1. INTRODUCTION
Lévy processes, processes with stationary independent
increments, became popular for modeling financial data during the last
decade. However, the earliest attempt to model the stock behavior by a
Lévy process, the Brownian motion, was by Bachelier (1900) in his Ph.D. thesis. More recently there has
been a focus on Lévy processes with jumps. Hyperbolic
Lévy motions (cf. Eberlein and Keller,
1995; Keller, 1997), generalized
hyperbolic Lévy motions (cf. Prause,
1999; Raible, 2000), normal inverse
Gaussian processes (cf. Barndorff-Nielsen,
1998; Rydberg, 1997), stable processes
(cf. Rachev and Mittnik, 2000), variance
gamma processes (cf. Madan and Senata, 1990),
and CGMY processes, also called truncated Lévy flights (cf.
Carr, Geman, Madan, and Yor, 2002) yield good
models for log-return processes of prices and exchange rates. These are
models of the form log St =
Xt, where St
is the price process. Models based on these processes are less
restrictive than the traditional ones; they allow jumps and include
both finite and infinite activity and also bounded and unbounded
variation. Furthermore, the empirical facts of excess kurtosis,
skewness, and fat tails can be modeled more realistically.
One parameter that is especially important for modeling is the
skewness parameter. The skewness of a distribution is modeled by
multiplying with an exponential term
.
For θ > 0 the resulting distribution is right/positive skewed
depending on the size of θ; i.e., the bigger θ is the more
weight is put on larger x. This parameter is an important
parameter in finance, because according to the empirical data the
distribution of the log-return prices is mostly skewed (cf. Prause, 1999). Carr et al. (2002) perform a detailed analysis of skewness,
finding that statistical data (i.e., the time series of stock returns)
are significantly skewed either right or left, whereas risk-neutral
data (i.e., data derived from option prices) are consistently left
skewed. The skewness parameter is of course not the same as the
financial term skewness, the appropriately normed third
centered moment of a distribution. It measures the same effect, namely,
the derivation from the symmetric distribution, but especially for
fitting data we need the skewness parameter itself.
The main problem for estimating parameters entering a Lévy
process Xt is that in general the process
is given by the Lévy–Khinchin formula or in other words
the characteristic function,
where μ denotes the drift, σ2 the diffusion, and
g the density w.r.t. ν of the Lévy measure,
satisfying ∫(1 ∧
x2)g(x,θ)ν(dx)
< ∞, and h(x) is some truncation function,
which behaves like x in the neighborhood of zero and ensures
integrability in the characteristic function. Common examples are
h(x) =
x1|x|≤1(x) and
h(x) = x/(x2 + 1).
However, the density of the process itself is unknown and cannot be
calculated explicitly, as for most stable, hyperbolic, generalized
hyperbolic and CGMY processes. Hence we have to find conditions on the
Lévy triplet
(μ(θ),σ2(θ),g(x,θ)ν(dx))
that allow us to construct efficient estimators explicitly.
We look at the special case where our unknown parameter is the
skewness, g(x,θ) =
eθxg(x), and focus
on the concept of asymptotic statistics. We show that under very mild
regularity conditions we obtain local asymptotic normality for the
skewness parameter, giving us the maximal rate of convergence of a
sequence of regular estimators and the minimal asymptotic variance,
which turns out to be fully explicit, only involving the quantities of
the Lévy triplet. Furthermore, we can then construct efficient
estimators and show the relation to martingale estimating functions.
For all financial applications (e.g., fitting of models and also
pricing derivatives and quantifying risk) it is important to have good
estimators of the underlying parameters when the data are given at
discrete time points,
XΔ,X2Δ,…,XnΔ,
because continuous data are hardly available in practice or not
economical to observe.
We face two different sampling schemes; either we let the distance
between the observations Δ be fixed and the number of observations
n tend to infinity, or we let nΔ → ∞
as Δ → 0 and n → ∞. The first sampling
scheme seems to be of more practical interest, because the distance of
the observations can be large. The second one is an approximation to
the continuously observed model. The third possible discrete sampling
scheme where nΔ = const. < ∞, n →
∞, and Δ → 0, which would be the classical framework of
high-frequency data, is not possible in our setting. Heuristically,
this can easily be seen when looking at the continuously observed
model. In the continuously observed model we obtain local asymptotic
normality for the skewness parameter (cf. Akritas
and Johnson, 1981) when the observed time T tends to
infinity. Hence we also need in the discretely observed model that
Δn → ∞, because T may be identified
with nΔ. This has some important implications. We cannot
infer the skewness parameter by high-frequency data over a fixed period
of time. On the other hand, it has the advantage that it makes sense to
infer the skewness parameter, even in the presence of other unknown
parameters such as diffusion or scale, because they have a faster rate
of convergence.
