346 K.K. Aase Insurance: Mathematics and Economics 27 2000 345–363
evaluations of contingent claims result, where only the short-term interest rate turns out to directly depend on preferences.
Given that a new asset pricing model is proposed, it seems natural to investigate how well it performs, and in particular if it is able to explain existing puzzles in the theory of asset pricing. This we attempt to do in Section 5,
where we derive a consumption-based capital asset pricing model CCAPM based on the NiG model, and calibrate it to the data in the Mehra and Prescott study of 1985. Although we do not perform any type of empirical investigation,
it turns out that the equity premiums may be explained for moderate values of the risk aversion and impatience rate in the market, yielding a possible explanation of the equity premium puzzle. Moreover, the survival hypothesis
of Brown et al. from 1995 can be tested in this framework, and a market crash probability may be endogenously derived, calibrated to the above data. Based on the given 90 years history of market data, a rather low market crash
probability was estimated.
The way we construct our model, it turns out to be related to stochastic volatility models of arbitrage pricing theory, the connection being that the aggregate consumption process in our framework plays the role of the exogenously
imposed stochastic volatility process in the arbitrage pricing theory. The apparent advantage with the equilibrium theory is that the randomness in the volatility parameter now comes as a result of more basic assumptions, not being
proposed as an extra exogenous source of uncertainty.
An early reference to a related model in finance is McCulloch 1978, who considered a return process with Cauchy-distributed marginals. Time series of daily returns usually do not exhibit significant autocorrelation, so it
may therefore seem natural to model the returns in non-overlapping periods of time to be independent and identically distributed, hence the Lévy process seems reasonable. This does not, however, imply a zero correlation structure of
each associated asset price. Since a Lévy process is also a semimartingale, we have to our disposal the associated stochastic calculus for this latter category of processes, a fact we make use of.
The first to introduce the NiG distribution to finance was Barndorff-Nielsen 1994, based on a paper of Eberlein and Keller 1994. The latter demonstrated that asset returns were well fitted by hyperbolic distributions. This class
of distributions turns out to outperform other distributions as models of returns of stocks in the German stock market. The hyperbolic distribution lacks the desirable feature of being closed under convolutions, and based on
this observation a stochastic process was constructed, built on a similar class of distributions, having the convolution property and termed the NiG distribution. Later work by, e.g., Rydberg 1996 has indicated that the NiG distribution
provides on its own a promising fit to returns on stocks.
This paper is organized as follows. The economic model is explained in Section 2.1, and the probability model of asset returns is given in Sections 2.2–2.4. In this part we model returns as cumulative logarithmic returns.
Preference-free evaluation is discussed in Section 3, and a special option pricing formula is derived in Section 4, where its relation to stochastic volatility models is briefly discussed. A CCAPM is derived in Section 5. In Section
6, we view returns as cumulative, simple returns by the use of the semimartingale exponential, in which case we can develop an approximation for the ruin intensity of a risky asset. In Section 7, the results of Section 6 are used
to discuss the survival hypothesis. In Section 8, we consider several risky assets, and Section 9 concludes.
