cu ) du], which can be computed from Propositions 1 and 2:
t+h
Tr(Cu ) du = t,h (0, 0, 0, C, 0, 0).
Et exp
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t
In a single asset framework (n = 1), the system (16)–(17)
become:
d log st = (μ + δσt2 ) dt + σt dWtS ,
d(σt2 )
= (kq
2
+ 2mσt2 ) dt + 2qσt dWtσ .
Thus, the volatility process is a CIR process and the model reduces to the specification introduced by Heston (1993). In this
case, the conditional Laplace transform of the log asset prices
is
γ log st+h )] = exp[
γ log st + B(h)t + b(h)],
Et [exp(
where
B(h) = B∗ + exp[2(m + 2q2 B∗ )h]
/ −(B∗ )−1 + q2 1 − exp[2(m + 2q2 B∗ )h]
/(m + 2q2 B∗ ) ,
b(h) = (
γ μ + kq2 B∗ )h
k
1 − exp[2(m + 2q2 B∗ )h]
− log1 −
,
2
m/(q2 B∗ ) + 2
and B∗ is a solution of
1 2
γ δ = 0.
+
2mB∗ + 2q2 B∗2 + γ
2
Thus, the multiasset model in Equations (16) and (17) can be
regarded as a multivariate extension of Heston’s model.
3.1.2 Derivative Pricing. The closed-form expression of
the conditional Laplace transform given in Proposition 2 can
be used to price derivatives written on the underlying assets
by using the transform analysis (Duffie, Pan, and Singleton
2000). It allows us to avoid numerical approximations based
on multibranches trees introduced in the multiasset framework
(see, e.g., Boyle 1988; Boyle, Evnine, and Gibbs 1989; Ho,
Stapleton, and Subrahmanyam 1995; Chen, Chung, and Yang
2002), or expansions around the constant volatility hypothesis
(see, e.g., Hull and White 1987). Without loss of generality, we
can assume a zero risk-free rate.
For derivative pricing, we need to discuss the form of the
joint process (log st , t ) under the risk-neutral distribution, and
the link between the historical and risk-neutral parameters. Under differentiability conditions on derivative prices, it is known
from Girsanov theorem that the change of density for period
(t, t + h) between the historical and risk-neutral distributions is
of the type
t+h
[γu′ d log su + Tr(Cu du )]
mt,t+h = exp
t
+
t+h
t
(γ0u du + c0u du) ,
where γu , Cu , γ0u , c0u denote predetermined coefficients. The
change of probabilities and thus the coefficients in the stochastic discount factor are constrained by both the unit mass restriction and the martingale condition on stock prices.
Proposition 6. Under the risk-neutral distribution, the joint
process (log st , t ) satisfies a stochastic differential system
with volatility equal to the historical volatility and a modified
drift:
Et∗ (d log st ) = Et (d log st ) + t γt dt
1
= − (e′1 t e1 , . . . , e′n t en )′ dt
2
1
= − (σ11,t , . . . , σnn,t )′ dt,
2
Et∗ (dt ) = Et (dt ) + Covt [Tr(Ct dt ), dt ]
(18)
= Et (dt ) + 2(t Ct Q′ Q + Q′ QCt′ t ) dt, (19)
where Et∗ denotes the conditional expectation under the riskneutral probability, and ei is the canonical vector with zero components except for the ith component which is equal to 1.
Proof. See Appendix C.
The risk premium on the Brownian motion of the return
equation is fixed by the martingale condition (see Appendix C),
whereas the risk premia corresponding to the volatilities–
covolatilities (that are Ct ) can be fixed arbitrarily as a consequence of market incompleteness (see, e.g., Garman 1977).
The stochastic system under the risk-neutral probability has
the same form as the stochastic system under the historical
probability if the volatility risk premia Ct = C∗ are constant.
Indeed, the former differential system corresponds to
Et∗ (d log st ) = μ∗ + [Tr(D∗i t )] dt,
where μ∗ = 0, D∗i = − 21 ei e′i , and
Et∗ (dt ) = (K ∗ Q∗′ Q∗ + M ∗ t + t M ∗′ ) dt,
where K ∗ Q∗′ Q∗ = KQ′ Q, M ∗ = M + 2Q′ QC∗′ . In this case, the
intercept in the volatility drift stays the same, whereas the matrix of “mean-reverting parameters” can be fixed arbitrarily.