The outline of the paper is the following. In Section 2 we will
review the concepts of local asymptotic normality and measure changes
in Lévy processes and the result for continuously observed
models. In Section 3 we will prove local asymptotic normality, and in
Section 4 we construct efficient estimators and view them in the
context of martingale estimating functions. Section 5 concludes.
2. PRELIMINARY RESULTS
The theory of local asymptotic statistics and the related efficiency
results are established by Le Cam (1960) and
Hájek (1972) and extended by
Jeganathan (1981, 1983) and others.
This concept provides answers to important questions in estimation
theory, e.g., how to characterize optimal estimators. Having local
asymptotic normality we can specify the maximal rate of convergence of
a sequence of estimators and the minimal asymptotic variance.
Furthermore, it not only allows us to decide if a given sequence of
estimators is efficient but also allows us to construct efficient
estimators from suboptimal estimators by a one-step improvement,
described, e.g., in Le Cam and Yang (1990).
Let us recall the definition of local asymptotic normality (LAN). Let
pn(X0,…,Xn;θ)
be the joint density of
(X0,…,Xn) under
θ ∈ Θ and ln(θ +
δn h,θ) the log-likelihood around
θ, i.e.,
where δn,h > 0.
DEFINITION (LAN). Let Θ be an open subset of
a sequence of experiments.
For fixed
is called LAN in θ, if
(i) There exist δn =
δn(θ) ↓ 0.
Γn symmetric, strictly positive definite,
such that for all
(ii) There exists a finite, positive semidefinite, nonrandom
matrix Γ, such that
as n → ∞.
LAN
that a sequence of estimators cannot converge to the true parameter value
θ0 at a rate faster than
δn−1 and the asymptotic
variance of a δn−1-consistent
estimator is bounded from below by Γ−1.
The LAN property is often established by proving
L2-convergence, which implies the appropriate
expansion of the log-likelihood function only involving first
derivatives. However, in our case of the skewness parameter it turns
out that it is easier to take a different way, looking at the Taylor
expansion of the log-likelihood function up to the second order,
because second derivatives exist. For the sampling scheme with fixed
distance of observations Δ we are in the framework of independent
and identically distributed (i.i.d.) random variables, which is well
studied (cf. Witting, 1985; Janssen, 1992; van der Vaart,
1998). For the other sampling scheme we are in the framework of
triangular arrays because of the dependence of the densities on
n through Δn, and we have to establish
an appropriate central limit theorem (CLT) and law of large numbers
(LLN) for the first and second terms of the Taylor expansion of the
log-likelihood function around the true parameter θ0.
For more details see the proof of Theorem 2 in Section 3.
Our aim is to prove LAN for discretely observed Lévy
processes. However, for continuously observed Lévy processes
there are some results known. Akritas and Johnson (1981) consider general purely discontinuous
Lévy processes. We will state their result for the special case
of the skewness parameter. We recall this result because the
continuously observed model builds a natural benchmark for the model
with discrete observations. Especially the sampling scheme with
nΔ → ∞ as Δ → 0 and n →
∞ may be well compared with the continuous model, as
nΔ corresponds to t.
However, the continuous model only provides an optimality bound when
the underlying measures are absolutely continuous. This is quite a
strong restriction to the possible variation of parameters as we can
see in the following theorem. Skorokhod (1957) derived this result first; for a detailed
account, see, e.g., Shiryaev (1999).
THEOREM 1. Let Xt be a Lévy
process with triplet
(μ,σ2,g(x)ν(dx))
under some probability measure P. Then the following two
conditions are equivalent.
(1) There is a probability measure
such that Xt is a Q-Lévy process
with triplet
(μ,σ2,g(x)ν(dx)).
(2) All of the following four conditions
hold.
[bull ] g(x)ν(dx)
= k(x)g(x)ν(dx) for some
Borel function :
.
[bull ] μ = μ +
∫h(x)(k(x) −
1)g(x)ν(dx) + σβ for some
.
