# Subordination, Self-Similarity, and Option Pricing

Authors: Mack L. Galloway; Craig A. Nolder
Journal: Journal of Applied Mathematics and Decision Sciences
 (2008)

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## Abstract

We use additive processes to price options on the Standard and Poor's
500 index (SPX) for the sake of comparison of pricing performance across
both model class and family of time-one distribution. Each of the additive processes in this study is defined using one of the following: subordination,
Sato's (2002) construction of self-similar additive processes from
self-decomposable distributions, or both. We find that during the year
2005: (1) for a given family of time-one distributions, four-parameter
self-similar additive models consistently yielded lower pricing errors than
those of four-parameter subordinated, and time-inhomogeneous additive
models, (2) for a given class of additive models, the time-one marginal
given by the normal inverse Gaussian distribution consistently yielded
lower pricing errors than those of the variance gamma distribution. Market
and model benchmarks for the additive models under consideration
are obtained via the bid-ask spreads of the options and Lévy stochastic
volatility model prices, respectively.valued generalizedw


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## Body

## 1. Introduction

As noted by Carr and Wu [1], Cont and Tankov [2, page 453], and others, Lévy
models are incapable of adequately fitting implied volatility surfaces of
equity options across both strike and maturity. Although stochastic volatility
models with jumps have success in pricing across both strike and maturity
[3], [2, page 470], [4, page 98], the use of such
models typically incurs the cost of computing with a significantly larger
number of parameters than in the Lévy case. By using certain additive models
with nonstationary increments, one obtains flexibility in pricing across both
strike and maturity, at the cost of one more parameter than a Lévy model of
like time-one distribution.One class of time-inhomogeneous additive processes
which has recently received much attention is the class of theH-self-similar
additive processes of Sato [5, Theorem 16.1(ii), page 99]. In 1998
Barndorff-Nielsen [6]
used Sato's results to construct a self-similar, time-inhomogeneous additive
process with marginal normal inverse Gaussian distributions. The financial
application of self-similar additive processes determined from
self-decomposable distributions was later recommended by Nolder [7]. In 2007, Carr,
Geman, Madan, and Yor (CGMY) [8] introduced the class of exponential H-self-similar additive models and showed
that these could accurately price SPX and equity options across both strike and
maturity with as few as four parameters. In their work, exponential
self-similar additive models were constructed and calibrated using option data
on several underlying assets. In other words, the self-similar additive model
class was fixed, and the in-sample model pricing error was observed with
respect to both the choice of the family of time-one distribution and the
choice of underlying asset. In contrast, our study fixes the underlying asset
(SPX) and observes pricing errors with respect to both the choice of the family
of time-one distribution and choice of the additive model class.The additive model classes used in this study
correspond to the Lévy processes,H-self-similar additive processes, and those
in which a Brownian motion is time-changed by an independent, increasing
H-self-similar additive process with time-one self-decomposable distribution.
For each of the three additive classes, two processes are constructed; one in
which the time-one marginal is a variance gamma distribution, and the other, a
normal inverse Gaussian distribution. The names for each of the additive
processes to be compared are given in Table 1.Table 1Process class∖time-one marginalVariance gammaNormal inverse GaussianLévyVGNIGH-self-similar additiveVGSSDNIGSSDSubordinated additiveVHGNHIGThe time-inhomogeneous additive processes discussed
here require only one more parameter than a Lévy model of like time-one
distribution. This additional parameter is known as the Hurst exponent,H. In
order that this research may be viewed in historical context, a brief history
of option pricing with additive processes now follows.


### 1.1. Brief History of Additive Processes in Option Pricing

The use of additive processes to price options dates
back to 1900 when Louis Bachelier, a student of Henri Poincaré, began the
development of the theory of Brownian motion [9]. In his dissertation, Théorie
de la Spéculation, changes in the price of a stock were modeled with what
is now known as a zero mean, Brownian motion [10], a 1/2-self-similar additive process. Mandelbrot's
exponential stable-Paretian processes [10] and Samuelson's “economic” (geometric) Brownian
motion [11] overcame
the problem of negative stock prices associated with a Brownian motion model.
The latter of the two processes was found extremely useful to the theory of
option pricing developed by Sprenkle [12], Samuelson [11], and others. Finally, in 1973 the famous closed-form
solution for the price of a European call option was introduced by Black and Scholes
[13]. In the same year
Merton [14] presented
another derivation of the Black-Scholes formula, using weaker assumptions than
in [13], and extended
the theory to include the case of stochastic interest rates.Not long after the introduction of the Black-Scholes
model, Merton [15], and later, Rubinstein [16], Bates [17], and others documented
inconsistencies of this model with market option prices. Merton [15] introduced the
jump-diffusion model in which the logarithm of the return on the underlying
asset follows a Lévy process whose one-dimensional marginals are given by the
convolution of a Normal distribution with a compound Poisson distribution.
Madan and Seneta [18] (resp., Madan et al. [19]) introduced the two (resp.,
three) parameter subordinated Lévy process, known as the variance gamma
process, in which a Brownian motion without (resp., with) drift is time changed
by an independent gamma process. Barndorff-Nielsen [6] introduced the three-parameter
normal inverse Gaussian process in which a Brownian motion is time changed by
an independent inverse Gaussian process. Other well-known Lévy processes
include the following: Meixner [20], KoBoL [21, 22] or equivalently, CGMY [23], hyperbolic [24, 25], generalized hyperbolic [25, 26], and nonparametric Lévy of
Cont and Tankov [27].Although Lévy processes with jumps could be used to
accurately price options across strike price for a given maturity, they were
inadequate to do so across both strike and maturity [1] and [2, page 453]. One answer to this
problem came in 2003 when CGMY [3] showed that one may subordinate a Lévy process by the
time integral of a mean-reverting, positive process in order to obtain a model
which was quite flexible across both strike and maturity. Such models have been
termed “Lévy stochastic volatility” models. Another answer to this problem came
with the simpler Finite Moment Log Stable (FMLS) model of Carr and Wu [1]. In this model the
logarithm of the return on the underlying followed a stable Paretian process in
which the skewness parameter, β,
was set to an extreme value of −1.
With β fixed and the drift term set so that the
martingale condition was satisfied, the FMLS model was able to fit relatively
well, with two free parameters, the prices observed in SPX option data from
April 1999 to May 2000. In an effort to find models which were more flexible
than the FMLS model and more parsimonious than the
stochastic volatility models, researchers pursued the development of
time-inhomogeneous additive processes.In 2003 Nolder [7] proposed the modeling of asset returns with the
H-self-similar additive processes of Sato. Carr et al. [8] constructed
exponential H-self-similar additive models and showed that they could
successfully price equity options of various underlyings across both maturity
and strike. Madan and Eberlein [28]
demonstrated that H-self-similar additive models could be useful for pricing
options on annualized realized variance and annualized realized volatility.
Such suitability is attributed to the dependence of the variance of realized
quadratic variation to time t (per unit time) on the Hurst exponent.


