ECONOMETRIC SPECIFICATIONS OF
STOCHASTIC DISCOUNT FACTOR
MODELS
C. GOURIEROUX
(1)
and A. MONFORT
(2)
July 19, 2001
We have beneted of helpful comments by O. Scaillet and J. Simonato.
1 CREST, CEPREMAP and University of Toronto
2 CNAM and CREST
1
Abstract
Econometric Specications of Stochastic Discount Factor Models
We consider the problem of derivative pricing when the stochastic discount factors are exponential-aÆne functions of underlying factors. In particular we discuss the conditionally gaussian framework and introduce semiparametric pricing methods for models with path dependent location and
scale parameters. This approach is also applied to more complicated frameworks, such as the pricing of a derivative written on an index, when the
interest rate is stochastic.
Keywords : Derivative Pricing, Stochastic Discount Factor, Implied Volatility, Variance-gamma Model.
Resume
Econometrie des modeles a facteurs d'escompte stochastique
Nous considerons le probleme de valorisation de produits derives, lorsque
les facteurs d'escompte stochastiques sont fonctions exponentielles-aÆnes de
facteurs sous-jacents. En particulier nous discutons le cas conditionnellement
gaussien et developpons des methodes de valorisation semi-parametriques,
pour des modeles ou les parametres d'echelle et de position dependent de
l'historique. Cette approche est egalement appliquee a des contextes plus
complexes, tel la valorisation d'un derive sur indice en presence de taux
d'inter^et stochastique.
Mots cles : Valorisation, facteur d'escompte stochastique, volatilite implicite, modele variance-gamma.
1
1. Introduction
The pricing of derivatives is generally based on continuous time models,
which may include latent stochastic factors, or jumps. This approach provides formulas for derivative prices, expressed as either conditional expectations, or solutions of partial dierential equations. However the formulas are
often diÆcult to implement, since both the data and the hedging strategies
are in discrete time. This explains why standard continuous time models
have a rather simple structure [see e.g. Black, Scholes (1973), Hull, White
(1987), Melino, Turnbull (1990), Heston (1993)]. In particular the number of
underlying factors is generally small, the risk premia associated with these
factors are assumed constant or even equal to zero, and the term structure of
interest rates is often treated independently of the index derivatives . However there restrictions, that simplify the implementation of the derivative
pricing formulas, often induce a poor t when historical data, or data on
derivative prices are considered.
The aim of this paper is to address the problem of derivating pricing
using the stochastic discount factor introduced in [Harrison, Kreps (1979),
Garman, Ohlson (1980), Hansen, Richard (1987)]. It is known that, if agents
make their investments at date t (t 2 1N) based on an information set Jt ,
the prices of actively traded assets satisfy a linear valuation formula. More
precisely there exists a stochastic discount factor (SDF) Mt;t function of
the updated information set Jt , such that the price of an asset that provides
the payo gt at date t + 1 is :
Ct (g ) = E [Mt;t gt jJt ] :
(1.1)
Since the market is incomplete in discrete time, there exists a multiplicity
of stochastic discount factors that are compatible with the valuation formula
(1.1) for the actively traded assets.
In this paper we consider a class of SDF, that are exponential functions
of an aÆne combination of factors. This specication corresponds to the
Esscher transform used in insurance [see e.g. Esscher (1932)], and used for
derivative pricing [see Buhlman et alli (1996), Shyraev (1999), Gourieroux,
Monfort (2001)]. Next we discuss the restrictions implied by the valuation
formula (1.1).
In section 2, we review standard pricing formulas derived from either
+1
+1
+1
+1
2
+1
an equilibrium condition [as the consumption based CAPM, or the model
with recursive utility], or the condition of no arbitrage in a continuous time
framework [see e.g. Hull, White (1987), Heston (1993)]. The aim is to justify the exponential-aÆne specication of the SDF and to give examples of
possible factors. In section 3, we consider a simple framework in which the
riskfree rate is constant (equal to zero) and the information set includes the
returns on the actively traded assets only. We prove that there exists a single
SDF compatible with the exponential-aÆne specication. Then we discuss
in section 4 the conditionally gaussian framework and the semi-parametric
models with path dependent location and scale parameters. These examples are used to derive a semi-parametric pricing method and to discuss the
patterns of implied (Black-Scholes) volatility surfaces in terms of departures
from time independence and conditional normality. Section 5 extends the
basic approach to the framework of stochastic interest rates and unobservable factors. As illustration we discuss the conditionally gaussian factors
framework and explain how to price a derivative written on an index when
the interest rate is stochastic. Statistical inference is discussed in section 6.
2. Examples of stochastic discount factors
The expressions of stochastic discount factors are generally derived under
either an equilibrium condition, or the arbitrage free conditions in a continuous time framework. We provide below some examples to show that
stochastic discount factors considered in the literature are often exponentialaÆne functions of underlying state variables.
Example 1 : Consumption based CAPM (CCAPM)
In the standard CCAPM, an agent maximizes his expected intertemporal
utility expressed in terms of a physical good. The intertemporal transfers
are ensured by means of investments on a nancial market. Let us denote by
U the utility function, by Æ the intertemporal psychological discount factor,
and assume the existence of a representative agent. At equilibrium we get
the relation :
2
3
dU
(Ct+1 ) 77
6
q
pt = Et 64pt+1 t Æ dc
;
qt+1 dU (C ) 5
dc t
3
where pt is the vector of prices of nancial assets, qt the price of the consumption good and Ct the quantity consumed at date t. Thus, for a power
utility function, the SDF associated with the consumption based CAPM is :
Mt;t+1
=
Ct+1 Æ
qt+1
Ct
qt
"
#
log qtq+1 + log CCt+1 :
t
t
= exp log Æ
It is an exponential-aÆne function of the two state variables, which are
the ination rate and the rate of change in consumption. It does not depend
on the asset returns. However the asset returns, that can be considered as
additional state variables, are generally included in the available information
set, and thus also inuence the derivative prices.
Example 2 : Recursive utility
Similar results can be derived, when the representative agent maximizes
a recursive utility [see Epstein-Zin (1991) and Weil (1989)]. If the utility is
a power function and the aggregator admits a Cobb-Douglas form, the SDF
is given by :
Mt;t+1
=
Æ =
= exp
(
Ct+1 Ct
log Æ
(1
)=
(1
Wt
Wt+1
)
!1 =
log CCt+1
t
qt+1
qt
(1
! =
W
=)log t+1
Wt
)
qt+1
log q :
t
The exponential-aÆne function now involves a third state variable log WWt ,
t
that represents the return on the market portfolio.
+1
Example 3 : Stochastic volatility model
In the literature standard stochastic volatility models [Hull, White (1987),
Heston (1993)] are of the type :
4
8
>
<
dSt = St dt + t St dWtS ;
df (t ) = a(t )dt + b(t )dWt ;
where St is the asset price, t the stochastic volatility and (WtS ); (Wt ) two
independent brownian motions. For a constant riskfree rate r, the discount
>
:
factor is deduced from Girsanov's theorem and given by :
(
Mt;t+1
= exp( r) exp (
r)
Z t+1
t
dWS
1 (
2
r)
2
Z t+1
t
d
2
)
Z t
1
exp t
2 t d ;
where is the risk premium associated with the stochastic volatility. This
premium can be selected arbitrarily as a function of the past due to the incomplete market framework. By approximating the integrals by their discrete
time counterpart, we note that :
Z t+1
+1
dW
2
)
1
1
1
Mt;t ' exp( r) exp
2 ( r) t t "t 2 t :
It is an exponential transformation of an aÆne function of the innovations
S
"t ; "t corresponding to both the return and volatility processes, with path
dependent coeÆcients.