2. Equilibrium pricing of derivative securities with Lévy process driven uncertainty
2.1. Introduction: the economic model In this section, we present the economic model. It consists of N
+ 1 primitive securities, where N of them are risky with return processes Rt
= R
1
t , . . . , R
N
t , paying no dividends and with associated price processes St
= S
1
t , . . . , S
N
t , where t is the time variable, 0 ≤ t ≤ T , and one is a risk-less security with real spot-price
unity, and real return rate equal to the short-term interest rate rt . There are m agents in this market each one being characterized by a nonzero consumptionendowment process e
i
, and by strictly increasing utility functions U
i
e
i
which are time-additive and of the von Neumann–Morgenstern type, i.e., U
i
e
i
= E{ R
T
u
i
e
i
t , t dt }. Special to
our model is that c
i
t is the total amount of consumption by time t for agent i. From this starting point we know
K.K. Aase Insurance: Mathematics and Economics 27 2000 345–363 347
the conditions under which there exists a capital market equilibrium with a representative agent felicity index u, in which all the agents are maximizing their utility functions subject to budget constrains such that markets clear. The
main analytic issue is then the determination of equilibrium price behavior. In the model to be specified below, we will find it convenient to let the felicity index u depend on accumulated
consumption c
t
during the time period [0, t ]. This may seem unusual to some readers. There is, however, a recent literature on deviations from the standard separable, additive utility representations. Following, e.g., Constantinides
1990, let us consider z
t
= c e
−at
+ d Z
t
e
−at−s
dc
s
, 1
where a and d are constants, a ≥ 0 and d 0, i.e., z
t
is a process of weighted past consumption derived from the consumption process c
t
, the latter denoting the total amount of consumption by time t , aggregated over all the consumers. Clearly past consumption matters more, the smaller the parameter a is. Let y be a process defined
analogous to 1 but with a different set of parameters. Then the felicity index u considered would be a function uz, y, t of two variables and time, say, where ∂u∂z 0, and ∂u∂y 0. The utility function of the market is
then given by U c = E{
R
T
uz
t
, y
t
, t dt }. In such a case we would have a situation with both habit formation
and durability. Habit formation because of the process y and the negative partial derivative of the felicity index with respect to its second argument, durability because of the process z and the positive partial derivative of u with
respect to its first argument see, e.g., Hindy et al., 1997. Returning to our model, we have a situation with only durability, where d
= 1 and a = 0. This is a strong form of durability, where past consumption clearly matters. Along a possible optimal consumption path c, a large value
of the total consumption for some t , would typically imply a smaller value at time t + with high probability.
Prices are then determined as follows. If q, D is any of the given securities with real price process q and accumulated dividend process D, the real market value q at each time t satisfies the following:
qt =
1 u
′
c
t
, t E
Z
T t
u
′
c
v
, v dDv + d[D, u
′
c]v |F
t
, 2
where u
′
signifies the marginal utility of the representative agent, and [D, u
′
c]t the square covariance pro- cess between accumulated dividends Dt and the marginal utility process u
′
c
t
see Aase, 1997. Here ct =
P
m i
=1
c
i
t =
P
m i
=1
e
i
t is the aggregate, accumulated consumption process in the market, and where F
t
is the information possibly available at time t , formally given by some filtration on a given probability space Ω, F , P
satisfying the usual hypotheses, where F = F
T
. The symbol E stands for the expectation operator on this space, and we will sometimes use E
t
to signify conditional expectation given F
t
.
1
By integrating to an intermediate time point s, we use iterated expectation and the pricing principle in 2, in which case Eq. 2 can be written as
q
t
= 1
u
′
c
t
, t E
Z
s t
u
′
c
v
, v dD
q
v + d[D, u
′
c]v + u
′
c
s
, sq
s
|F
t
, t
≤ s ≤ T . 3
If there are no intermediate dividends before the final time point T , the pricing relation is q
t
= 1
u
′
c
t
, t E
t
{u
′
c
T
, T q
T
}, t
≤ T . 4
In what follows, we shall assume that the intertemporal marginal utility function of the market is given by u
′
c, t =
e
−ρt
e
−ηc
, where ρ is the market’s impatience rate and η the intertemporal coefficient of absolute risk aversion in the market. Thus, ρ, η will be referred to as the preference parameters. This marginal utility function may result
in equilibrium if all the agents have different utility functions of this class with individual preference parameters
1
Conceptually, 2 corresponds to Eq. 6 in Lucas 1978.
348 K.K. Aase Insurance: Mathematics and Economics 27 2000 345–363
ρ
i
, η
i
, i = 1, 2, . . . , m, in which case η
−1
= P
m i
=1
η
−1 i
, i.e., the risk tolerance of the market is the sum of the agents’ individual risk tolerances.