### 1.2. Purpose and Organization

The purpose of this paper is to compare the option
pricing performance of certain additive models with respect to the choice of
time-one distribution (VG or NIG) and with respect to the choice of additive
model class (Lévy, self-similar, and subordinated additive). We use SPX option
data from the year 2005 to calibrate the exponential additive models
corresponding to the VG, VGSSD, VHG, NIG, NIGSSD, and NHIG processes, along
with the benchmark exponential VG-OU-Γ and NIG-OU-Γ Lévy stochastic volatility models. The latter
are constructed by using the time-integral of a process of O-U type (with
associated stationary gamma distribution) to time change an independent VG or
NIG process.The following describe the organization of this paper.
In Section2 the six additive models used in this study are defined. The
moments of the increments corresponding to the variance gamma and normal
inverse Gaussian families of processes are given in Section 3. Also included is
a discussion of the dependence of increment variance on the Hurst exponent for
both time-inhomogeneous additive classes. In Section 4 the characteristic
functions of the exponential additive models are given, along with those of the
Lévy stochastic volatility models. Algorithmic and data specifications for the
calibration procedure are discussed in Section 5. The calibration results are
presented in Section 6. The conclusion is presented
in Section 7.


## 1.1. Brief History of Additive Processes in Option Pricing

The use of additive processes to price options dates
back to 1900 when Louis Bachelier, a student of Henri Poincaré, began the
development of the theory of Brownian motion [9]. In his dissertation, Théorie
de la Spéculation, changes in the price of a stock were modeled with what
is now known as a zero mean, Brownian motion [10], a 1/2-self-similar additive process. Mandelbrot's
exponential stable-Paretian processes [10] and Samuelson's “economic” (geometric) Brownian
motion [11] overcame
the problem of negative stock prices associated with a Brownian motion model.
The latter of the two processes was found extremely useful to the theory of
option pricing developed by Sprenkle [12], Samuelson [11], and others. Finally, in 1973 the famous closed-form
solution for the price of a European call option was introduced by Black and Scholes
[13]. In the same year
Merton [14] presented
another derivation of the Black-Scholes formula, using weaker assumptions than
in [13], and extended
the theory to include the case of stochastic interest rates.Not long after the introduction of the Black-Scholes
model, Merton [15], and later, Rubinstein [16], Bates [17], and others documented
inconsistencies of this model with market option prices. Merton [15] introduced the
jump-diffusion model in which the logarithm of the return on the underlying
asset follows a Lévy process whose one-dimensional marginals are given by the
convolution of a Normal distribution with a compound Poisson distribution.
Madan and Seneta [18] (resp., Madan et al. [19]) introduced the two (resp.,
three) parameter subordinated Lévy process, known as the variance gamma
process, in which a Brownian motion without (resp., with) drift is time changed
by an independent gamma process. Barndorff-Nielsen [6] introduced the three-parameter
normal inverse Gaussian process in which a Brownian motion is time changed by
an independent inverse Gaussian process. Other well-known Lévy processes
include the following: Meixner [20], KoBoL [21, 22] or equivalently, CGMY [23], hyperbolic [24, 25], generalized hyperbolic [25, 26], and nonparametric Lévy of
Cont and Tankov [27].Although Lévy processes with jumps could be used to
accurately price options across strike price for a given maturity, they were
inadequate to do so across both strike and maturity [1] and [2, page 453]. One answer to this
problem came in 2003 when CGMY [3] showed that one may subordinate a Lévy process by the
time integral of a mean-reverting, positive process in order to obtain a model
which was quite flexible across both strike and maturity. Such models have been
termed “Lévy stochastic volatility” models. Another answer to this problem came
with the simpler Finite Moment Log Stable (FMLS) model of Carr and Wu [1]. In this model the
logarithm of the return on the underlying followed a stable Paretian process in
which the skewness parameter, β,
was set to an extreme value of −1.
With β fixed and the drift term set so that the
martingale condition was satisfied, the FMLS model was able to fit relatively
well, with two free parameters, the prices observed in SPX option data from
April 1999 to May 2000. In an effort to find models which were more flexible
than the FMLS model and more parsimonious than the
stochastic volatility models, researchers pursued the development of
time-inhomogeneous additive processes.In 2003 Nolder [7] proposed the modeling of asset returns with the
H-self-similar additive processes of Sato. Carr et al. [8] constructed
exponential H-self-similar additive models and showed that they could
successfully price equity options of various underlyings across both maturity
and strike. Madan and Eberlein [28]
demonstrated that H-self-similar additive models could be useful for pricing
options on annualized realized variance and annualized realized volatility.
Such suitability is attributed to the dependence of the variance of realized
quadratic variation to time t (per unit time) on the Hurst exponent.