The examples above show :
i) the convenience of exponential-aÆne specications, which ensure both the
positivity and the tractability of the SDF ;
ii) the various candidates for state variables, including nancial returns, innovations on nancial return or on volatilities, rate of change in consumption,
ination rate, etc. These state variables can be observable, or latent;
iii) the multiplicity of specications in the incomplete market framework due
to the arbitrary choice of the risk premium for the non traded factors.
(
+1
+1
S
( r) "t+1
t
+1
5
2
2
+1
2
In the next section, we adopt the approach, in which we specify a priori
the admissible forms of the SDF. Then this set is reduced by taking into
account the arbitrage free restrictions.
3. SDF modelling : the principle
To present the modelling principle, we rst consider the framework of a
riskfree asset with zero riskfree rate and several risky assets with prices
pj;t; j = 1; : : : ; J and (geometric) returns : rj;t = log (pj;t =pj;t). We
assume that these assets are actively traded on the markets, and that the
dierent prices are observable for the investor.
3
+1
+1
3.1 The historical distribution
The (conditional) historical distribution of the return rt = (r ;t ; : : : ; rJ;t )0
is dened by means of its (conditional) Laplace transform, (also called moment generating function) supposed to belong to a parametric set :
E [exp u0rt jrt ]
= exp t (u; );
(say):
The Laplace transform is dened on a convex set, that depends on the
tails of the conditional distribution. We assume below that this convex set
is not reduced to one point located at the origin.
+1
1 +1
+1
+1
3.2 The stochastic discount factor
We assume a priori that the SDF can be written under an exponentialaÆne form :
Mt;t+1 = exp (t0 rt+1 + t ) ;
(3.1)
with rj;t ; j = 1; : : : ; J as the J state variables and coeÆcients t and t ,
that can be path dependent, that is function of the past : rt = (rt ; rt ; : : :).
+1
1
3 If
the future evolution of the riskfree rate is known at date t, it is possible to get a
zero riskfree rate by a deterministic change of numeraire. The case of a predetermined
stochastic interest rate is considered in the next section.
6
By writing the pricing formula for the riskfree asset and the J risky
assets, we get J + 1 restrictions on the relationship between the SDF and
the historical distribution. More precisely the constraints induced by the
arbitrage free conditions are :
8
>
E
M
j
r
>
t;t
t = 1;
>
>
+1
<
"
>
>
>
>
:
p
E Mt;t+1 j;t+1 jrt
pj;t
8
>
>
<
E
() >>
h
#
=E
h
Mt;t+1 exp rj;t+1 jrt
i
= 1; j = 1; : : : ; J;
i
exp (t0 rt + t ) jrt = 1;
+1
h
i
exp t0 rt + e0j rt + t jrt = 1; j = 1; : : : ; J;
where ej = (0; : : : 0; 1; 0; : : : 0)0, with 1 as component of order j ,
8
>
< exp [ t (t ; ) + t ] = 1;
() >:
exp [ t (t + ej ; ) + t] = 1; 8j = 1; : : : ; J;
8
>
t (t ; );
< t =
() >:
t (t + ej ; )
t (t ; ) = 0; 8j = 1; : : : ; J:
This system generally admits a unique solution :
8
>
< t = (rt ; );
>
:
t = (rt ; ) = t [(rt ; ); ]; say:
Then we deduce a unique form of the SDF (3.1) that satises the condition of no arbitrage. The associated risk neutral distribution Q admits the
conditional Laplace transform :
:
E
+1
+1
7
Qh
E
=
E
exp u0rt jrt
i
+1
h
exp(t0 rt + t ) exp u0rt jrt
+1
i
+1
= exp [ t (t + u; ) + t ]
= exp [ t (t + u; ) t (t; )] :
This result is summarized in the proposition below.
Proposition 1 : For a zero riskfree rate and an exponential-aÆne SDF with
state variable rt , there exists in general a unique admissible risk-neutral
distribution. Its Laplace transform is given by :
+1
Qh
i
exp u0rt jrt = exp [ t (t + u; ) t (t; )] ;
where t is the solution of :
t (t + ej ; ) = t (t ; ); j = 1; : : : ; J;
and t is the conditional historical Log-Laplace transform.
In discrete time we are in an incomplete market framework. Therefore
there exists an innity of admissible risk neutral distributions. The uniqueness condition in Proposition 3.1 is due to the constrained exponential-aÆne
form of the SDF .
We will explain in the next section how to introduce a multiplicity of SDF
by means of unobservable stochastic factors with parametrized risk premia.
By using the SDF Mt;t = Mt;t (rt ; ), we can propose a price for any
E
+1
4
4 Some
5
+1
+1
+1
other constraints could have been imposed on M +1. For instance [Hansen,
Jagannathan (1997), Cochrane (2000)] assume an aÆne form M +1 = 0 r +1 + . This
specication
does not ensure the positivity of the underlying state prices.
5 The dynamics of the return under the historical and risk neutral distributions can be
very dierent. For instance they can feature stationarity or nonstationarity under the risk
neutral distribution whereas the historical world is stationary. This question is diÆcult
to study in the general framework of this paper [see the discussion in Gourieroux-Monfort
(2001)b for aÆne models of interest rates].
t;t
t;t
8
t
t
t
derivative written on rt . Let us denote by g(rt h; t + h); h = 1; : : : ; H the
payos provided at t + h; h = 1; : : : ; H . A derivative price is :
6
Ct (g ) =
H
X
h=1
+
h
i
E Mt;t+1 (rt+1 ; ) : : : Mt;t+h (rt+h ; )g (rt+h ; t + h)jrt :
(3.2)
In practice this price cannot be computed analytically and is approximated by simulations. These simulations can be performed under the historical probability, or under a modied probability [see Gourieroux, Jasiak
(2001), chapter 13, section 5.2, for a discussion].
3.3 Information and time aggregation
The assumption of exponential-aÆne SDF depends on the selected information set and time horizon. To illustrate this dependence we consider
below two examples.
i) If we are interested on derivatives based on a subset of risky assets with
returns rt , the basic pricing formula (1.1) applied to g(rt ) can be written
as :
Ct (g ) = E [Mt;t g (rt )jrt ]
= E [Mt;t g(rt )jrt];
where Mt;t = E [Mt;t jrt ; rt]
This modied SDF does not admit in general an exponential-aÆne expression Mt;t = exp(t0 rt + t); even if the initial SDF does.
ii) Moreover the assumption is not stable by a change of time unit. Let
us consider the pricing of a european derivative at horizon 2. The price is
given by :
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
6 On the market the derivatives are usually written on prices and not directly on returns.
However, let us consider a european call for instance, with payo (S +1+ K )+ at date t +1.