This theorem implies that we cannot have the LAN property in the
continuous model when we aim to estimate the diffusion, or a scalar
factor in front of an infinite Lévy measure, because the
underlying measures cannot be absolutely continuous. However, for the
skewness parameter, i.e., k(x) =
exp{θx}, we can have an absolutely continuous change of
measures and LAN for the continuously observed model with
,
where T is the observed time that tends to infinity and σ = 0
(cf. Akritas and Johnson, 1981).
3. LOCAL ASYMPTOTIC NORMALITY
Let us now assume that we are given a Lévy process with a
skewed Lévy measure
eθxg(x)ν(dx)
and we aim to estimate the skewness parameter θ. First of all we
are interested how the multiplicative term in the Lévy measure
changes the distribution of the underlying Lévy process. We
obtain the following lemma. Though we shall only look at Lévy
processes with unbounded support, the same calculations hold for
processes with bounded support, especially for subordinators.
LEMMA 1. Denote by pt(x) the
density of the Lévy process w.r.t. m, given by the triplet
(μ,σ2,g(x)ν(dx)) and
by pt(x,θ) the density
corresponding to the process with the skewed Lévy measure
eθxg(x)ν(dx).
Furthermore, assume that for all θ ∈
U(θ0) ⊂ Θ, where
U(θ0) is a neighborhood of the true parameter
θ0,
Then we obtain
and the corresponding drift is μ = μ +
∫h(x)(eθx −
1)g(x)ν(dx).
Remark. (1) For processes with finite variation, i.e.,
∫(|x| ∧
1)g(x)ν(dx) < ∞, e.g., all
compound Poisson processes and subordinators, we may take
h(x) = 0 and obtain μ = μ.
(2) We consider densities pt w.r.t. to
some measure m. However, of most practical interest, except
for compound Poisson processes, is when m equals the Lebesgue
measure, as is the case in our examples. Even though the density
pt might not be known explicitly,
conditions on the Lévy measure may ensure its existence; e.g.,
as was shown in Tucker (1962), infiniteness
of the Lévy measure together with a Lebesgue density of the
Lévy measure already ensures the existence of a Lebesgue density
pt. However, the relation between the
existence of a density pt and the
existence of a density of the corresponding Lévy measure and its
mass near zero is much more complex. A detailed outline is given in
Sato (1999).
Proof. Under the assumption (1) the denominator in (2) is well defined.
We assume that the characteristic function corresponding to
pt is
Hence we can calculate the characteristic function of the skewed
distribution
which yields the desired result. █
Of course this result is not new. The principle of multiplying with
an exponential factor is well established in different disciplines of
stochastics but is named differently. In statistics the family of
distributions or processes derived by varying the skewness parameter is
called the natural exponential family (cf. Janssen,
1992; Küchler and Sørensen,
1997). In finance the the distribution obtained by adding the
skewness is called the Esscher transform (cf. Shiryaev, 1999). This concept is also an easy way
of explicitly calculating equivalent martingale measures.
Using (2) we can prove LAN for the skewness parameter. Though we
start with a general Lévy process with unknown density, we
obtain a fully explicit result of the minimal asymptotic variance,
depending only on the quantities of the Lévy triplet, for both
sampling schemes. Furthermore, our minimal asymptotic variance for the
sampling scheme with Δ → 0 turns out to be the same as in the
continuously observed model.
THEOREM 2. Assume condition (1) for θ ∈
U(θ0) ⊂ Θ, a neighborhood of
θ0; then we obtain LAN
Proof. Because we have i.i.d. increments as a result of the structure
of the Lévy processes, we can use the results for i.i.d. random
variables.
First, we have to calculate the derivatives. We may interchange
integration and differentiation if θ is in the interior of Θ
and obtain
(i) For the sampling scheme with fixed distance of observations
Δ, we are in the framework of i.i.d. random variables with
densities; hence conditions under which LAN in θ0 holds
are well known (see, e.g., Witting, 1985, p.
179; Le Cam and Yang, 1990, Ch. 6.3). The
conditions are that the density is positive m a.s. and continuously
differentiable w.r.t. the unknown parameter θ in a neighborhood
U(θ0). Then we obtain LAN with maximal rate of
convergence
and
which is the inverse of the minimal asymptotic variance, provided
Γ(θ) is finite for all θ ∈
U(θ0) and continuous in θ0.
This result applies as for θ ∈
U(θ0),
pt(x,θ) is continuously
differentiable in θ and
pt(x,θ) > 0
[m] . We obtain LAN with
and
because Γ(θ) is obviously finite for θ ∈
U(θ0) and continuous in θ0.