∗ (γ , γ ,
The risk-neutral conditional Laplace transform t,h
0
γ , C, c0 , C) is defined as in Equation (4), with Et replaced by
Et∗ . If Ct = C∗ , it can be computed in closed form under the parameter restrictions in Proposition 2, after replacing the historical parameters by the risk-neutral ones. As explained in Duffie,
Pan, and Singleton (2000), the risk-neutral conditional Laplace
transform is the basis for derivative pricing since it can be used
to obtain explicit or quasi-explicit prices for various derivatives.
∗ (0, 0,
For example, t,h
γ , 0, 0, 0) is the price at time t of a
European derivative with exponential payoff exp[
γ ′ log st+h ] at
Gourieroux and Sufana: Derivative Pricing With Wishart Multivariate Stochastic Volatility
time t + h. European options and other derivatives with more
complex payoffs are priced using Fourier inversion techniques,
where the Fourier transform is simply the Laplace transform
evaluated at pure imaginary arguments.
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3.2 Credit Risk Model
The multivariate stochastic volatility model can also be applied to credit risk analysis by considering the asset values and
liabilities of the firms as the basic contingent claims. It defines
a multifactor model for a simultaneous pricing of risky bonds
and credit default swaps.
In the standard framework of the firm value model introduced
by Merton (1974), the potential time to default h, say, is predetermined, and the values of the stock, bonds, and credit default
swaps corresponding to a given firm i are defined from the conditional risk-neutral distribution of its asset Ai,t+h and liability
Li,t+h at date t + h. More precisely, assuming a zero risk-free
rate, the value at date t of a zero-coupon bond with residual
maturity h issued by the firm i is
Ai,t+h
Bi (t, t + h) = Et∗
1Ai,t+h Li,t+h ,
Li,t+h
where Et∗ denotes the conditional expectation with respect to
the risk-neutral probability and the first component takes into
account the recovery rate when default occurs. The value at date
t of the equity is
Si,t = Et∗ [(Ai,t+h − Li,t+h )+ ],
whereas the value of the credit default swap with residual maturity h is
CDSi (t, t + h) = Et∗ 1Ai,t+h
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Derivative Pricing With Wishart Multivariate
Stochastic Volatility
Christian Gourieroux & Razvan Sufana
To cite this article: Christian Gourieroux & Razvan Sufana (2010) Derivative Pricing With
Wishart Multivariate Stochastic Volatility, Journal of Business & Economic Statistics, 28:3,
438-451, DOI: 10.1198/jbes.2009.08105
To link to this article: http://dx.doi.org/10.1198/jbes.2009.08105
Published online: 01 Jan 2012.
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Date: 12 January 2016, At: 00:29
Derivative Pricing With Wishart Multivariate
Stochastic Volatility
Christian G OURIEROUX
CREST, 92245 Malakoff Cedex, France, CEPREMAP, 75013 Paris, France, and Department of Economics,
University of Toronto, Toronto, Ontario M5S 3G7, Canada (c.gourieroux@utoronto.ca)
Razvan S UFANA
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Department of Economics, York University, Toronto, Ontario M3J 1P3, Canada (rsufana@yorku.ca)
This paper deals with the pricing of derivatives written on several underlying assets or factors satisfying a
multivariate model with Wishart stochastic volatility matrix. This multivariate stochastic volatility model
leads to a closed-form solution for the conditional Laplace transform, and quasi-explicit solutions for
derivative prices written on more than one asset or underlying factor. Two examples are presented: (i) a
multiasset extension of the stochastic volatility model introduced by Heston (1993), and (ii) a model for
credit risk analysis that extends the model of Merton (1974) to a framework with stochastic firm liability,
stochastic volatility, and several firms. A bivariate version of the stochastic volatility model is estimated
using stock prices and moment conditions derived from the joint unconditional Laplace transform of the
stock returns.
KEY WORDS: Credit default swap; Credit risk; Derivative pricing; Stochastic volatility; Wishart process.
1. INTRODUCTION
The standard Black–Scholes model (Black and Scholes
1973) is not flexible enough to reproduce some stylized facts
observed on derivative prices such as the smile effect, that is
a U-shaped relationship between the implied Black–Scholes
volatility and the strike price (for any given residual maturity).