This derivative is S times the derivative with payo (exp r +1 k ) , where k = K=S
t
is the moneyness strike. Thus we are just assuming a preliminary transformation of the
payo.
t
t
9
t
t
t
Ct (g )
= E [Mt;t Mt ;t g(rt )jrt ]
= E [E [Mt;t ; Mt ;t jrt ; rt]jrt]:
Neither Mt;t Mt ;t , nor E [Mt;t Mt ;t jrt ; rt] is exponential-aÆne
in general even if Mt;t is.
Therefore in practice it will be necessary to check for the right information
set and horizon, before applying the approach of exponential-aÆne SDF.
Various test procedures are described in section 6.
+1
+1 +2
+1
+1
+1 +2
+1
7
+1
+2
+1 +2
+2
+1 +2
+2
4. Examples
4.1 The conditionally gaussian framework
If the conditional historical distribution of rt given rt is gaussian, with
mean mt and variance-covariance matrix t , the conditional log-Laplace
transform is given by :
1 0
0
(4.1)
t (u; ) = u mt ( ) + u t ( )u:
2
The risk correction term t satises :
(t0 + e0j )mt () + 12 (t0 + e0j )t ()(t + ej )
1 0 () = 0; 8j = 1; : : : ; J;
t0 mt ()
2 t t t
() e0j mt () + e0j t ()t + 12 e0j t ()ej = 0; 8j = 1; : : : J;
+1
() mt() + t ()t + 21 vdiag t () = 0;
where vdiag t () is the vector, whose components are the diagonal elements
of t (). We deduce that :
7 In the standard continuous time stochastic volatility model (see example 3), the SDF
is exponential-aÆne at innitesimal horizon, and is not exponential-aÆne for more realistic
horizons.
10
t () [mt () + 21 vdiag t ()]:
(4.2)
Then the conditional log-Laplace transform of the risk-neutral distribution given in Proposition 1 is :
t (t + u; )
t (t ; )
= u0 [mt () + t ()t ] + 21 u0t ()u
(4.3)
= 12 u0 vdiag t () + 12 u0t ()u:
As usual the risk-neutral distribution is also conditionally gaussian, with
the same variance-covariance matrix as the conditional historical distribution,
and with a conditional mean function of t (), which is introduced to ensure
the arbitrage-free condition. These formulas can be applied to a large class of
models including for instance the conditionally gaussian multivariate ARCH
models [see e.g. Duan (1995)]. In particular they do not require a Markov
condition for (rt ), that is the fact that mt and t depend on the past
through rt only.
t =
1
+1
4.2 Variance-gamma model
This model has been introduced in Madan, Seneta (1990), Madan, Milne
(1991), Madan, Carr, Chang (1998). There is only one risky asset and the
historical log-Laplace transform of its return is :
t2
!
(u) = t log 1 umt u 2 ;
where t > 0; it corresponds to a time deformed gaussian model. The correcting factor is :
!
1
t
t =
mt +
t
2 ;
whereas the risk-neutral log-Laplace transform is :
2
t
2
2
Q
t
(u) = t log 1
11
um
t
u
2
t2
2
!
;
where : mt =
1
mt + t t2
t2 t2
; t2 =
2
t
t2 t2
:
1 t m t 2
2
Thus the (conditional) parameter t , that determines the time deformation is unchanged, whereas mt and t are modied. It is easy to check that
the Laplace transforms of the historical and risk neutral distributions are
both dened on non degenerate intervals.
t mt
t
2
2
4.3 Path dependent location and scale parameters
In this subsection we consider a single risky asset (J = 1) and assume
that the return satises :
rt = mt + t "t ; t > 0;
(4.4)
where mt and t are the location and scale parameters respectively, that may
depend on lagged values of the return and ("t) is a sequence of i.i.d. variables
with Laplace transform :
E (exp u"t ) = exp (u);
(say):
(4.5)
This model is convenient to show the consequence of the conditional normality assumption. Indeed it includes the case of gaussian errors, but may
also allow for other types of conditional distributions with heavier tails, such
as Student or Laplace distribution.
+1
+1
+1
i) The stochastic discount factor and the risk neutral distribution.
The conditional log-Laplace transform of rt is given by :
t (u) = mt u + (t u):
From Proposition 1, the log-Laplace transform of the risk-neutral distribution is given by :
Q
t (t + u)
t (t )
t (u) =
= mt u + [t (t + u)] [t t ];
where t is solution of :
+1
12
(t + 1) = t (t)
() mt = [t (t + 1)]
t
[t t ]:
(4.6)
There exists a unique solution t to equation (4.6), if
the log-Laplace transform is strictly convex and tends to innity at the
boundaries of its domain.
Proof : Indeed the mapping ! [t ( +1)] (t ) is continuous, strictly
increasing, with range ( 1; +1). The result follows directly.
Proposition 2 :
QED
Various examples are provided in Table 1. They include standard models
such as the Black-Scholes model with the gaussian error, the binomial tree
with dichotomous errors, or the Laplace models [Gourieroux, Monfort (2001)]
and a number of other specications.
13
Table 1 : Historical and risk neutral distributions
Distribution
p.d.f.
N (0; 1)
p1
2
1
symmetrical
2
exponential
exp
exp
"2
u2
2
2
d2
du2
1>0
u 2 IR
jxj
u2 )
log(1
u2[
;
2
1 1]
1+
1
t
1
2
u2
>0
u2
mt
t2
1
if
2
mt = 0
t2
2
1
1
2
2
if
1
Æ+1 + Æ 1
2
P [" = 1] =
if
shu
u
u 2 IR
[ 1;+1] (x)
1l
1
1
2
log
1
u2
log
sh2 u
>0
2
1
2
ch(u)
ch(u)]
[
(1
t2
1
2
j
=1
(*)
t =
P [" = "j ] = pj
J
X
log(
j
1
2 t
n
log
exp(
exp(
t )
mt )
=1
mt ) o
;
exp(t )
exp(
exp(
exp(
J
X
t
=1
pj;t =
1
+
2
y;
)
jmt j < t .
t )
t )
t )g
exp(
mt )
exp( t )
pj exp(t t "j )
J
X
=1
pj exp(t t "j )
ii) Semi-parametric pricing
The approach above can in particular be applied to an empirical distribution of the residuals from a given model of returns. As an illustration, let
us consider a parametric specications of the location and scale parameters :
mt = m(rt ; ); t = (rt ; );
(say);
(4.7)
and let us leave unspecied the distribution of the error term. The available
observations on the returns are denoted by r ; : : : ; rT .
1
y;
)
pj;t exp u(mt + t "j )]
j
14
1
2
where :
pj exp u"j )
if
) exp(
1
where :
j
pj Æ"j
t
pt ) exp u(mt
log[
J
X
1
) exp(
fpt exp u(mt + t )
pt =
u 2 IR
4
y 0:
+(1
(*)
4
y 0:
log
2>0
(1
(y)
t2
with p.d.f. :
=
u[ 1;1]
2
u + t u2
skewed Laplace distribution
=
t < 2
Q
t
R.N. Distribution
Step 1 : Calibration
The parameter can be consistently estimated from historical data by
applying a quasi-maximum likelihood method; the estimator is given by :
)
T (
X
1
[
r
m
(
r
;
)]
t
t
^T = arg max
log (rt ; ) 2 (r ; ) :
t
t
1
1
2
=1
2
1
Step 2 : Estimation of the error distribution
Then we compute the residuals :
r m(r ; ^T )
; = 2; : : : ; T;
"^ = (r ; ^T )
and the empirical distribution of the residuals :
T
^P = 1 X Æ" ;
T 1
where Æ" is the point mass at ".