The question whether the pricing rule 2 can be used also for assets other than the given primitive ones represented by S, is related to the complete market issue. One way to complete such a market is to introduce trading of nonlinear
contracts, i.e., contracts that cannot be written as stochastic integrals of self-financing portfolios with respect to the given primitive securities. Because of the above assumptions we have on preferences, we can take another path: In
our model the aggregation property is satisfied, so exchange efficiency holds for our model. Here we appeal to the aggregation theorem of Borch 1962 see also Wilson, 1968; Rubinstein, 1974. It relies on the observation that
under certain conditions, even though the market may be incomplete, the equilibrium prices will be determined as if there were an otherwise similar complete Arrow–Debreu market.
We shall, however, interpret the model as a representative agent equilibrium in which all assets are priced by the relation 2. The introduction of a new asset may change the allocations and hence lead to a new equilibrium,
where all assets, old and new satisfy Eq. 2. For other treatments where the consumption and dividend processes are semimartingales, see, e.g., Back 1991, Naik and Lee 1990, Aase 1993a,b.
2.2. Probability distributional assumptions on consumption We now turn to the distributional assumptions, and by this we mean marginal probability distributions at each
time t , and we start with the aggregate, cumulative consumption process ct , denoted c
t
, of all the agents at time t .
Assumption 1. c
t
has an inverse Gaussian probability density at each time t . This means that its density function is given by
p
t
x =
x
−32
t δ √
2π e
−12xtδ−x √
ψ
2
, x 0, t 0,
5 and p
t
x = 0 for x ≤ 0 and all t, where δ 0 and ψ = α
2
− β
2
≥ 0 are parameters, δ being a scale parameter. The inverse Gaussian process c
= c
t ≤t≤T
with parameters δ, α and β is now defined as the homogeneous Lévy process for which the probability density of c
t
is given by Eq. 5. We remind our readers that an adapted process X
= X
t ≤t≤T
is a Lévy process if i X has independent increments, ii X has stationary increments, and iii X
t
is continuous in probability. Our assumption implies that the increment c
T
− c
t
given F
t
has the same conditional distribution as c
T −t
at time T
− t given F , assuming that c
= 0. Thus, we know that c is a strong Markov process with Feller transition functions P
t
y, dx = p
t
x − y dx, where P
t
y, A = P c
s +t
∈ A|c
s
= y for any Borel set A of reals. Let K
t
v = ln Ee
−vc
t
. 6
From 5, it then follows that K
t
v = tδ{
p ψ
− p
ψ + 2v}.
7 From this expression, we can find the cumulants and consequently, e.g., the two first moments: Ec
t
= tδ √
ψ , and varc
t
= tδψ
32
for all t ≥ 0.
Note that the increments are here non-negative, and c
t
is interpreted as cumulative consumption over the time interval [0, t ], in addition to also representing an aggregate quantity over all the investors. Because of the time
homogeneity of c, we may define the average consumption c
t
= c
t
t in the period [0, t ]. The probability density p
t c
x of c
t
is then given by p
t c
x = p
t
txt, where p
t
x is given in 5. The process c is known to make an infinity of jumps in any finite time interval but the set of jump times has
Lebesgue measure zero almost surely. The inverse Gaussian distribution has many applications see, e.g., Seshadri,
K.K. Aase Insurance: Mathematics and Economics 27 2000 345–363 349
1993. Bachelier 1900 discussed in his thesis “Theorie de la spéculation” a problem leading to the inverse Gaussian distribution. This distribution can accommodate a wide variety of shapes and, being a member of the exponential
family, often lends a meaningful interpretation that provides satisfactory physical support to empirical fit. It is likely that this also may be the case in the present application to financial economics.