It is well known that a smile can be created by introducing stochastic volatility in the Black–Scholes model. This approach
was introduced by Hull and White (1987) and improved by
Heston (1993), and Ball and Roma (1994), who changed the
volatility dynamics to ensure a positive volatility.
Another stylized fact widely documented in the empirical literature and not captured in the Black–Scholes framework is the
skewness of the univariate implied volatilities as a function of
the stock price, called the leverage effect. The leverage effect
can be recovered at least in two different ways:
(i) It can be created by considering a stochastic volatility
model with an instantaneous correlation between the stock return and volatility shock (see, e.g., Heston 1993). In this extension the instantaneous correlation is conditional on a rather
small information set including the past values of the asset return and volatility shock only.
(ii) An alternative way for creating a leverage effect consists
in increasing the information set. The conditional correlation
depends on the selected information set, and the asset return
and stochastic volatility can be conditionally uncorrelated for a
large information set and conditionally correlated for a smaller
information set as a consequence of the covariance analysis
equation. More precisely, let us consider two information sets at
time t: It and Jt , with It ⊂ Jt . The covariance analysis equation
written for the change in return r and volatility σ is
Thus, we can have simultaneously Cov(drt , dσt |Jt ) = 0 and
Cov(drt , dσt |It ) = 0. This case arises when a common unobservable factor is introduced in the drifts of drt and dσt [i.e., in
E(drt |Jt ) and E(dσt |Jt )] and then integrated out. Loosely speaking, a leverage effect observed on a single asset can be recovered by considering an appropriate multifactor framework and
integrating out the effect of some of the factors.
The present paper introduces a multivariate model in which
the stochastic volatility matrix follows the Wishart autoregressive (WAR) process introduced by Bru (1991) and studied in
Gourieroux, Jasiak, and Sufana (2009) (see also Gourieroux
2006 for a survey). The Wishart process is a multivariate extension of the CIR process (Cox, Ingersoll, and Ross 1985), and
allows us to jointly model not only the dynamics of volatilities,
but also the evolution of covolatilities. The WAR specification
ensures by construction the positive definiteness of the stochastic volatility matrix.
An advantage of the multivariate stochastic volatility model
presented in this paper is the closed-form expression of the
conditional Laplace transform (moment generating function).
Since the conditional Laplace transform is the basis for pricing derivatives (Duffie, Pan, and Singleton 2000), the model
implies quasi-explicit solutions for derivative prices. The multivariate stochastic volatility model for asset returns provides
a more accurate analysis of the joint risks captured in the covolatilities, a focus on the risk premia corresponding to these
covolatilities, and is needed for pricing derivatives written on
several underlying assets or factors. The multivariate stochastic volatility model can also be applied with some unobservable
Cov(drt , dσt |It )
= Cov[E(drt |Jt ), E(dσt |Jt )|It ] + E[Cov(drt , dσ t |Jt )|It ].
438
© 2010 American Statistical Association
Journal of Business & Economic Statistics
July 2010, Vol. 28, No. 3
DOI: 10.1198/jbes.2009.08105
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Gourieroux and Sufana: Derivative Pricing With Wishart Multivariate Stochastic Volatility
factors in order to capture the observed leverage effect in the alternative way mentioned above, or to model default dependence
and price basket derivatives in credit risk. The introduction of
unobservable common factors is required by the Basel 2 regulation concerning internal risk models (Basel Committee on Bank
Supervision 2001). The cost of using the multivariate model is
the large number of parameters, which increases the complexity of the model and represents the usual curse of dimensionality encountered in multivariate stochastic volatility models.
It can be solved by considering WAR factor models or by introducing appropriate parameter restrictions (Bonato, Caporin,
and Ranaldo 2008; Gourieroux, Jasiak, and Sufana 2009).
Two examples of multivariate stochastic volatility models
illustrate the flexibility and tractability of the general framework. The first model is a multiasset extension of the stochastic
volatility model introduced by Heston (1993), that specifies a
price equation with a volatility-in-mean effect to capture the
risk premia on each asset. By introducing interactions between
covolatilities and expected returns, it will also capture the tendency for the volatility and stock price to move together (i.e.,
the leverage effect) without assuming an instantaneous correlation between the stock return and the volatility-covolatility
innovations. The second example of the general multivariate
model is a model for credit risk analysis that extends the model
of Merton (1974) to a framework with stochastic firm liability, stochastic volatility, and more than one firm. The extended
model allows for default dependence and a joint pricing of risky
bonds and credit default swaps.