Its log-Laplace transform is given by :
!
T
^T (u) = log 1 X exp u"^ ; u varying;
T 1
and is a consistent estimator of the unknown log-Laplace transform .
1
1
^
=2
=2
Step 3 : Determination of the risk correction factor
The correcting term t associated with the true distribution can be
consistently approximated by the correcting term ^t associated with ^T .
The correcting term ^t is the solution of :
T
X
=2
exp[(rt ; ^T )^t"^ ]fexp[m(rt ; ^T ) + (rt ; ^T )^" ] 1g = 0:
Step 4 : Determination of the SDF
The true underlying SDF is approximated by :
15
M^ t;t+1
h
= exp ^t [rt
+1
m(rt ; ^T )]
= exp ^t[rt
+1
= m^ t;t (rt
+1
+1
m(rt ; ^T )]
; rt );
^T (^t ^t)i
"
T
1
T
X
1 exp ^t ^t"^
#
1
:
=2
(say):
Step 5 : Pricing
Finally the price of a derivative can easily be approximated by simulations
based on the empirical distribution. For instance the price of a derivative
with residual maturity 1 and payo g(rt ) is approximated by :
+1
C^t (g )
= E^ [m^ t;t (rt ; rt)g(rt )]
n
o
= E^ g[m(rt; ^T ) + (rt; ^T )"t ] exp[^t (rt ; ^T )"t ]
"
#
T
1 X
exp ^t^t "^
T 1
+1
+1
+1
+1
+1
1
=2
T
X
=
=2
exp(^t^t "^ )g(m^ t + ^t "^ )
T
X
=2
exp(^t ^t "^ )
;
by the last line of Table 1.
When the derivative to be priced corresponds to a maturity H much
larger than 1, dierent steps of the algorithm above have to be modied. For
instance the correcting factors and the SDF's have to be computed for the
dates t; t + 1; : : : t + H 1, and the pricing formula modied according to
(3.2). Finally the expectation in the price formula has to be approximated
by an empirical average over simulated paths rt ; : : : ; rt H drawn from
the empirical distribution. Indeed an average performed on all admissible
empirical paths would involve a number of points increasing exponentially
with the maturity H .
+1
16
+
1
iii) Implied volatility surfaces
In practice the implied Black-Scholes volatilities depend on the moneyness
strike and residual maturity contrary to what is assumed in the Black-Scholes
model. They can feature smiles, asymmetries or sneers. These patterns
are due to the misspecication of the Black-Scholes model, which assumes
i.i.d. gaussian returns. The aim of this section is to consider conditionally
heteroscedastic models of the type :
rt = (rt )"t ;
and to describe how the implied volatility surface depends on the functional
form of the volatility and on the error distribution. We consider the following
four experiments :
experiment (1) : ARCH model
(rt ) = (10 + 0:95rt ) = ; "t N (0; 1)
experiment (2) : Symmetrical piecewise linear volatility
(rt ) = (0:1 1lr 0 + 0:01 1lr 0 ) = ; "t N (0; 1)
experiment (3) : Asymmetrical piecewise linear volatility
(rt ) = (0:005 1ljr j 0 05 + 0:001 1ljr j 0 05 ) = ; "t N (0; 1)
experiment (4) : (rt ) = 0:1; "t i.i.d. symmetrical exponential.
As expected, we observe smile or sneer eects. The importance of smile
eects is a function of how the volatility depends on the past and on the
magnitude of the tails of the error distribution. A sneer eect is obtained by
introducing either an asymmetric volatility function, or a skewed distribution
of the error. Finally the smile and sneer eects diminish, when maturity
increases. This is a consequence of the central limit theorem, which can be
applied to the return between t and t + H , given by :
rt;t H = rt + rt + : : : + rt H ;
for return processes, that are stationary under the risk neutral distribution.
+1
3
+1
2 1 2
t>
t<
t> :
+1
1 2
t < :
1 2
+1
+
+1
+2
17
+
[Insert Figure 1 and 1 bis : Black-Scholes implied volatilities,
ARCH model]
18
[Insert Figure 2 and 2 bis : Black-Scholes implied volatilities,
asymmetrical piecewise linear conditional variance]
19
[Insert Figure 3 and 3 bis : Black-Scholes implied volatilities,
symmetrical piecewise linear conditional variance]
20
[Insert Figure 4 and 4 bis : Black-Scholes implied volatilities,
symmetrical exponential]
21
5. General specication
In this section the basic approach to SDF modelling is extended in two
directions. First, the information used by the agents to x their asset demands does not only include the asset returns, but also variables from the
real sector of the economy and additional factors. These factors are not directly observedf by the econometrician. Second we introduce a time varying
riskfree rate rt . This rate is predetermined, that is known at time t, but
assumed stochastic and in particular unknown at time t 1. This allows for
a joint analysis of the derivatives written on the risky assets and on the short
term interest rate. In particular we are interested in the analysis of the term
structure of interest rates and in the eect of the stochastic interest rate on
the price of a european call with maturity H written on a stock index, for
instance .
+1
8
5.1 The investor's information and the SDF
The investors' information at date t is denoted by Jt. It includes the data
used by the investors, when they rebalance their portfolios, submit their
orders and decide the volumes to be traded. This information includes :
i) the current riskfree rate and its lagged values, rtf , say;
ii) the lagged returns on J risky assets, rt, say ;
iii) the lagged changes of real economic variables, xt , say, that may be
macrovariables as the GNP, the retail price index..;
iv) the values of additional factors, ft, say.
At this stage the additional factors cannot be interpreted. However they
will be partly recovered through their eect on the prices of basic and derivative assets, as discussed below.
+1
8 The standard nancial literature treats separately the analysis of the
term structure
and the pricing of options written on a stock. Typically the term structure is constructed
using only the information contained in the past interest rates, whereas the standard
option pricing formula assumes that the interest rate r +1 is deterministic, that is known
in advance for all future dates.
f
t
22
Thus the information is :
Jt = rtf+1 ; rt ; xt ; ft :
The sizes of the dierent vectors are 1; J; L and K , respectively.
(5.1)
In the sequel we are interested in pricing derivatives written onf the basic
riskfree and risky assets . If the derivative provides a payo g rt ; rt at
date t + 1, its price at date t is given by . :
h
i
Ct (g ) = E Mt;t g rtf ; rt jJt ;
(5.2)
where the stochastic discount factor Mt;t depends on Jt . As before the
SDF is constrained to be exponential-aÆne :
Mt;t = exp o;t rtf + t0 rt + t0 xt + Æt0 ft + t ;
(5.3)
where the dierent coeÆcients depend on Jt . We denote :
h
i0
Ft = rtf ; x0t ; ft0 ; t = (o;t ; t0 ; Æt0 )0 ;
such that the SDF becomes :
Mt;t = exp [t0 rt + 0t Ft + t ] :
(5.4)
9
+2
+1
10
+1
+1
+2
+1
+1
+1
+2
+1
+2
+1
+1
+1
+1
+1
+1
+1
+1
5.2 The historical distribution
This distribution has to be considered for all variables in the investor's
information set. The conditional Laplace transform of (rt0 ; Ft0 )0 given Jt
is :
+1
+1
9 It would have been possible to also include real variables in the contractual payo as
for Treasury Bonds indexed on ination.