For the last equation we use the well-known moment representation, by
the derivatives of the characteristic function,
.
(ii) For the sampling scheme with Δ → 0 we need a few more
conditions to ensure the LAN property, which results basically from
conditions needed to perform the CLT and LLN for triangular schemes
(see Woerner, 2001; Gnedenko and Kolmogorov, 1968). The conditions are
as follows. Assume that there exists a neighborhood
U(θ0) of θ0 such that the
density pt(x,θ) w.r.t.
m of the Lévy process is two times continuously
differentiable w.r.t. the unknown parameter θ and regular (i.e.,
)
for all θ ∈ U(θ0). Denote by
θn = θn(x) a
measurable function from
.
Furthermore, assume that for all ε > 0 as n →
∞ and Δ → 0, such that nΔ → ∞,
Then we obtain LAN with
for the sampling scheme with Δ → 0 as n → ∞.
Finding dominating functions to ensure these conditions is
straightforward because of the special simple structure that θ
only enters in the exponential term and Δ only as a polynomial
factor, by applying the moment representation to the integrals w.r.t.
pΔ(x,θ)m(dx).
We look at the details of condition (i); the others can be checked
analogously.
as n → ∞, Δ → 0, and nΔ
→ ∞. █
Example 1 (Normal inverse Gaussian process)
The normal inverse Gaussian process is characterized by the
Lévy triplet
where K1 denotes the modified Bessel function of
third order with index 1 and
.
Because as |x| → 0
the process is of unbounded variation and possesses infinitely many
jumps. Hence it also has a density w.r.t. the Lebesgue measure,
As pointed out in Barndorff-Nielsen (1998)
this process is used both for modeling turbulence, in particular when
the Reynolds number is high, and in finance. This is due to some
special properties of the normal inverse Gaussian process, such as
possible asymmetry modeled by the skewness parameter β, unbounded
variation, and semiheavy tails; namely, as |x|
→ ∞
The parameter β may be estimated according to Theorem 2. Here
assumption (1) is satisfied. Hence for the sampling scheme with Δ
fixed and n → ∞ we have
and
which is indeed the same result as using the density and the results
for i.i.d. random variables.
Example 2 (Gamma process)
The gamma process is characterized by the Lévy triplet
with α,β,x > 0. Hence the process is a
subordinator and only possesses nonnegative jumps. The density w.r.t.
the Lebesgue measure can be calculated, and we can see that the name
reflects the fact that the increments are distributed according to a
gamma function,
Analogously to Example 1 β can be estimated, and we obtain for
the sampling scheme with Δ fixed and
and
which is again the same result as using the density and the i.i.d.
results.
Example 3 (Hyperbolic Lévy motion)
The hyperbolic Lévy motion which was introduced by
Barndorff-Nielsen (1977) for modeling
mass-size distributions of aeolin sand deposits, has also been applied
to some other areas of interest, e.g., turbulence data (cf. Barndorff-Nielsen, 1996) and to financial data (cf.
Eberlein and Keller, 1995; Keller, 1997; Rydberg,
1997; Prause, 1999; Raible, 2000).
The hyperbolic Lévy motion may be characterized by the
Lévy triplet
where J1 denotes the Bessel function of the first
order with index 1 and Y1 the Bessel function of
the second order with index 1. Furthermore, we have α,δ > 0
and 0 ≤ |β| < α. Keller (1997) has established that the density of the
Lévy measure behaves like x−2 at the
origin; hence the process is of unbounded variation. Though the process
possesses a density, we cannot calculate it analytically. Only the
distribution for t = 1 can be written down explicitly; it is
the hyperbolic distribution
where K1 denotes the modified Bessel function of
third order with index 1. We can estimate β analogously to Example
1 and obtain for the sampling scheme with Δ fixed and
and
where
using Keller (1997).
Example 4 (CGMY process)
The CGMY process, named after Carr, Geman, Madan, and Yor, or in
physical literature called truncated stable or truncated Lévy
flight, is given by the Lévy triplet
where C > 0, G ≥ 0, M ≥ 0, and
Y < 2. This class of processes is a flexible model for
index dynamics and also for the dynamics of individual stocks, because
by varying the parameters it allows all features of finite and infinite
activity, bounded and unbounded variation, and also skewness to be
modeled directly by the characteristic function (cf. Carr et al., 2002). The variance gamma process (cf.