The paper is organized as follows. Section 2 introduces the
factor model with multivariate stochastic volatility. Then we derive a closed-form solution for the conditional Laplace transform of the joint process of integrated factors, volatilities, and
covolatilities. This derivation requires the analysis of the Riccati differential system associated with the WAR process. We
prove that the structure of the Riccati equations allows for an
explicit solution of this differential system. Section 3 studies
two examples of multivariate stochastic volatility models, a
multiasset model and a credit risk model, and explains how to
obtain solutions for derivative prices. In Section 4, a bivariate
version of the model is estimated by the generalized method
of moments, using daily stock prices and moment conditions
derived from the joint unconditional Laplace transform of the
stock returns, and illustrates the differences relative to traditional multivariate models. Section 5 concludes. Proofs and
other technical results are gathered in appendices.
2. FACTOR MODEL WITH MULTIVARIATE
STOCHASTIC VOLATILITY
In this section we introduce and discuss a model for the joint
evolution of a vector of stochastic factors, and of their associated stochastic volatility matrix.
2.1 The Factor Model
Let us consider n positive stochastic factors whose values at
time t are represented in an n-dimensional vector yt . The (n, n)
439
infinitesimal stochastic volatility matrix of the factors is denoted by t . The joint dynamics of log yt and t is given by
the stochastic differential system:
d log yt = μ + (Tr(D1 t ), . . . , Tr(Dn t ))′ dt
1/2
+ t
dt =
dWtS ,
(1)
1/2
(KQ′ Q + Mt + t M ′ ) dt + t dWtσ (Q′ Q)1/2
1/2
+ (Q′ Q)1/2 (dWtσ )′ t
(2)
,
where WtS and Wtσ are a n-dimensional vector and a (n, n)
matrix, respectively, whose elements are independent unidimensional standard Brownian motions, μ is a constant ndimensional vector, K is a scalar such that K > n − 1, and Di ,
i = 1, . . . , n, M, Q are (n, n) matrices with Q′ Q invertible. Tr
1/2
denotes the trace operator and t is the positive symmetric
square root of the volatility matrix t .
The dynamics of the volatility matrix t in Equation (2)
represents a continuous-time process of stochastic, symmetric, positive definite matrices. It corresponds to the continuoustime WAR introduced by Bru (1991) and studied in Gourieroux,
Jasiak, and Sufana (2009) and Gourieroux (2006) as a tool for
modeling the dynamics of volatility matrices. By construction,
the Wishart specification of the volatility matrix ensures that,
at any point in time, the factor volatilities and covolatilities satisfy the constraints imposed by the positive definiteness of the
volatility matrix (see Section A.1). An alternative specification
of the volatility matrix assumes that the inverse of t follows
a Wishart process. This extends the inverted gamma distribution assumed in one-dimensional stochastic volatility models to
get a closed form expression for the return density and to study
its tail magnitude (see, e.g., Praetz 1972; Clark 1973; Blattberg
and Gonedes 1974). The direct Wishart specification used in
our framework is more appropriate for deriving closed form expressions for the moment generating functions and for pricing
derivatives.
The drift in Equation (1) is an affine function of factor volatilities and covolatilities:
Et (d log yi,t ) = [μi + Tr(Di t )] dt,
i = 1, . . . , n,
(3)
where Et denotes the expectation conditional on the information
available at time t. Without loss of generality we can restrict Di
to be symmetric, since Tr(Di t ) = Tr( 21 (Di + D′i )t ).
Even for a rather small dimension n, the model involves a
large number of parameters equal to n2 (n + 5)/2 + n + 1. For
instance, there are 40 and 77 parameters for n = 3 and n = 4,
respectively. This is the curse of dimensionality encountered in
multivariate volatility models. It explains the flexibility of the
model, but can create difficulties for estimation. A reduction
in the number of parameters can be achieved by considering
WAR-in-mean models with factors as discussed in Gourieroux,
Jasiak, and Sufana (2009), or by introducing appropriate restrictions on the parameters as in Bonato, Caporin, and Ranaldo
(2008).