10 The role of conditioning information is important in derivative pricing (see e.g.
Hansen, Richard (1987)). For instance the pricing formula (4.2) can also be written as :
i
h
C (g) = E E (M +1 jr +2 ; r +1 ; J )g(r +2 ; r +1 )jJ
f
t
t;t
h
f
t
t
t
t
t
i
t
= E M +1g(r +2 ; r +1)jJ :
Clearly M +1 is another admissible SDF, but it does not admit an exponential-aÆne
expression with respect to r +1; r +2 , once the factors x +1; f +1 have been integrated out.
Thus the constraint of exponential-aÆne function is associated with the basic traders'
information J +1:
f
t;t
t
t
t
t;t
f
t
t
t
t
23
t
E [exp(u0 rt+1 + v 0 Ft+1 )jJt ] = exp
(u; v; ):
(5.5)
It depends on the lagged values of the asset returns rt; of the real variables
xt , of the unobservable factors ft and on the current and lagged values of
the riskfree rate rtf . is the vector of parameters that characterizes this
conditional distribution.
It has to be noted that the riskfree rate has a particular status. It is
not only predetermined, but also constrained to be positive, whereas rt and
xt are generally of any sign due to their interpretations as changes (see the
example of CCAPM for the real variables) .
t
+1
5.3 Constraints on the S.D.F.
By writing the pricing conditions for the riskfree asset and the J risky
assets, we get the equations :
8
h
i
f
>
>
< E Mt;t exp rt jJt = 1;
+1
>
>
:
8
>
<
+1
E [Mt;t+1 exp rj;t+1 jJt ] = 1; 8j = 1; : : : ; J;
(t ; t ; ) + t + rtf = 0;
() >:
t (t + ej ; t ; ) + t = 0; 8j = 1; : : : ; J;
the solutions of which are :
8
>
< t = (t ; rt ; Ft ; );
(5.6)
>
:
t = (t ; rt ; Ft ; ):
Since the additional variables do not correspond to returns on assets
traded at t, their coeÆcient t can be xed arbitrarily as a function of the
information set. Thus the undeterminacy of the risk correction t still exists,
despite the a priori constraint on the specication of the SDF. The results
are summarized in the proposition below.
Proposition 3 : For an exponential-aÆne SDF with state variables rt ; Ft ,
there exists a multiplicity of admissible risk-neutral distributions. Their conditional Laplace transforms are given by :
t
+1
+1
24
+1
Q
E [exp(u0 rt+1 + v 0 Ft+1 )jJt ]
= E [Mt;t exp rtf exp(u0rt + v0Ft )jJt ]
= exp f t (t + u; t + v; ) t (t; t ; )g ;
where t is the solution of :
f
t (t + ej ; t ; )
t (t ; t ; ) rt = 0; 8j = 1; : : : ; J;
and the risk correction t can be chosen arbitrarily.
The dimension of incompleteness is equal to the dimension of t . It
corresponds to the numberf of real economy state variables (xt ) plus the future
prices of traded assets (rt ) introduced in the SDF.
5.4 The conditionally gaussian framework.
Since the riskfree interest rate rtf admits positive values, the conditionally gaussian framework can only be applied if the future path of the riskfree
rate is completely known at date t. Thus this subsection extends subsection
4.1, when there are additional unobservable factors.
When rt0 ; Ft0 0 are conditionally gaussian under the historical distribution, the log-Laplace transform is :
!
1
u
0
0
0
0
t (u; v ; ) = (u ; v )mt ( ) + (u ; v )( ) v :
2
The log-Laplace transforms of the risk-neutral distributions are :
+1
+1
+1
+1
+1
+2
+1
+1
(t ; t; )
!
"
!#
1
u
t
0
0
0
0
= (u ; v ) mt () + t () t + 2 (u ; v )t () v :
They correspond to gaussian distributions with the same conditional
variance-covariance matrix as the historical distribution and a conditional
Q
t
(u; v; ) =
+1
t
(t + u; t + v; )
25
t
mean corrected for the risk. The expression of this mean depends on the
solution t of the pricing restrictions.
Let us decompose the conditional mean and variance as :
0
1
0
1
mr;t
rr;t rF;t
C
B
C
mt = B
@
A ; t = @
A:
mF;t
F r;t F F;t
The restrictions of proposition 3 are :
"
!#
t
(e0j ; 0) mt + t
t
+ 21 (e0j ; 0)t e0j
!
rtf+1 = 0; 8j = 1; : : : ; J;
() e0j mr;t + e0j rr;tt + e0j rF;tt + 12 e0j rr;t ej rtf = 0; 8j;
() mr;t + rr;t t + rF;t t + 21 vdiag rr;t rtf e = 0;
+1
+1
where e = (1; : : : ; 1)0:
We deduce that :
t = rr;t mr;t
rtf+1 e
1
1
+ 2 vdiag rr;t + rF;tt ;
(5.7)
which extends equation (3.4).
Thus the conditional mean of the risk-neutral distribution is :
m
+
0
1
m + + A
= @
m + + 2
1 vdiag r e
6 +1
2
= 664
m
1 m r +1 e + 12 vdiag t
t
r;t
F;t
t
t
rr;t
F r;t
t
r;F;t
t
f
F F;t
t
t
3
rr;t
t
f
F;t
F r;t
rr;t
r;t
t
26
rr;t
+ F F;t
1
F r;t
rr;t
rF;t
7
7
7
5
t
It is easy to recognize the residual conditional mean mF;t F r;trr;tmr;t
and variance F F;t F;r;trr;t rF;t. The correction for risk is performed on
the returns of traded assets, and there is no correction at all on the residual
part, i.e. on the component of Ft which is not conditionally linked with
rt . This result can be considered as a direct extension of the pricing formula
derived by Stapleton, Subrahmanyam (1984).
1
1
+1
+1
Example 4 : Stochastic volatility model
It is interesting to discuss the denition of the factors in the framework
of the discretized stochastic volatility model :
8
>
St = St + t St "St ;
< St
>
:
f (t ) f (t ) = a(t ) + b(t )"t :
A possible expression of the SDF is given in example 3 as an exponentialaÆne function of "St and "t . However this S.D.F. can also be expressed
as an exponential-aÆne function of St ; f (t ), or of (St St )=St;
(f (t f (t ))=f (t ) with path dependent coeÆcients. Therefore the factors are not uniquely dened. In the example, we can choose as factors either
the innovations, or the levels of variables, or the associated returns. In particular by selecting the innovation processes, we can assume some white noise
properties of the factors under the historical distribution.