The distribution seems well suited to the modeling of cumulative consumption, which by assumption must be non-negative. Since the quantity in question is accumulated over time, an aggregate over agents, and also over
goods and services in empirical applications, by the central limit theorem we could then expect something Gaussian looking. This can be accomplished by the present family of distributions, since the inverse Gaussian distribution
tends towards normality as the parameter δt √
ψ increases.
2
This can be inferred from the above cumulant generating function K
t
v in 7.
3
2.3. Probability distributional assumptions on asset returns We start with the case N
= 1, with one risky and one riskless security. We denote the accumulated return process of the risky asset by ˜
R, and assume it to be a homogeneous Lévy process.
Assumption 2. The conditional distribution of the accumulated return on the risky asset given the aggregate consumption c
t
, we require to be normal with 1. mean tµ
+ βc
t
and 2. variance σ c
t
, where σ is a positive constant. This assumption is here crucial in order to obtain the desired marginal distributions on assets returns derived below.
However, it is not as far-fetched as it may appear at first glance. Recall the classical asset pricing model by Black and Scholes 1973. It is possible to alternatively derive their results in a representative agent model with one consumption
good much the same way as we do in our model see, e.g., Bick, 1987 or Aase, 1998. This can be done using, e.g., a negative exponential utility function for the representative agent, if the return of the risky security is assumed to
be conditionally normally distributed given c
t
, where c
t
is normally distributed see e.g., Aase, 1997, example 2. The variance assumption in 2 is more peculiar to this model. It leads to a relationship with stochastic volatility
models, since c is a random process. We will concentrate on the case σ = 1 for matters of simplicity. Assumptions
1 and 2 lead directly to the marginal distribution for the accumulated return of the risky asset in the time period [0, t ], denoted by ˜
R
t
, to be of the desired NiG type with density function q
t
x = aα, β, tµ, tδq
x − tµ t δ
−1
K
1
q x − tµ
t δ t δα
e
βx
, t
≥ 0, x ∈ R, 8
where aα, β, t µ, t δ = 1πα e
t δ √
ψ −βµt
, qx =
√ 1
+ x
2
, and K
1
x the modified Bessel function of the third-order and index 1. An integral representation of K
1
x is given in Eq. 39. Here 0 ≤ |β| ≤ α, where β is a
symmetry parameter, α a steepness parameter, δ a scale parameter and µ a location parameter. Consider the accumulated return ˜
R
t +s
− ˜ R
s
in the time interval s, s + t] of a risky asset. Assuming that the
probability distribution of this quantity is given by 8 for all s and t ≥ 0, ˜
R = ˜
R
t ≤t≤T
is a strong Markov process with Feller transition probabilities Q
t
x, dy = q
t
y − x dy, and the moment generating function of the
increments is Mu
; α, β, tµ, tδ = E e
u ˜ R
t +s
− ˜ R
s
= exp t δ
q α
2
− β
2
− q
α
2
− β + u
2
+ tµu .
9
2
This is in the usual formal manner of convergence in distribution, and there will of course not be any probability mass on the negative real for any finite value of this parameter.
3
To our knowledge this is the first time the inverse Gaussian distribution has been suggested as a model for the aggregate consumption. An empirical investigation would eventually be needed to test the goodness of this hypothesis.
350 K.K. Aase Insurance: Mathematics and Economics 27 2000 345–363
The NiG process is accordingly defined as the homogeneous Lévy process ˜ R
= ˜ R
t ≤t≤T
for which the moment generating function of ˜
R
t +s
− ˜ R
s
is given by M
t
u ; α, β, µ, δ = Mu; α, β, µ, δ
t
, 10
where Mu ; α, β, µ, δ is given in 9 with t = 1. Thus
M
t
u ; α, β, µ, δ = Mu; α, β, tµ, tδ.