440
Journal of Business & Economic Statistics, July 2010
2.2 Affine Property and Conditional Laplace Transform
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The joint process (log yt , t ) is an affine process, that admits
drift and volatility functions which are affine functions of log yt
and t (see Duffie and Kan 1996 and Duffie, Filipovic, and
Schachermayer 2003 for the definitions and analysis of standard
affine processes). The results in Section A.2 show that the affine
property is satisfied for the drift and volatility of (log yt , t ).
The general theory of affine processes implies that the conditional Laplace transform (moment generating function) of the
joint process (log yt , t ) is exponential affine. The conditional
Laplace transform of the joint process (log yt , t ) and of its integrated values is defined by
γ , C, c0 ,
C)
t,h (γ , γ0 ,
t+h
= Et exp
(γ ′ log yu + γ0 ) du +
γ ′ log yt+h
+
t
t+h
Tr(Cu + c0 ) du + Tr(Ct+h ) ,
(4)
where the coefficients γ , γ0 ,
γ , c0 , and the symmetric matrices
C,
C can be real or complex whenever the expectation exists.
The property below is a direct consequence of general results
on affine processes (see Duffie, Filipovic, and Schachermayer
2003).
Proposition 1. The conditional Laplace transform of the
affine process (1)–(2) is
γ , C, c0 ,
C)
t,h (γ , γ0 ,
= exp a(h)′ log yt + Tr(B(h)t ) + b(h) ,
(5)
where a, b, and the symmetric matrix B satisfy the system of
Riccati equations:
da(h)
= γ,
dh
dB(h)
= B(h)M + M ′ B(h) + 2B(h)Q′ QB(h)
dh
n
1
′
ai (h)Di + C,
+ a(h)a(h) +
2
(6)
(7)
i=1
db(h)
= a(h)′ μ + K Tr[B(h)Q′ Q] + γ0 + c0 ,
dh
with initial conditions: a(0) =
γ , B(0) =
C, b(0) = 0.
(8)
Proof. See Appendix B.
The differential system (6)–(8) involves parameters γ , γ0 ,
γ ,
C define the initial conditions. The differenC, c0 , whereas
tial equation for a admits the explicit solution
a(h) = γ h +
γ.
Proposition 2. Let us assume γ = 0, and the existence of a
symmetric matrix B∗ which satisfies
n
1 ′
M ′ B∗ + B∗ M + 2B∗ Q′ QB∗ + γ
γi Di + C = 0.
γ +
2
(9)
i=1
The solution B(h) is given by
B(h) = B∗ + exp[(M + 2Q′ QB∗ )h]′
h
× (
C − B∗ )−1 − 2
exp[(M + 2Q′ QB∗ )u]Q′ Q
0
t
This is an example of multidimensional Riccati equation with
a fundamental system of solutions (see Walcher 1986, 1991 for
a general definition, and Grasselli and Tebaldi 2008 for a discussion of quadratic models, which do not include the present
WAR process).
Then, system (6)–(8) can be solved recursively. The Riccati
Equation (7) provides the matrix B(h) and finally the form of
b(h) is obtained substituting a(h) and B(h) by their expressions
in Equation (8).
In general, Riccati equations do not admit closed-form solutions and have to be solved numerically. The main result of
the paper is the following proposition which provides a closedform solution for Riccati Equation (7) in some special cases.
× exp[(M + 2Q′ QB∗ )u]′ du
× exp[(M + 2Q′ QB∗ )h].
−1
(10)
The expression of b(h) can be derived by integrating Equation (8):
h
′
′
b(h) = (
γ μ + γ0 + c0 )h + K Tr
B(u) du Q Q . (11)
0
Proof. See Appendix D.
The expression of B(h) in Proposition 2 admits a closed
form. Indeed, up to an appropriate change of basis, the matrix
exp[(M + 2Q′ QB∗ )u] can be written under a diagonal or triangular form, whose elements are simple functions of u (exponential or exponential times polynomial), which can be explicitly
integrated between 0 and h.
The existence of a symmetric solution B∗ of the quadratic
system (9) in Proposition 2 implies joint restrictions on the parameters and on the argument of the conditional Laplace transform.
Proposition 3. Under the conditions of Proposition 2, if
γ, C
are real, there exists an orthonormal basis ei , i = 1, . . . , n, such
that
n
′ 1
2
′ ′
′
′
(ei Mei ) − 2ei Q Qei ei
γi Di + C ei ≥ 0,
γ
γ +
2
i=1
i = 1, . . . , n.