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
5.5 Models with stochastic interest rate
We rst consider a model without an unobservable factor, where the
short term zero-coupon bond is the only tradeable asset. By selecting a
compound autoregressive model for the historical distribution of the riskfree
rate (Darolles, Gourieroux, Jasiak (2001)), we get a direct extension of the
Cox-Ingersoll-Ross model (Cox, Ingersoll, Ross (1985)). Then we discuss the
pricing of an index derivative, when the interest rate is stochastic.
i) Model with a stochastic interest rate only
When the short term zero-coupon bond is the only tradable asset and the
SDF is given by :
Mt;t = exp(o rtf + t );
(5.8)
+1
+2
27
the arbitrage free condition becomes :
t = rtf
where : t (u) = E [exp urtf jrtf ].
t
+1
+2
(o);
(5.9)
+1
As an illustration let us assume that the conditional distribution of rtf
is a noncentered gamma distribution with log-Laplace transform :
u f
(5.10)
t (u) = log(1 uc) +
1 uc rt ;
where ; c; are positive parameters. We deduce that :
+2
11
+1
t
=
rtf+1
=
rtf+1 + log(1 o c)
t
(o)
o f
1 o c rt+1;
o f
1 + 1 c rt+1 + log(1 oc) :
o
= exp
Finally the log-Laplace transform of the risk-neutral distribution is given
by :
o rtf+2
Mt;t+1
Q
E t [exp urtf+2 ]
= exp[ t (u + o) t (o)]
!
1
(
u + o )c
= exp[ log 1 c + 1 (u(u++o))c rtf
o
o
+1
= exp[ log(1
11 The
uc ) +
u f
1 uc rt+1 ];
o f
1 o c rt+1]
variable r +2 follows the gamma distribution with degrees of freedom , scale
parameter c and noncentrality parameter r +1 =c, if and only if :
r +2 =c follows a gamma distribution ( + Z ); where Z is drawn in the Poisson distribution P [r +1=c]. It is easily checked that this conditional distribution corresponds to
a discretized version of the Cox, Ingersoll, Ross model, if 0 < < 1; (see e.g Gourieroux,
Jasiak (2000)), and that its log-Laplace transform is given by (4.10).
f
t
f
t
f
t
t
f
t
28
t
where : c = c=(1 oc); = =(1 o c) .
For this risk neutral distribution to be dened, the risk correcting factor
o has to be chosen such that 1 o c > 0 () o < 1=c.
Thus the risk-neutral conditional distributions belong to the same family
of noncentered gamma distributions as the historical conditional distribution. The degrees of freedom are the same for the historical and risk-neutral
distributions, whereas the parameters c and depend on the risk correction
o , associated with the future short term interest rate. The standard Cox,
Ingersoll, Ross, model corresponds to a zero risk premium o = 0.
Let us now consider the derivatives with exponential payos exp(urtf ); u
varying. Since the Laplace transform characterizes the distribution, these
derivatives dene a generating system for all european derivatives.
The price Ct (u; 1) of the european derivative providing the payo exp(urtf )
at date t + 1 is :
2
+2
+2
Q
Ct (u; 1) = exp( rtf+1 ) E t (exp urtf+2 ):
This price depends on rtf+1 and o :
Ct (u; 1) = (o ; rtf+1 ; u)
(say):
It is easy to check that we get a semi-interval of admissible
prices, when
f
o varies : (t (u); +1), where : t (u) = min < =c (o ; rt ; u).
Explicit formulas can also be derived for pricing european derivatives at
any horizon. We still consider the generating system of derivatives with
exponential payos.
Proposition 4 : Let us denote by Ct (u; h) the price at t of the european
derivative, that provides the payo exp(urtf h ) at t + h. We get :
Ct (u; h) = exp[a(h; u)rtf + b(h; u)];
where the functions a and b satisfy the recursive equations :
o
+ +1
+1
29
1
+1
1 1 c + 1 [++a(ah(h 1;1u; )u)]c ;
b(h; u) = log(1 o c) log[1 [o + a(h 1; u)]c] + b(h 1; u);
for h 2:
Proof : We get :
a(h; u) =
Ct (u; h)
0
0
0
=
=
Et [Mt;t+1 Ct+1 (u; h
1)]
Et exp[o rtf+2
1 + 1 o rtf+1 + log(1 oc)
o o
+a(h 1; u)rtf+2 + b(h 1; u)]
= exp
Et
=
0
n
o f
1 + 1 c rt+1 + log(1 oc) + b(h
o
exp[o + a(h 1; u)]rtf
1; u)
o
+2
o f
exp 1 + 1 c rt+1 + log(1 oc) log[1
o
)
o + a(h 1; u)
f
+b(h 1; u) + 1 ( + a(h 1; u))c rt+1 :
o
(o + a(h 1; u))c]
The result follows by identication.
QED
The nonlinear recursive equation does not depend on the argument u, that
is on the derivative to be priced . This argument
has an eect through the
u
initial condition only, since : a(1; u) = 1+ 1 uc ; b(1; u) = log(1 uc ).
Once the functions a and b have been computed, the risk neutral distribution
at horizon h admits the log-Laplace transform :
12
12 This property is analogous to a standard property for continuous time models, saying
that the price of a european derivative satises a partial dierential equation, which is
independent of the payo.
30
Q
t;h
(u) = log Ct(u; h) log Ct(0; h)
= [a(h; u) a(h; 0)]rtf + b(h; u) b(h; 0):
It is possible to get the explicit expressions of the coeÆcient a(h; u) (and
b(h; u)). Indeed the series a(h; u) satises a rational recursive equation, which
is equivalent to :
a(h; u) 1 + a(h 1; u) ;
=
a(h; u) 1 + a(h 1; u) where and are distinct real roots of the second degree polynomial :
c + [ + c 1] 1 = 0.
Thus we get :
"
#
a(h; u) 1
+
h a(1; u) = 1+
:
a(h; u) a(1; u) +1
1
1
2
1
2
2
1
1
2
2
2
1
1
1
2
2
1
1
2
2
ii) Risky asset and stochastic interest rate
The approach above can be extended to the framework of a risky asset
with return rt and a stochastic interest rate rtf . A simple specication
of the historical
distribution assumes independence between thef riskfree rate
f
process (rt ) and the excess return process (rt = rt rt ). Then by
selecting appropriate distributions for both components, we can take into
account the positivity constraint on the riskfree rate.
As an illustration, we consider a gaussian autoregressive model for (rt )
and an autoregressive gamma model for (rtf ). Thus the conditional logLaplace transform is :
+1
+1
+1
+1
+1
+1
+1
+1
t
(u; v) = log E [exp(urt + vrtf )jJt]
= log E (exp urt jrt) + urtf + log E (exp vrtf jrtf )
= u(art + b) + 2u + urtf log(1 vc) + 1 vvc rtf
= t (u) + urtf + t (v);
(say);
(5.11)
+1
+2
+1
+1
2
+1
1
+1
+2
+1
2
2
31
+1
where art + b denotes the one-step ahead prediction of the excess return.