## 1.2. Purpose and Organization

The purpose of this paper is to compare the option
pricing performance of certain additive models with respect to the choice of
time-one distribution (VG or NIG) and with respect to the choice of additive
model class (Lévy, self-similar, and subordinated additive). We use SPX option
data from the year 2005 to calibrate the exponential additive models
corresponding to the VG, VGSSD, VHG, NIG, NIGSSD, and NHIG processes, along
with the benchmark exponential VG-OU-Γ and NIG-OU-Γ Lévy stochastic volatility models. The latter
are constructed by using the time-integral of a process of O-U type (with
associated stationary gamma distribution) to time change an independent VG or
NIG process.The following describe the organization of this paper.
In Section2 the six additive models used in this study are defined. The
moments of the increments corresponding to the variance gamma and normal
inverse Gaussian families of processes are given in Section 3. Also included is
a discussion of the dependence of increment variance on the Hurst exponent for
both time-inhomogeneous additive classes. In Section 4 the characteristic
functions of the exponential additive models are given, along with those of the
Lévy stochastic volatility models. Algorithmic and data specifications for the
calibration procedure are discussed in Section 5. The calibration results are
presented in Section 6. The conclusion is presented
in Section 7.


## 2. Construction of additive processes

We first begin with a few definitions and facts
associated with self-similar additive processes.Definition 2.1.
Anℝd-valued stochastic process {Xt:t≥0} on (Ω,ℱ,ℙ) is an additive process if(i)
X0=0 a.s.(ii)
For any choice ofn∈ℕ and 0≤t0<t1<⋯<tn,
the random variables Xt0,Xt1−Xt0,Xt2−Xt1,…,Xtn−Xtn−1 are independent.(iii)
{Xt} is stochastically continuous.(iv)
There exists aΩ0∈ℱ with ℙ(Ω0)=1 such that, for every ω∈Ω0,Xt(ω) is right-continuous in t≥0 and has left limits in t>0.Remark 2.2.
A Lévy process is a time-homogeneous additive process—an additive
process which satisfies(2.1)Xt+δ−Xt=dXδ∀t,δ≥0.Definition 2.3.
A stochastic process{Xt:t≥0} is self-similar if, for any a>0,
there exists b>0,
such that {Xat:t≥0} and {bXt:t≥0} are identical in law.Furthermore, if{Xt:t≥0} is an ℝd-valued, nontrivial, self-similar process
which is stochastically continuous and X0= constant a.s., then there exists H>0 such that b=aH [5, Theorem 13.11, page 73].Definition 2.4.
Ford≥1 a probability measure μ on ℝd is self-decomposable if, for all b>1,
there exists a probability measure ρb on ℝd such that (2.2)μ^(z)=μ^(zb)ρb^(z), where μ^(z) is the characteristic function of μ.Since a self-decomposable distributionμ is infinitely divisible, it generates a Lévy
process {Xt:t≥0}.
In this case, the characteristic function of the distribution of Xt is μ^t(z) for all t≥0 [5, Lemma 7.10(ii), page 35]. The following theorem shows
that a self-decomposable distribution also generates a self-similar additive
process.Theorem 2.5 (Sato [5, Theorem 16.1(ii), page 99]).
Ifμ is a nontrivial self-decomposable distribution
on ℝd,
then, for any H>0,
there exists, uniquely in law, a nontrivial H-self-similar additive process {Xt:t≥0} such that ℙX1=μ.Remark 2.6 (Sato [5, Proof of Theorem 16.1, page 100]).
In the previous theorem, the characteristic function ofXt is given by (2.3)ΦXt(ξ)=μ^(tHξ),t≥0,ξ∈ℝ.The following representation theorem is useful for the
purpose of checking self-decomposability of a distribution.Theorem 2.7 (Sato [5, Corollary 15.11, page 95]).
A probability measureμ on ℝ is self-decomposable if and only if (2.4)μ^(z)=exp(−12Az2+iγz+∫−∞∞(eizx−1−izx1[−1,1](x))k(x)|x|dx), where A≥0,γ∈ℝ,k(x)≥0,∫−∞∞(|x|2⋀1)(k(x)/|x|)dx<∞,
and k(x) is increasing on (−∞,0) and decreasing on (0,∞).The following terminology will be used in the
application of this theorem.Definition 2.8.
Ak-function of the self-decomposable
distribution μ [5, page 403] is an ℝ+¯-valued function on ℝ∖{0} which is decreasing and left-continuous on ℝ+ and which is increasing and right-continuous
on ℝ− with the Lévy density of μ given by k(x)/|x|,x∈ℝ∖{0}.The notation below will be used in the theorems and
derivations which follow.Notation 2.9.
Given anℝ-valued process X≡{Xt:t≥0},
we define the following: (2.5)Characteristic  function:ΦXt(ξ)=E[eiξXt],ξ∈ℝCharacteristic  exponent:ηXt(ξ)=log(ΦXt(ξ)),ξ∈ℝ.
IfX is ℝ+¯-valued, then we also have (2.6)Laplace  exponent:ΨXt(ξ)=log(E[eξXt]),ℜ(ξ)≤0.Remark 2.10.
Given a Lévy processX and an independent, increasing additive
process Z,
the characteristic function of the time-t marginal of X time changed by Z can easily be computed by conditioning on Zt,
yielding (2.7)E[eiξ(XZt)]=exp(ΨZt(ηX1(ξ))),ξ∈ℝ. This computation is done in
[2, page 109], where
the directing process Z is assumed to be time homogeneous. However,
the derivation of (2.7) does not use the property of time homogeneity of Z.In the remainder of this section, three sets of
additive processes will be defined. For each set, one process will have a
time-one marginal which is variance gamma distributed. The other process will
have a time-one marginal which is normal inverse Gaussian distributed.