Proof. See Appendix E.
The restriction in Proposition 3 is related to the existence of
the real conditional Laplace transforms t,h in Equation (4).
Since the stochastic volatility matrix is positive definite, the
t+h
function exp t Tr(Cu ) du is integrable for any negative definite matrix C. However, the presence of stochastic volatility
will increase the tails of the factor return distribution and therefore the argument
γ cannot be too large.
Under additional conditions on the parameters, we can derive explicit expressions for the solution B∗ of the system (9),
and for the coefficients B(h) and b(h) in the conditional Laplace
transform (5) without considering a diagonal or triangular decomposition of exp[(M + 2Q′ QB∗ )u].
Gourieroux and Sufana: Derivative Pricing With Wishart Multivariate Stochastic Volatility
γ′ +
Proposition 4. Let Ŵ = 21 γ
that:
n
γi D i
i=1
+ C, and assume
(i) (Q′ Q)−1 M is a symmetric matrix,
(ii) M ′ (2Q′ Q)−1 M − Ŵ is positive definite.
Then, we get
B∗ = −(2Q′ Q)−1 M
+ (2Q′ Q)−1/2 (2Q′ Q)1/2 (M ′ (2Q′ Q)−1 M − Ŵ)
1/2
× (2Q′ Q)1/2
(2Q′ Q)−1/2 .
(12)
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Proof. See Appendix F.
In particular, we get the following corollary, by noting that
(Q′ Q)−1 M = (Q′ Q)−1/2 M(Q′ Q)−1/2 is symmetric when M is
symmetric and M and (Q′ Q)1/2 commute.
γ
γ ′ + ni=1
γi D i +
Corollary 5. Let P = (2Q′ Q)1/2 , Ŵ = 21
C, and assume that:
(i) M ′ = M,
(ii) PM = MP,
(iii) M 2 − PŴP is positive definite.
Then, denoting G = −(M 2 − PŴP)1/2 , the matrix B∗ is given
by
B∗ = P−1 [G − M]P−1 ,
and the expressions of B(h) and b(h) are
CP − G + M)−1
B(h) = P−1 G − M + exp(Gh) (P
−1
1
−1
+ (Id − exp(2Gh))G
exp(Gh) P−1 ,
2
K
b(h) =
γ ′ μ + γ0 + c0 + Tr(G − M) h
2
1
K
CP − G + M)
− log det Id + (P
2
2
−1
× (Id − exp(2Gh))G
.
(13)
(14)
(15)
Proof. See Appendix G.
When the parameters do not satisfy the restrictions of Proposition 4, the solution B∗ of Equation (9) has to be found numerically or the Riccati Equation (7) can be solved by applying the
linearization technique considered in Fonseca, Grasselli, and
Tebaldi (2005) (see also Buraschi, Cieslak, and Trojani 2007,
appendix A.5).
3. MULTIVARIATE EXTENSIONS OF HESTON
AND MERTON MODELS
In this section we present two illustrations of models with
multivariate stochastic volatility. The first illustration is an extension to the multiasset framework of the stochastic volatility
model introduced by Heston (1993). The second illustration is
a model for credit risk analysis.
441
3.1 A Stochastic Volatility Model for Asset Prices
3.1.1 The Model. Let us consider a market with one riskfree asset and n risky assets. The risk-free rate r is assumed to
be constant, whereas the infinitesimal geometric returns of the
risky assets are represented in a n-dimensional vector d log st ,
with st being the vector of asset prices at time t. The (infinitesimal) stochastic volatility matrix of the risky returns is denoted
by t .
The model for the n risky assets is obtained from the multivariate stochastic volatility model in Section 2 by replacing yt
with st . Thus, the joint dynamics of log st and t is given by the
stochastic differential system:
d log st = μ + (Tr(D1 t ), . . . , Tr(Dn t ))′ dt
1/2
+ t
dWtS ,
(16)
1/2
dt = (KQ′ Q + Mt + t M ′ ) dt + t
1/2
+ (Q′ Q)1/2 (dWtσ )′ t
dWtσ (Q′ Q)1/2
,
(17)
where the notation is explained in Section 2.