The stochastic discount factor is set as :
Mt;t = exp[o rtf + t rt + t ]
(5.12)
f
f
= exp[o rt + t(rt rt ) + t]:
Then the arbitrage free conditions imply :
8
f
>
< Et (Mt;t exp rt ) = 1;
+1
+2
+1
+1
+2
+1
>
:
()
+1
+1
Et [Mt;t+1 exp(rt+1 + rtf+1 )] = 1;
8
>
<
1
;t
(t ) +
;t
2
(o) + rtf + t = 0;
+1
(t + 1) + ;t (o) + rtf + t = 0:
Thus the risk correcting factor t satises :
t (t + 1) = t (t )
() t = 12 art+ b :
The risk-neutral log-Laplace transforms are given by :
>
:
1
;t
2
+1
1
1
(5.13)
2
(u; v) = t (u + t ) t (t) + urtf + t (v + o) t (v): (5.14)
The processes (rt ) and (rtf ) are still independent in the risk neutral
world, for any value of o. However the joint distribution of (rt ; rtf ) has
to be used, for pricing
a standard european call, since the payo depends
jointly on rt and rtf : :
(exp rt k) = [exp(rt + rtf ) k] :
Thus this standard call can be considered as a quanto-option, for which
the riskfree rate provides the exchange rate between the money units of dates
t and t + 1, respectively.
Q
t
1
1
+1
2
2
+1
+1
+1
+1
+1
+1
+
+1
6. Statistical inference
32
+1
+
+1
The stochastic discount factors are suitable for an analysis based jointly
on the prices of basic assets and derivatives. The observations and the econometric specication are described in the rst subsection. Then we discuss the
estimation methods based on either the basic variables, or both the basic variables and the derivative prices. Finally we introduce dierent specication
tests.
6.1 The econometric model
The observations concern :
i) the basic returns rt ; rtf ; t = 1; : : : ; T ,
ii) the real sector variables xt ; t = 1; : : : ; T
iii) the prices zi;t; i = 1; : : : ; n of n derivatives with stationary cash-ows.
More precisely zi;t is the price at t of a european derivative providing the
cash-ow gi(rtf H ; rt H ) at date t + Hi.
As usual the econometric specication is based on a latent model, which
denes the dynamics of all state variables and the stochastic discount factor.
More precisely the parameterized latent model admits three components :
i) the historicalf distribution, which is characterized by the conditional distribution of rt ; rt ; xt; ft given the information set Jt . The associated p.d.f.
is denoted by :
l(rt ; rtf ; xt ; ft jrt ; rtf ; ft ; )
(6.1)
= l(yt; ftjyt ; ft ; );
where : yt = (rt0 ; rtf ; x0t )0:
ii) The stochastic discount factor constrained by the no arbitrage condition :
+1
+
+
i +1
i
1
+1
1
+1
1
1
1
+1
= expf(t; rt; Ft ; )rt + 0tFt + (t; rt; Ft ; )g
= mt;t (yt ; ft ; ; t); (say):
iii) A parameterized risk premium :
Mt;t+1
+1
+1
+1
+1
33
+1
(6.2)
t = (yt; ft ; );
(6.3)
where is a given function and is an additional vector of parameters.
Thus the latent model includes two types of parameters, where characterizes the historical dynamics, whereas allows for the choice of a riskneutral distribution among the admissible ones.
Under the specication above, the derivative prices are :
zi;t
=
E
n
H
i
=1
mt+
;t 1 +
h
yt+ ; ft+ ; ; (yt+ 1 ; ft+
o
i
1
; )
(6.4)
(
; rt H )jJt ) ; i = 1; : : : ; n:
They are known functions of the current and past values of the state
variables and of the two types of parameters. We denote this relation by :
(6.5)
zt = q (yt ; ft ; yt ; ft ; ; ):
gi rtf+Hi
1
+
i
13
1
1
6.2 Estimation
To avoid deterministic relationships between the observed variables [see
the discussion in Gourieroux, Jasiak (2001)], we assume that the number of
unobservable factors K is larger than the number n of observed derivative
prices :
Assumption : K n.
Then the econometric model is a standard nonlinear state space model,
where :
i) the nonlinear transition equation is dened by the parametrized conditional
distribution :
l(yt ; ft jyt ; ft ; );
(6.6)
1
1
13 The function q has generally a complicated expression and its values have to be com-
puted numerically.
34
ii) the nonlinear measurement equation is :
8
>
< yt = yt
(6.7)
>
:
zt = q (yt ; ft jyt ; ft ; ; ):
When K > n, the likelihood function has a complicated expression that
includes multiple integrals of order T (K n), due to the K n factors
which cannot be recovered from the observable variables (or equivalently to
the dimension of incompleteness). However the model (5.6), (5.7) is easy to
simulate for given value of parameters and initial conditions. This allows for
simulation based estimation methods, as the maximum simulated likelihood
approach (MSL) [see e.g. Gourieroux, Monfort (1996) chap 3, Gourieroux,
Jasiak (2001), chap 13].
When K = n, the estimation problem becomes simple. Indeed we can
invert the measurement equation to express ft as a function of yt; zt ; yt ; ft
and by recursive substitution as a function of yt; zt ; yt ; zt :
ft = q~(yt ; zt jyt ; zt ; ; ) = q~t (; ); say:
(6.8)
Then the likelihood function is directly derived by applying the Jacobian
formula; for date t, its component is given by :
1
1
1
1
1
= jdet
1
1
1
L(yt ; zt jyt 1 ; zt 1 ; ; )
@ q~
(y ; z jy ; z ; ; )jl[yt; q~t (; )jyt 1; q~t
@zt0 t t t 1 t 1
1
(; ); ; ]: (6.9)
This approach has been initially proposed by Renault, Touzi (1996), Pastorello, Renault, Touzi (2000).
6.3 Specication tests
Besides the standard procedures, several specication tests can be performed in this derivative pricing framework. They concern the specication of
the risk premia and the compatibility between the historical and risk-neutral
densities.
6.3.1 Constant risk premium
35
We can check the constancy of the parameter by considering the alternative hypothesis, where this parameter is time dependent with two regimes :
Æ
H = ft = ; if t To ; t = ; if t > To g:
Under the alternative the price of a derivative depend on both levels
and , if the period ft; t + H g includes the threshold To . The testing
procedure can be based on a Lagrange multiplier approach.
1
1
0
1
6.3.2 Coherency between the historical and risk-neutral distributions.
For expository purpose, the approach
is presented for K = n, where
the information is : Jt = yt ; zt : The econometric specication can be
nested in a more general one, with dierent parameter levels in the historical
distribution and in the SDF. The alternative is dened by :
i) the pdf : l(yt ; ftjyt ; ft ; o);
ii) the SDF : mt;t (yt ; ft ; ; t);
iii) the risk premium : t = (yt; ft; ).
Thus the nesting model depends on three types of parameters ; and
, and the null hypothesis is Ho = ( = ).
This approach has been rst proposed in the econometric literature on
the term structure of interest rates [see e.g. De Munnik, Schotman (1994),
De Jong (1997), Bams, Schotman (1997), Bams (1998)]. They propose to
test the null hypothesis in the following way.
i) Firstly, the parameter o is estimated by using the time series of observed basic variables. The estimator is :
+1
1
1
+1
+1
1
0
0
^o = arg max
o
T
X
t=1
1
1
log l(ytjyt ; o):
1
ii) Secondly, and are jointly estimated by considering the so-called
cross-sectional approach, that is by optimizing :
1
36
T
X
(^ ; ^) = arg max log l(zt jyt; yt
1
1 ; t=1
1
; zt
1
; ^o; ; ):
1
ii) Finally the estimates ^o and ^ are compared by means of a standard
specication test.