Madan and Senata, 1990) is a special case of
the CGMY process. In general the density of the process is not known
explicitly, but as in the previous examples we can infer the parameters
M or G in the special one-sided case, when either
G = ∞ or M = ∞, only by the knowledge of
the Lévy measure. When G = ∞ we obtain
(0,0,C(exp{−M|x|}/|x|1+Y)1x>0)
and for the sampling scheme with Δ fixed and
and
4. EFFICIENT ESTIMATORS
With this explicit result for the minimal asymptotic variance we can
now try to find a sequence of estimators that is efficient. Because the
continuous model is a benchmark for the discretely observed model, we
look at the continuous likelihood function to get an idea of what
discrete estimation functions may look like. In the continuous model
the likelihood function is given by
where Yt denotes the process under the
measure
pt(x)m(dx).
Changing to the process Xt under the
measure
eθxpt(x)m(dx)/(∫eθxpt(x)m(dx))
and discretizing the time, i.e., t = nΔ, leads to
the log-likelihood function
which indeed provides an appropriate estimating function.
We obtain that only the knowledge of the last observation is
important for the estimation procedure. In other words, the last
observation contains all necessary information. This is however not
surprising, because from the theory of exponential families it is well
known that Yt or
XnΔ, respectively, is sufficient for
{Ptθ,θ ∈
Θ}.
THEOREM 3 (Optimal estimators).
Proof.
(i) Using X0 = 0 and
nΔEθ X1 =
Eθ XnΔ, we
can rewrite
This yields asymptotic normality by inserting
(ii) This is analogous to using the CLT for triangular schemes
as in Theorem 2 for the convergence in distribution and calculating the
explicit form of
as in (i). █
With Theorem 3 equation (3) now provides an easily computable
estimating function that indeed leads to efficient estimators. It is
especially simple because it only involves the observations and the
first moment. Depending on the form of the first moment the equation
can sometimes even be solved analytically.
Example 5 (Gamma process)
For the gamma process equation (3) is
Hence we obtain
for the sequence of efficient estimators as n → ∞ for
the sampling scheme with fixed Δ.
Equation (4) can also be viewed as the simplest form of a martingale
estimating function, having the general form
where Yi =
XiΔ −
X(i−1)Δ i.i.d. distributed
according to pΔ(.,θ). We can now show that
our sequence of estimators is not only optimal in the sense of local
asymptotic statistics but also in the sense of Godambe and Heyde (1987), which is the classical optimality concept
for martingale estimating functions.
Let us first give a short review on the concept of martingale
estimating functions and the definition of optimality by Godambe and
Heyde (1987).
The basic problem is that we want to draw inference for discretely
observed stochastic processes when the likelihood function is unknown.
Because in general the maximum likelihood estimator performs quite
well, the idea is to approximate the unknown score function to obtain
an approximate maximum likelihood estimator. Using an approximation,
the problem might occur that we perhaps do not have mean zero and hence
eventually will get biased estimates, especially when the distance of
observations Δ is bounded away from zero. A solution is to
approximate the score function by a zero mean martingale w.r.t. the
filtration generated by the observations. This implies that we obtain
consistent and asymptotically normal estimators.
The optimality concept of Godambe and Heyde (1987) and Heyde (1988)
formalizes the heuristics that the optimal element in a given class of
martingale estimating functions is the one whose
L2-distance to the true score function is minimal,
or in the partial order of nonnegative matrices the distance to the
score function is minimal.
DEFINITION (OA-optimality).
Let
where Gn denotes the
compensator of
and 〈G,GT〉 the
quadratic characteristic of Gn. Then
is called OA-optimal in
if and only if
As a result of the special structure of i.i.d. increments, all
considerations simplify greatly for our model compared to general
stochastic processes. By straightforward calculations we obtain
f*(x) = x for the optimal G*. Hence
we also have optimality in the sense of Godambe and Heyde (1987) for the estimators in Theorem 3.
5. CONCLUSION
We derived local asymptotic normality for the skewness parameter of
Lévy processes that are observed at discrete time points only.
This provides an efficiency criterion in terms of the maximal rate of
convergence and the minimal asymptotic variance for a sequence of
estimators. Furthermore, we obtained easily computable estimating
functions that lead to efficient estimators both in the sense of
asymptotic statistics and in the sense of Godambe and Heyde (1987) for martingale estimating functions. Hence
our results enable us to optimally infer the skewness parameter from
discrete observations for the popular Lévy process models, such
as generalized hyperbolic, normal inverse Gaussian, and CGMY, even when
other unknown parameters are involved.