The volatility matrix in the drift of the price Equation (16)
accounts for a risk premium. Section A.3 proves that, if the
matrix Di is symmetric positive semidefinite for any i, the risk
premium is positive: Tr(Di ) ≥ 0, and that the risk premium
is an increasing function of risk: Tr(Di ) ≥ Tr(Di ∗ ), for any
≫ ∗ , and for any i, where ≫ denotes the standard ordering on symmetric matrices. Moreover, despite the assumption
of independence between WtS and Wtσ , this multivariate model
is able to capture the leverage effect observed separately on a
single asset. The reason is that by introducing for instance the
covolatility between assets 1 and 2 in the expressions of the expected return and volatility of asset 1, we can introduce a tendency for the volatility and return of asset 1 to move together
when this unobserved stochastic covolatility is integrated out.
This argument is valid for any number of additional assets introduced in the system. By increasing their number, we get a
larger number of unobserved covolatility factors.
As a consequence of Propositions 1 and 2, a closed-form solution can be derived for the conditional Laplace transform of
γ ′ log st+h )], which fully characterthe log asset prices Et [exp(
izes the conditional distribution of the asset returns:
γ ′ log st+h )] = t,h (0, 0,
γ , 0, 0, 0).
Et [exp(
As usual, the introduction of stochastic volatility increases the
tail magnitude of the stock returns. In the present framework,
it is easily checked that the Laplace transform admits a series
expansion in a neighborhood of zero and the power moments
of stock returns exist at any nonnegative order. Thus, the tail
increase due to Wishart stochastic volatility does not imply the
nonexistence of some moments of stock returns.
The associated transition of the stock returns can be derived
by inverting the Fourier transform, which is the Laplace transform evaluated at pure imaginary arguments, or in a more direct
way. Indeed, for a given volatility path, the return process is
multivariate Gaussian. The conditional distribution of log st+h
442
Journal of Business & Economic Statistics, July 2010
given (log st , t ) is normal with mean
t+h
log st + μh + Tr D1
u du , . . . ,
t
Tr Dn
t+h
u du
t
t+h
′
,
and variance–covariance matrix t u du. Thus the transition of log st+h given (log st , t ) is deduced by integrating
t+h
out the cumulated volatility t u du given t . The distrib
t+h
ution of the integrated volatility t u du is characterized by
t+h
Et [exp t Tr(Cu ) du], which can be computed from Propositions 1 and 2:
t+h
Tr(Cu ) du = t,h (0, 0, 0, C, 0, 0).
Et exp
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:29 12 January 2016
t
In a single asset framework (n = 1), the system (16)–(17)
become:
d log st = (μ + δσt2 ) dt + σt dWtS ,
d(σt2 )
= (kq
2
+ 2mσt2 ) dt + 2qσt dWtσ .
Thus, the volatility process is a CIR process and the model reduces to the specification introduced by Heston (1993). In this
case, the conditional Laplace transform of the log asset prices
is
γ log st+h )] = exp[
γ log st + B(h)t + b(h)],
Et [exp(
where
B(h) = B∗ + exp[2(m + 2q2 B∗ )h]
/ −(B∗ )−1 + q2 1 − exp[2(m + 2q2 B∗ )h]
/(m + 2q2 B∗ ) ,
b(h) = (
γ μ + kq2 B∗ )h
k
1 − exp[2(m + 2q2 B∗ )h]
− log1 −
,
2
m/(q2 B∗ ) + 2
and B∗ is a solution of
1 2
γ δ = 0.
+
2mB∗ + 2q2 B∗2 + γ
2
Thus, the multiasset model in Equations (16) and (17) can be
regarded as a multivariate extension of Heston’s model.
3.1.2 Derivative Pricing. The closed-form expression of
the conditional Laplace transform given in Proposition 2 can
be used to price derivatives written on the underlying assets
by using the transform analysis (Duffie, Pan, and Singleton
2000). It allows us to avoid numerical approximations based
on multibranches trees introduced in the multiasset framework
(see, e.g., Boyle 1988; Boyle, Evnine, and Gibbs 1989; Ho,
Stapleton, and Subrahmanyam 1995; Chen, Chung, and Yang
2002), or expansions around the constant volatility hypothesis
(see, e.g., Hull and White 1987). Without loss of generality, we
can assume a zero risk-free rate.