### 2.1. Lévy Processes

The three-parameter variance gamma process of Madan et al.  [19]
can be defined as a Brownian motion with drift which is time changed by an
independent gamma process. Let σ,ν>0 and μ∈ℝ.
On the probability space (Ω,ℱ,ℙ) let X≡{Xt:t≥0} be an ℝ-valued Brownian motion, where X1 is normal(μ,σ2) distributed. On the same probability space let Z≡{Zt:t≥0} be an ℝ+¯-valued, independent gamma process where, for
any t≥0, Zt is gamma(t/ν,1/ν) distributed. The variance gamma process, {VGt},
can be constructed as (2.8)VGt(ω)=XZt(ω)(ω),∀t≥0,ω∈Ω,which characterizes the
distributions of a variance gamma process. Equation (2.7) of Remark 2.10, Remark 2.2,
and the Laplace exponent of Zt, (2.9)ΨZt[1/ν,1/ν](ξ)=log[(1−ξν)−t/ν],fort≥0,ℜ(ξ)≤0 are used to obtain the
characteristic function of the random variable VGt as follows: (2.10)E[eiξ(VGt)]=exp(ΨZt(ηX1(ξ)))=exp(log[(1−ηX1(ξ)ν)−t/ν])=(1+ν[−iμξ+12σ2ξ2])−t/ν.The normal inverse Gaussian process of
Barndorff-Nielsen [6]
can be defined as a Brownian motion with drift which is time changed by an
independent inverse Gaussian process. Fix δ,γ>0,
and
β∈ℝ.
On the probability space (Ω,ℱ,ℙ) let X≡{Xt:t≥0} be an ℝ-valued Brownian motion, where X1 is normal(βδ2,δ2) distributed. On the same probability space let Z≡{Zt:t≥0} be an ℝ+¯-valued, independent inverse Gaussian process
where, for any t≥0, Zt is IG(t,δγ) distributed. The normal inverse Gaussian
process, {NIGt},
can be constructed as (2.11)NIGt(ω)=XZt(ω)(ω),∀t≥0,ω∈Ω. Since the Laplace exponent of Zt is given by (2.12)ΨZt[1,δγ](ξ)=−t{−2ξ+(δγ)2−δγ},fort≥0,ℜ(ξ)≤0, we have, similar to the previous
case, the characteristic function of the distribution of NIGt: (2.13)E[eiξ(NIGt)]=exp(ΨZt(ηX1(ξ)))=exp(−t{−2ηX1(ξ)+(δγ)2−δγ})=exp(−δt{ξ2−2iβξ+γ2−γ}).


### 2.2.H-Self-Similar Additive Processes

#### 2.2.1. Self-Decomposability of Time-One Distributions

H-self-similar additive processes are constructed
using Remark 2.6. Consequently, self-decomposability of the one-dimensional
marginal distribution (at time one) must be verified for the variance gamma and
normal inverse Gaussian processes.Variance Gamma Case
As mentioned in the construction of the variance gamma
process in Section2.1, X1 is Normal(μ,σ2) distributed while Zt is gamma(t/ν,1/ν) distributed for all t≥0. Let ϱZ1 denote the Lévy density of the distribution of Z1 (see Table 2). By the subordination theorem for Lévy processes [5, Theorem 30.1, page 197], the
Lévy density of the distribution of the time-one variance gamma random
variable, VG1,
is given by (2.14)ϱVG1(x)=∫(0,∞)dμXsdx(x)ϱZ1(s)ds=∫(0,∞)12πσ2sexp(−(x−μs)22σ2s)1νsexp(−sν)ds=12πσνexp(xμσ2)[2(μ2/σ2+2/ν|x|σ)1/2K1/2(|x|σμ2σ2+2ν)]=1ν|x|exp(−1σ[μ2σ2+2ν|x|−μσx]), where the representation of the
modified Bessel function of the second kind of order ν˜ in (2.15) [2, page 500] is used in the third step with ν˜=1/2 and α,β>0. (2.15)∫(0,∞)exp(−α2s2−β22s)s−(1+ν˜)ds=2(αβ)ν˜Kν˜(αβ). Identity (2.16) [29, page 109] is used in the final
step. (2.16)K1/2(z)=π2ze−z,z>0. The product given by x↦|x|ϱVG1(x) is positive, decreasing on ℝ+ and increasing on ℝ− if ν>0. By Theorem 2.7, the k-function given
by (2.17)k(x)=1νexp(−1σ[μ2σ2+2ν|x|−μσx]),x∈ℝ∖{0} verifies self-decomposability of
the time-one distribution of the variance gamma process, provided ν>0. For an alternative proof of
self-decomposability using the (C,G,M) parameterization of the variance gamma process,
see CGMY [8].Table 2
Lévy
densities of subordinator distributions.
DistributionLévy DensityGamma(a,b)ϱ(x)=aexp(−bx)x−1,x>0Inverse Gaussian(a,b)ϱ(x)=12πax−3/2exp(−12b2x),x>0Normal Inverse Gaussian Case
Using the same construction as in Section2.1 for the
development of the NIG process, X1 is normal(βδ2,δ2) distributed while Zt is IG(t,δγ) distributed. Denoting the Lévy density of the
distribution of Z1 by ϱZ1 (Table 2), the Lévy density of the
distribution of the time-one normal inverse Gaussian random variable, NIG1,
is given by (2.18)ϱNIG1(x)=∫(0,∞)dμXsdx(x)ϱZ1(s)ds=∫(0,∞)12πδ2sexp(−(x−βδ2s)22δ2s)12πs−3/2exp(−12δ2γ2s)ds=12πδexp(βx)∫(0,∞)exp(−[x2δ22s+(β2+γ2)δ22s])s−2ds. Using (2.15) with ν˜=1 yields (2.19)ϱNIG1(x)=12πδexp(βx)[2(δβ2+γ2|x|/δ)K1(δβ2+γ2|x|δ)]=δβ2+γ2π1|x|exp(βx)K1(β2+γ2|x|),x∈ℝ∖{0}. Since x↦|x|ϱNIG1(x) is decreasing on ℝ+ and increasing on ℝ− if and only if x↦e(β/β2+γ2)xK1(|x|) is decreasing on ℝ+ and increasing on ℝ−,
we make use of identity (2.20) [30, equation (9.6.24)]: (2.20)Kν˜(t)=∫(0,∞)e−tcosh(z)cosh(ν˜z)dz,ℜ(t)>0. We have, in a manner similar to
Laforgia [31], (2.21)e(β/β2+γ2)xK1(|x|)=∫(0,∞)e−|x|cosh(z)+x(β/β2+γ2)cosh(z)dz=∫(0,∞)e−|x|(cosh(z)−signum(x)(β/β2+γ2))cosh(z)dz,x∈ℝ∖{0}. It follows that the product of ϱNIG1 and |x| is positive, decreasing on ℝ+ and increasing on ℝ− for all δ>0,β∈ℝ.
By Theorem 2.7, the k-function given by (2.22)k(x)=δβ2+γ2πexp(βx)K1(β2+γ2|x|),x∈ℝ∖{0} verifies self-decomposability of
the time-one distribution of the normal inverse Gaussian process. For an
alternative proof of self-decomposability with this parameterization see
Barndorff-Nielsen [6]. CGMY [8] address self-decomposability using a different
parameterization. Sato's construction of H-self-similar additive processes from
self-decomposable distributions (Remark 2.6) may now be applied to the variance
gamma and normal inverse Gaussian cases.