1
37
REFERENCES
Ait-Sahalia, Y. (1996) : "Nonparametric Pricing of Interest Rate Derivative Securities", Econometrica, 64, 527-560.
Bams, D. (1998) : "An Empirical Comparison of Time Series and CrossSectional Information in the Longsta-Schwartz Term Structure Models",
Maastricht Univ. DP.
Bams, D., and P., Schotman (1997) : "A Panel Data Analysis of AÆne
Term Structure Models", Maastricht Univ. DP.
Black, F., and M., Scholes (1973) : "The Pricing of Options and Corporate Liabilities", Journal of Political Economy, 81, 637-659.
Breeden, D., Glosten, L., and R., Jagannathan (1989) : "Economic Significance of Predictable Variations in Stock Index Returns ", Journal of Finance,
44, 1177-1189.
Brown, D., and M., Gibbons (1985) : "A Single Econometric Approach
for Utility-Based Asset Pricing Models", Journal of Finance, 40, 359-381.
Buhlman, H., Delbaen, F., Embrechts, P., and A., Shyraev (1996) : "No
Arbitrage, Change of Measure and Conditional Esscher Transforms in a SemiMartingale Model of Stock Processes", CWI Quarterly, 9, 291-317.
Campbell, J., Lo, A., and C., McKinlay (1997) : "The Econometrics of
Financial Markets", Princeton Univ. Press.
Clark, S. (1993) : "The Valuation Problem in Arbitrage Price Theory",
Journal of Mathematical Economics, 22, 463-478.
Cochrane, J. (2001) : "Asset Pricing", Princeton University Press.
Cochrane, J., and L., Hansen (1992) : "Asset Pricing Explorations for
Macroeconomics", in NBER Macroeconomics Annual, MIT Press, 115-165.
Cox, J., Ingersoll, J., and S., Ross (1985) : "A Theory of the Term
Structure of Interest Rates", Econometrica, 53, 385-408.
38
Darolles, S., Gourieroux, C., and J., Jasiak (2001) : "Compound Autoregressive Models", CREST, DP.
De Jong, F. (1997) : "Time Series and Cross Section Information in AÆne
Term Structure Models", DP.
De Munnik, J., and P., Schotman (1994) : "Cross-Sectional Versus Time
Series Estimation of Term Structure Models : Empirical Results for the
Dutch Bond Market", Journal of Banking and Finance, 18, 997-1025.
Duan, J.C. (1995) : "The GARCH Option Pricing Model", Mathematical
Finance, 5, 13-32.
Engle, R. and C., Mustafa (1992) : "Implied ARCH Models from Option
Prices", Journal of Econometrics, 52, 289-311.
Epstein, L., and S., Zin (1991) : "Substitution, Risk Aversion and the
Temporal Behavior of Consumption and Asset Returns I : A Theoretical
Framework", Econometrica, 57, 937-969.
Esscher, F. (1932) : "On the Probability Function in the Collective Theory of Risk", Skandinavisk Aktuariedskrift, 15, 165-195.
Garman, M., and J., Ohlson (1980) : "Information and the Sequential
Valuation of Assets in Arbitrage Free Economics", Journal of Accounting
Research, 18, 420-440.
Gourieroux, C., and J., Jasiak (2000) : "Autoregressive Gamma Processes", CREST, DP.
Gourieroux, C., and J., Jasiak (2001) : "Financial Econometrics", Princeton University Press.
Gourieroux, C., and A., Monfort (1996) : "Simulation Based Econometric
Methods", Oxford Univ. Press.
Gourieroux, C., and A., Monfort (2001)a : "Pricing with Splines", CREST
DP.
Gourieroux, C., and A., Monfort (2001)b : "AÆne Models of Interest
39
Rates", CREST DP.
Hansen, L., Heaton, J. and E., Luttmer (1995) : "Econometric Evaluation
of Asset Pricing Models", The Review of Financial Studies, 8, 237-274.
Hansen, L., and R., Jagannathan (1997) : "Assessing Specication Errors
in Stochastic Discount Factor Models", Journal of Finance, 52, 557-590.
Hansen, L., and S., Richard (1987) : "The Role of Conditioning Information in Deducing Testable Restrictions Implied by Dynamic Asset Pricing
Models", Econometrica, 55, 587-613.
Hansen, L., and K., Singleton (1982) : "Generalized Instrumental Variables Estimation of Nonlinear Rational Expectations Models", Econometrica,
50, 1269-1286.
Harrison, M., and D., Kreps (1979) : "Martingales and Arbitrage in
Multiperiod Securities Markets", Journal of Economic Theory, 20, 381-408.
Heston, S. (1993) : "A Closed Form Solution for Options with Stochastic
Volatility with Application to Bond and Foreign Currency Options", Review
of Financial Studies, 6, 327-343.
Hull, J., and A., White (1987) : "The Pricing of Options on Assets with
Stochastic Volatilities", Journal of Finance, 42, 281-300.
Kallsen, J., and M., Taqqu (1998) : "Option Pricing in ARCH Type
Models", Mathematical Finance, 58, 13-26.
Madan, D., Carr, P., and E., Chang (1998) : "The Variance Gamma
Process and Option Pricing", European Finance Review, 2, 79-105.
Madan, D., and F., Milne (1991) : "Option Pricing with VG Martingale
Components", Mathematical Finance, 1, 39-48.
Madan, D., and E., Seneta (1990) : "The Variance-Gamma Model for
Share Market Returns", Journal of Business, 63, 511-524.
Melino, A., and S., Turnbull (1990) : "Pricing Foreign Currency Options
with Stochastic Volatility", Journal of Econometrics, 45, 239-265.
40
Noh, J., Engle, R., and A., Kane (1995) :"Forecasting Volatility and Option Prices of the S& P Index", in Engle, R. ed. "ARCH Selected Readings",314331, Oxford Univ. Press.
Pastorello, S., Renault, E., and N., Touzi (2000) : "Statistical Inference
for Random Variance Option Pricing", Journal of Business and Economic
Statistics, 18, 358-367.
Renault, E. (1999) :"Dynamic Factor Models in Finance", CORE Lecture,
Oxford Univ. Press, forthcoming.
Renault, E., and N., Touzi (1996) : "Option Hedging and Implied Volatilities in a Stochastic Volatility Model", Mathematical Finance, 6, 279-302.
Shyraev, A. (1999) : "Essentials of Stochastic Finance : Facts, Models,
Theory", World Scientic Publishing, London.
Stapleton, R., and M., Subrahmanyam (1984) : "The Valuation of Multivariate Contingent Claims in Discrete Time Models", The Journal of Finance,
39, 207-228.
Weil, P. (1989) : "The Equity Premium Puzzle and the Risk-Free Rate
Puzzle", Journal of Monetary Economics, 24, 401-421.
41