For derivative pricing, we need to discuss the form of the
joint process (log st , t ) under the risk-neutral distribution, and
the link between the historical and risk-neutral parameters. Under differentiability conditions on derivative prices, it is known
from Girsanov theorem that the change of density for period
(t, t + h) between the historical and risk-neutral distributions is
of the type
t+h
[γu′ d log su + Tr(Cu du )]
mt,t+h = exp
t
+
t+h
t
(γ0u du + c0u du) ,
where γu , Cu , γ0u , c0u denote predetermined coefficients. The
change of probabilities and thus the coefficients in the stochastic discount factor are constrained by both the unit mass restriction and the martingale condition on stock prices.
Proposition 6. Under the risk-neutral distribution, the joint
process (log st , t ) satisfies a stochastic differential system
with volatility equal to the historical volatility and a modified
drift:
Et∗ (d log st ) = Et (d log st ) + t γt dt
1
= − (e′1 t e1 , . . . , e′n t en )′ dt
2
1
= − (σ11,t , . . . , σnn,t )′ dt,
2
Et∗ (dt ) = Et (dt ) + Covt [Tr(Ct dt ), dt ]
(18)
= Et (dt ) + 2(t Ct Q′ Q + Q′ QCt′ t ) dt, (19)
where Et∗ denotes the conditional expectation under the riskneutral probability, and ei is the canonical vector with zero components except for the ith component which is equal to 1.
Proof. See Appendix C.
The risk premium on the Brownian motion of the return
equation is fixed by the martingale condition (see Appendix C),
whereas the risk premia corresponding to the volatilities–
covolatilities (that are Ct ) can be fixed arbitrarily as a consequence of market incompleteness (see, e.g., Garman 1977).
The stochastic system under the risk-neutral probability has
the same form as the stochastic system under the historical
probability if the volatility risk premia Ct = C∗ are constant.
Indeed, the former differential system corresponds to
Et∗ (d log st ) = μ∗ + [Tr(D∗i t )] dt,
where μ∗ = 0, D∗i = − 21 ei e′i , and
Et∗ (dt ) = (K ∗ Q∗′ Q∗ + M ∗ t + t M ∗′ ) dt,
where K ∗ Q∗′ Q∗ = KQ′ Q, M ∗ = M + 2Q′ QC∗′ . In this case, the
intercept in the volatility drift stays the same, whereas the matrix of “mean-reverting parameters” can be fixed arbitrarily.
∗ (γ , γ ,
The risk-neutral conditional Laplace transform t,h
0
γ , C, c0 , C) is defined as in Equation (4), with Et replaced by
Et∗ . If Ct = C∗ , it can be computed in closed form under the parameter restrictions in Proposition 2, after replacing the historical parameters by the risk-neutral ones. As explained in Duffie,
Pan, and Singleton (2000), the risk-neutral conditional Laplace
transform is the basis for derivative pricing since it can be used
to obtain explicit or quasi-explicit prices for various derivatives.
∗ (0, 0,
For example, t,h
γ , 0, 0, 0) is the price at time t of a
European derivative with exponential payoff exp[
γ ′ log st+h ] at
Gourieroux and Sufana: Derivative Pricing With Wishart Multivariate Stochastic Volatility
time t + h. European options and other derivatives with more
complex payoffs are priced using Fourier inversion techniques,
where the Fourier transform is simply the Laplace transform
evaluated at pure imaginary arguments.
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:29 12 January 2016
3.2 Credit Risk Model
The multivariate stochastic volatility model can also be applied to credit risk analysis by considering the asset values and
liabilities of the firms as the basic contingent claims. It defines
a multifactor model for a simultaneous pricing of risky bonds
and credit default swaps.
In the standard framework of the firm value model introduced
by Merton (1974), the potential time to default h, say, is predetermined, and the values of the stock, bonds, and credit default
swaps corresponding to a given firm i are defined from the conditional risk-neutral distribution of its asset Ai,t+h and liability
Li,t+h at date t + h. More precisely, assuming a zero risk-free
rate, the value at date t of a zero-coupon bond with residual
maturity h issued by the firm i is
Ai,t+h
Bi (t, t + h) = Et∗
1Ai,t+h Li,t+h ,
Li,t+h
where Et∗ denotes the conditional expectation with respect to
the risk-neutral probability and the first component takes into
account the recovery rate when default occurs. The value at date
t of the equity is
Si,t = Et∗ [(Ai,t+h − Li,t+h )+ ],
whereas the value of the credit default swap with residual maturity h is
CDSi (t, t + h) = Et∗ 1Ai,t+h