#### 2.2.2. Construction ofH-Self-Similar Additive Processes

Using the parameterizations for this study, we
reproduce the results of CGMY [8] and Barndorff-Nielsen [6] (NIG case only) for the
characteristic functions of the time-t marginal distributions of the
self-similar additive analogs of the variance gamma and normal inverse Gaussian
processes (denoted in [8] as “VGSSD” and “NIGSSD,” resp.). Let σ,ν,H>0, μ∈ℝ, and m=ℙVG1 be the self-decomposable variance gamma(σ,μ,ν) distribution on ℝ.
By Theorem 2.5 and Remark 2.6, the characteristic function of VGSSDt is given by (2.23)E[eiξ(VGSSDt)]=m^(tHξ;σ,μ,ν)=(1+ν[−iμtHξ+12σ2t2Hξ2])−1/ν.For the characteristic function of the distribution ofNIGSSDt,
let δ,γ,H>0,β∈ℝ, and m=ℙNIG1 be the self-decomposable normal inverse
Gaussian(δ,β,γ) distribution on ℝ.
Similar to the previous case, the characteristic function of the distribution
of NIGSSDt is given by (2.24)E[eiξ(NIGSSDt)]=m^(tHξ;β,δ,γ)=exp(−δ{t2Hξ2−2iβtHξ+γ2−γ}).  By construction, the
distributions given by (2.10) and (2.23) agree at time one. The same is also true for
(2.13) and (2.24).


### 2.3. Subordinated Additive Processes

The subordinated, time-inhomogeneous additive
processes in this study are a special case of the class of subordinated
processes discussed by Madan and Yor [32]. Such processes are defined by time-changing a
Brownian motion by an independent, increasing Markov process with independent
and time-inhomogeneous increments. The first process to be constructed is a
Brownian motion time changed by an independent H-self-similar additive gamma process (VHG),
while the second is a Brownian motion time changed by an independent H-self-similar additive inverse Gaussian
process (NHIG). In order to construct the self-similar additive directing
processes, self-decomposability of the gamma and inverse Gaussian distributions
must first be verified. The Lévy densities for these distributions are given in
Table 2 where a,b>0 for both distributions [4]. By Theorem 2.7, the k-functions given by the following equations:
(2.25)kΓ(x)=aexp(−bx),x>0,kIG(x)=12πax−1/2exp(−12b2x),x>0 verify self-decomposability of
the time-one distributions of the gamma and inverse Gaussian processes,
respectively.The characteristic function of the distribution ofVHGt is determined as follows. Let σ,ν,H>0,
and μ∈ℝ.
On the probability space (Ω,ℱ,ℙ) let X≡{Xt:t≥0} be an ℝ-valued Brownian motion where X1 is normal(μ,σ2) distributed. On the same probability space,
using Theorem 2.5, define Z≡{Zt:t≥0} to be an independent, increasing H-self-similar additive process such that Z1 is gamma(1/ν,1/ν)  distributed. The subordinated,
time-inhomogeneous additive analog of the VG process, {VHGt},
can be constructed as (2.26)VHGt(ω)=XZt(ω)(ω),∀t≥0,ω∈Ω. Using (2.7) in Remark 2.10, Remark 2.6,
and the Laplace exponent of Z1, (2.27)ΨZ1[1/ν,1/ν](ξ)=log[(1−ξν)−1/ν],ℜ(ξ)≤0, the characteristic function of
the distribution of the random variable VHGt is given by (2.28)E[eiξ(VHGt)]=exp(ΨZt(ηX1(ξ)))=exp(ΨZ1(tHηX1(ξ)))=exp(log[(1−tH[−12σ2ξ2+iμξ]ν)−1/ν])=(1+tHν[−iμξ+12σ2ξ2])−1/ν. Consequently, the distributions
given by (2.10), (2.23), and (2.28) all agree at time one.The characteristic function of the distribution ofNHIGt is similarly defined. Fix δ,γ,H>0,
and β∈ℝ.
On the probability space (Ω,ℱ,ℙ) define X≡{Xt:t≥0} to be an ℝ-valued Brownian motion where X1 is normal(βδ2,δ2) distributed. On the same probability space,
using Theorem 2.5, define Z≡{Zt:t≥0} to be an independent, increasing H-self-similar additive process such that Z1 is IG(1,δγ) distributed. The subordinated,
time-inhomogeneous additive analog of the NIG process, {NHIGt},
can be constructed as (2.29)NHIGt(ω)=XZt(ω)(ω),∀t≥0,ω∈Ω. The Laplace exponent of Z1 is given by (2.30)ΨZ1[1,δγ](ξ)=−{−2ξ+(δγ)2−δγ},ℜ(ξ)≤0. Similar to the previous case, the
characteristic function of the distribution of NHIGt is obtained as follows: (2.31)E[eiξ(NHIGt)]=exp(ΨZt(ηX1(ξ)))=exp(ΨZ1(tHηX1(ξ)))=exp(−{−2tHηX1(ξ)+(δγ)2−δγ})=exp(−δ{tH[ξ2−2iβξ]+γ2−γ}). Note that (2.31) agrees at time one
with (2.13) and (2.24).