11 From this, it follows that if ˜
R
1
, ˜ R
2
, . . . , ˜ R
n
are independent NiG-distributed random variables with common para- meters α and β but having individual location and scale parameters µ
i
and δ
i
, then ˜ R
= P
n i
=1
˜ R
i
is again distributed accordingly to the NiG law with parameters α, β,
P µ
i
, P δ
i
. Furthermore, if ˜ R
t
− ˜ R
is distributed according to 8, then it has mean and variance
E ˜ R
t
− ˜ R
= tµ + tδ β
α p
1 − βα
2
, 12
var ˜ R
t
− ˜ R
= t δ
2
α1 − βα
2 32
. 13
In particular ˜ R
− bt is a martingale, where the constant b is found from 12. A multivariate extension of the NiG distribution is possible to construct in view of the normal mixture representation for details, see Barndorff-Nielsen,
1977, 1994. We give a brief discussion of several variables in Section 8. The NiG process has the following properties: for β
= 0, α → ∞ and δα = σ
2
, the process tends to the Brownian motion with drift µ and diffusion coefficient σ sometimes called the Bachélier process. The process
˜ R
t
− ˜ R
may be represented via a random time change of a Bachélier process B
t
as ˜ R
t
− ˜ R
= B
c
t
+ µt, where B
t
has drift β and diffusion coefficient 1, and where c
t
is the inverse Gaussian process with parameters δ and √
Ψ , c being independent of B. Furthermore, it can be shown from the Lévy representation of ˜
R that the small jumps dominate the behavior of the process ˜
R. The distribution 8 can be used to closely approximate the hyperbolic distribution, which in its turn has displayed
close fits to return data for common stocks Eberlein and Keller, 1994; Küchler et al., 1994. However, the hyperbolic distribution does not have the convolution property. As mentioned above, the NiG distribution itself has proved to
provide a close fit to stock return data. Both classes belong to the family of generalized hyperbolic distribution, and all of these are the so-called normal variance-mean mixtures. In the case of the latter class the mixing distribution
is inverse Gaussian, which accounts for the terminology. Of all the properties of the processes ˜
R and c listed above, many of which are mentioned in Barndorff-Nielsen 1994, the conditional normality of ˜
R given c is of particular use to in Section 4.
2.4. A representation for the return process Before we proceed, we consider a representation of the return process defined above. An expression for the
characteristic function of the NiG process follows from conditioning on c
t
: E e
iτ ˜ Rt
= exp{iτ tµ+K
t 1
2
τ
2
−iτβ}, where K
t
· is given in Eq. 7.
4
It is now possible to derive the Lévy decomposition of the NiG process, which is ˜
R = θ + µt +
Z
|y|1
y {Nt; dy − tνdy} +
Z
|y|≥1
yNt ; dy,
14 where θ
= 2π
−1
δα R
1
sinhβxK
1
αx dx and the Lévy measure νdy is νdy
= π
−1
δα |y|
−1
e
βy
K
1
α |y| dy.
15
4
Compare this expression to the moment generating function in 9.
K.K. Aase Insurance: Mathematics and Economics 27 2000 345–363 351
Here N t ; dy is a random Poisson measure, meaning that N
A t
= R
A
N t ; dy is a Poisson process with parameter
νA =
R
A
νdy, and N
A t
and N
B t
are independent if A and B are disjoint sets of reals. Thus, ˜ R is a process with
paths of bounded variation. Any P , F
˜ R
t
-martingale M can be represented as Mt
= M0 + Z
t
Z
−∞,∞
θ s, y {Nds; dy − νdy ds}.
16 This representation property is obviously different from a representation in the form of linear spanning. Thus, our
model is incomplete. The price process of a risky asset having return process ˜
R
t
we assume to be given by the following: St
= S e
˜ R
t
, ≤ t T .
17 In other words, we interpret ˜
R
t
as the cumulative logarithmic returns by time t . In Section 8, we alternatively model S by the semimartingale exponential, implying that we view the term ˜
R
t
as the cumulative simple returns for the period [0, t ].
3. Preference-free equilibrium pricing of derivative securities