## 2.1. Lévy Processes

The three-parameter variance gamma process of Madan et al.  [19]
can be defined as a Brownian motion with drift which is time changed by an
independent gamma process. Let σ,ν>0 and μ∈ℝ.
On the probability space (Ω,ℱ,ℙ) let X≡{Xt:t≥0} be an ℝ-valued Brownian motion, where X1 is normal(μ,σ2) distributed. On the same probability space let Z≡{Zt:t≥0} be an ℝ+¯-valued, independent gamma process where, for
any t≥0, Zt is gamma(t/ν,1/ν) distributed. The variance gamma process, {VGt},
can be constructed as (2.8)VGt(ω)=XZt(ω)(ω),∀t≥0,ω∈Ω,which characterizes the
distributions of a variance gamma process. Equation (2.7) of Remark 2.10, Remark 2.2,
and the Laplace exponent of Zt, (2.9)ΨZt[1/ν,1/ν](ξ)=log[(1−ξν)−t/ν],fort≥0,ℜ(ξ)≤0 are used to obtain the
characteristic function of the random variable VGt as follows: (2.10)E[eiξ(VGt)]=exp(ΨZt(ηX1(ξ)))=exp(log[(1−ηX1(ξ)ν)−t/ν])=(1+ν[−iμξ+12σ2ξ2])−t/ν.The normal inverse Gaussian process of
Barndorff-Nielsen [6]
can be defined as a Brownian motion with drift which is time changed by an
independent inverse Gaussian process. Fix δ,γ>0,
and
β∈ℝ.
On the probability space (Ω,ℱ,ℙ) let X≡{Xt:t≥0} be an ℝ-valued Brownian motion, where X1 is normal(βδ2,δ2) distributed. On the same probability space let Z≡{Zt:t≥0} be an ℝ+¯-valued, independent inverse Gaussian process
where, for any t≥0, Zt is IG(t,δγ) distributed. The normal inverse Gaussian
process, {NIGt},
can be constructed as (2.11)NIGt(ω)=XZt(ω)(ω),∀t≥0,ω∈Ω. Since the Laplace exponent of Zt is given by (2.12)ΨZt[1,δγ](ξ)=−t{−2ξ+(δγ)2−δγ},fort≥0,ℜ(ξ)≤0, we have, similar to the previous
case, the characteristic function of the distribution of NIGt: (2.13)E[eiξ(NIGt)]=exp(ΨZt(ηX1(ξ)))=exp(−t{−2ηX1(ξ)+(δγ)2−δγ})=exp(−δt{ξ2−2iβξ+γ2−γ}).


## 2.2.H-Self-Similar Additive Processes

### 2.2.1. Self-Decomposability of Time-One Distributions

H-self-similar additive processes are constructed
using Remark 2.6. Consequently, self-decomposability of the one-dimensional
marginal distribution (at time one) must be verified for the variance gamma and
normal inverse Gaussian processes.Variance Gamma Case
As mentioned in the construction of the variance gamma
process in Section2.1, X1 is Normal(μ,σ2) distributed while Zt is gamma(t/ν,1/ν) distributed for all t≥0. Let ϱZ1 denote the Lévy density of the distribution of Z1 (see Table 2). By the subordination theorem for Lévy processes [5, Theorem 30.1, page 197], the
Lévy density of the distribution of the time-one variance gamma random
variable, VG1,
is given by (2.14)ϱVG1(x)=∫(0,∞)dμXsdx(x)ϱZ1(s)ds=∫(0,∞)12πσ2sexp(−(x−μs)22σ2s)1νsexp(−sν)ds=12πσνexp(xμσ2)[2(μ2/σ2+2/ν|x|σ)1/2K1/2(|x|σμ2σ2+2ν)]=1ν|x|exp(−1σ[μ2σ2+2ν|x|−μσx]), where the representation of the
modified Bessel function of the second kind of order ν˜ in (2.15) [2, page 500] is used in the third step with ν˜=1/2 and α,β>0. (2.15)∫(0,∞)exp(−α2s2−β22s)s−(1+ν˜)ds=2(αβ)ν˜Kν˜(αβ). Identity (2.16) [29, page 109] is used in the final
step. (2.16)K1/2(z)=π2ze−z,z>0. The product given by x↦|x|ϱVG1(x) is positive, decreasing on ℝ+ and increasing on ℝ− if ν>0. By Theorem 2.7, the k-function given
by (2.17)k(x)=1νexp(−1σ[μ2σ2+2ν|x|−μσx]),x∈ℝ∖{0} verifies self-decomposability of
the time-one distribution of the variance gamma process, provided ν>0. For an alternative proof of
self-decomposability using the (C,G,M) parameterization of the variance gamma process,
see CGMY [8].Table 2
Lévy
densities of subordinator distributions.
DistributionLévy DensityGamma(a,b)ϱ(x)=aexp(−bx)x−1,x>0Inverse Gaussian(a,b)ϱ(x)=12πax−3/2exp(−12b2x),x>0Normal Inverse Gaussian Case
Using the same construction as in Section2.1 for the
development of the NIG process, X1 is normal(βδ2,δ2) distributed while Zt is IG(t,δγ) distributed. Denoting the Lévy density of the
distribution of Z1 by ϱZ1 (Table 2), the Lévy density of the
distribution of the time-one normal inverse Gaussian random variable, NIG1,
is given by (2.18)ϱNIG1(x)=∫(0,∞)dμXsdx(x)ϱZ1(s)ds=∫(0,∞)12πδ2sexp(−(x−βδ2s)22δ2s)12πs−3/2exp(−12δ2γ2s)ds=12πδexp(βx)∫(0,∞)exp(−[x2δ22s+(β2+γ2)δ22s])s−2ds. Using (2.15) with ν˜=1 yields (2.19)ϱNIG1(x)=12πδexp(βx)[2(δβ2+γ2|x|/δ)K1(δβ2+γ2|x|δ)]=δβ2+γ2π1|x|exp(βx)K1(β2+γ2|x|),x∈ℝ∖{0}. Since x↦|x|ϱNIG1(x) is decreasing on ℝ+ and increasing on ℝ− if and only if x↦e(β/β2+γ2)xK1(|x|) is decreasing on ℝ+ and increasing on ℝ−,
we make use of identity (2.20) [30, equation (9.6.24)]: (2.20)Kν˜(t)=∫(0,∞)e−tcosh(z)cosh(ν˜z)dz,ℜ(t)>0. We have, in a manner similar to
Laforgia [31], (2.21)e(β/β2+γ2)xK1(|x|)=∫(0,∞)e−|x|cosh(z)+x(β/β2+γ2)cosh(z)dz=∫(0,∞)e−|x|(cosh(z)−signum(x)(β/β2+γ2))cosh(z)dz,x∈ℝ∖{0}. It follows that the product of ϱNIG1 and |x| is positive, decreasing on ℝ+ and increasing on ℝ− for all δ>0,β∈ℝ.
By Theorem 2.7, the k-function given by (2.22)k(x)=δβ2+γ2πexp(βx)K1(β2+γ2|x|),x∈ℝ∖{0} verifies self-decomposability of
the time-one distribution of the normal inverse Gaussian process. For an
alternative proof of self-decomposability with this parameterization see
Barndorff-Nielsen [6]. CGMY [8] address self-decomposability using a different
parameterization. Sato's construction of H-self-similar additive processes from
self-decomposable distributions (Remark 2.6) may now be applied to the variance
gamma and normal inverse Gaussian cases.


### 2.2.2. Construction ofH-Self-Similar Additive Processes

Using the parameterizations for this study, we
reproduce the results of CGMY [8] and Barndorff-Nielsen [6] (NIG case only) for the
characteristic functions of the time-t marginal distributions of the
self-similar additive analogs of the variance gamma and normal inverse Gaussian
processes (denoted in [8] as “VGSSD” and “NIGSSD,” resp.). Let σ,ν,H>0, μ∈ℝ, and m=ℙVG1 be the self-decomposable variance gamma(σ,μ,ν) distribution on ℝ.
By Theorem 2.5 and Remark 2.6, the characteristic function of VGSSDt is given by (2.23)E[eiξ(VGSSDt)]=m^(tHξ;σ,μ,ν)=(1+ν[−iμtHξ+12σ2t2Hξ2])−1/ν.For the characteristic function of the distribution ofNIGSSDt,
let δ,γ,H>0,β∈ℝ, and m=ℙNIG1 be the self-decomposable normal inverse
Gaussian(δ,β,γ) distribution on ℝ.
Similar to the previous case, the characteristic function of the distribution
of NIGSSDt is given by (2.24)E[eiξ(NIGSSDt)]=m^(tHξ;β,δ,γ)=exp(−δ{t2Hξ2−2iβtHξ+γ2−γ}).  By construction, the
distributions given by (2.10) and (2.23) agree at time one. The same is also true for
(2.13) and (2.24).


## 2.2.1. Self-Decomposability of Time-One Distributions

H-self-similar additive processes are constructed
using Remark 2.6. Consequently, self-decomposability of the one-dimensional
marginal distribution (at time one) must be verified for the variance gamma and
normal inverse Gaussian processes.Variance Gamma Case
As mentioned in the construction of the variance gamma
process in Section2.1, X1 is Normal(μ,σ2) distributed while Zt is gamma(t/ν,1/ν) distributed for all t≥0. Let ϱZ1 denote the Lévy density of the distribution of Z1 (see Table 2). By the subordination theorem for Lévy processes [5, Theorem 30.1, page 197], the
Lévy density of the distribution of the time-one variance gamma random
variable, VG1,
is given by (2.14)ϱVG1(x)=∫(0,∞)dμXsdx(x)ϱZ1(s)ds=∫(0,∞)12πσ2sexp(−(x−μs)22σ2s)1νsexp(−sν)ds=12πσνexp(xμσ2)[2(μ2/σ2+2/ν|x|σ)1/2K1/2(|x|σμ2σ2+2ν)]=1ν|x|exp(−1σ[μ2σ2+2ν|x|−μσx]), where the representation of the
modified Bessel function of the second kind of order ν˜ in (2.15) [2, page 500] is used in the third step with ν˜=1/2 and α,β>0. (2.15)∫(0,∞)exp(−α2s2−β22s)s−(1+ν˜)ds=2(αβ)ν˜Kν˜(αβ). Identity (2.16) [29, page 109] is used in the final
step. (2.16)K1/2(z)=π2ze−z,z>0. The product given by x↦|x|ϱVG1(x) is positive, decreasing on ℝ+ and increasing on ℝ− if ν>0. By Theorem 2.7, the k-function given
by (2.17)k(x)=1νexp(−1σ[μ2σ2+2ν|x|−μσx]),x∈ℝ∖{0} verifies self-decomposability of
the time-one distribution of the variance gamma process, provided ν>0. For an alternative proof of
self-decomposability using the (C,G,M) parameterization of the variance gamma process,
see CGMY [8].Table 2
Lévy
densities of subordinator distributions.
DistributionLévy DensityGamma(a,b)ϱ(x)=aexp(−bx)x−1,x>0Inverse Gaussian(a,b)ϱ(x)=12πax−3/2exp(−12b2x),x>0Normal Inverse Gaussian Case
Using the same construction as in Section2.1 for the
development of the NIG process, X1 is normal(βδ2,δ2) distributed while Zt is IG(t,δγ) distributed. Denoting the Lévy density of the
distribution of Z1 by ϱZ1 (Table 2), the Lévy density of the
distribution of the time-one normal inverse Gaussian random variable, NIG1,
is given by (2.18)ϱNIG1(x)=∫(0,∞)dμXsdx(x)ϱZ1(s)ds=∫(0,∞)12πδ2sexp(−(x−βδ2s)22δ2s)12πs−3/2exp(−12δ2γ2s)ds=12πδexp(βx)∫(0,∞)exp(−[x2δ22s+(β2+γ2)δ22s])s−2ds. Using (2.15) with ν˜=1 yields (2.19)ϱNIG1(x)=12πδexp(βx)[2(δβ2+γ2|x|/δ)K1(δβ2+γ2|x|δ)]=δβ2+γ2π1|x|exp(βx)K1(β2+γ2|x|),x∈ℝ∖{0}. Since x↦|x|ϱNIG1(x) is decreasing on ℝ+ and increasing on ℝ− if and only if x↦e(β/β2+γ2)xK1(|x|) is decreasing on ℝ+ and increasing on ℝ−,
we make use of identity (2.20) [30, equation (9.6.24)]: (2.20)Kν˜(t)=∫(0,∞)e−tcosh(z)cosh(ν˜z)dz,ℜ(t)>0. We have, in a manner similar to
Laforgia [31], (2.21)e(β/β2+γ2)xK1(|x|)=∫(0,∞)e−|x|cosh(z)+x(β/β2+γ2)cosh(z)dz=∫(0,∞)e−|x|(cosh(z)−signum(x)(β/β2+γ2))cosh(z)dz,x∈ℝ∖{0}. It follows that the product of ϱNIG1 and |x| is positive, decreasing on ℝ+ and increasing on ℝ− for all δ>0,β∈ℝ.
By Theorem 2.7, the k-function given by (2.22)k(x)=δβ2+γ2πexp(βx)K1(β2+γ2|x|),x∈ℝ∖{0} verifies self-decomposability of
the time-one distribution of the normal inverse Gaussian process. For an
alternative proof of self-decomposability with this parameterization see
Barndorff-Nielsen [6]. CGMY [8] address self-decomposability using a different
parameterization. Sato's construction of H-self-similar additive processes from
self-decomposable distributions (Remark 2.6) may now be applied to the variance
gamma and normal inverse Gaussian cases.


## 2.2.2. Construction ofH-Self-Similar Additive Processes

Using the parameterizations for this study, we
reproduce the results of CGMY [8] and Barndorff-Nielsen [6] (NIG case only) for the
characteristic functions of the time-t marginal distributions of the
self-similar additive analogs of the variance gamma and normal inverse Gaussian
processes (denoted in [8] as “VGSSD” and “NIGSSD,” resp.). Let σ,ν,H>0, μ∈ℝ, and m=ℙVG1 be the self-decomposable variance gamma(σ,μ,ν) distribution on ℝ.
By Theorem 2.5 and Remark 2.6, the characteristic function of VGSSDt is given by (2.23)E[eiξ(VGSSDt)]=m^(tHξ;σ,μ,ν)=(1+ν[−iμtHξ+12σ2t2Hξ2])−1/ν.For the characteristic function of the distribution ofNIGSSDt,
let δ,γ,H>0,β∈ℝ, and m=ℙNIG1 be the self-decomposable normal inverse
Gaussian(δ,β,γ) distribution on ℝ.
Similar to the previous case, the characteristic function of the distribution
of NIGSSDt is given by (2.24)E[eiξ(NIGSSDt)]=m^(tHξ;β,δ,γ)=exp(−δ{t2Hξ2−2iβtHξ+γ2−γ}).  By construction, the
distributions given by (2.10) and (2.23) agree at time one. The same is also true for
(2.13) and (2.24).


## 2.3. Subordinated Additive Processes

The subordinated, time-inhomogeneous additive
processes in this study are a special case of the class of subordinated
processes discussed by Madan and Yor [32]. Such processes are defined by time-changing a
Brownian motion by an independent, increasing Markov process with independent
and time-inhomogeneous increments. The first process to be constructed is a
Brownian motion time changed by an independent H-self-similar additive gamma process (VHG),
while the second is a Brownian motion time changed by an independent H-self-similar additive inverse Gaussian
process (NHIG). In order to construct the self-similar additive directing
processes, self-decomposability of the gamma and inverse Gaussian distributions
must first be verified. The Lévy densities for these distributions are given in
Table 2 where a,b>0 for both distributions [4]. By Theorem 2.7, the k-functions given by the following equations:
(2.25)kΓ(x)=aexp(−bx),x>0,kIG(x)=12πax−1/2exp(−12b2x),x>0 verify self-decomposability of
the time-one distributions of the gamma and inverse Gaussian processes,
respectively.The characteristic function of the distribution ofVHGt is determined as follows. Let σ,ν,H>0,
and μ∈ℝ.
On the probability space (Ω,ℱ,ℙ) let X≡{Xt:t≥0} be an ℝ-valued Brownian motion where X1 is normal(μ,σ2) distributed. On the same probability space,
using Theorem 2.5, define Z≡{Zt:t≥0} to be an independent, increasing H-self-similar additive process such that Z1 is gamma(1/ν,1/ν)  distributed. The subordinated,
time-inhomogeneous additive analog of the VG process, {VHGt},
can be constructed as (2.26)VHGt(ω)=XZt(ω)(ω),∀t≥0,ω∈Ω. Using (2.7) in Remark 2.10, Remark 2.6,
and the Laplace exponent of Z1, (2.27)ΨZ1[1/ν,1/ν](ξ)=log[(1−ξν)−1/ν],ℜ(ξ)≤0, the characteristic function of
the distribution of the random variable VHGt is given by (2.28)E[eiξ(VHGt)]=exp(ΨZt(ηX1(ξ)))=exp(ΨZ1(tHηX1(ξ)))=exp(log[(1−tH[−12σ2ξ2+iμξ]ν)−1/ν])=(1+tHν[−iμξ+12σ2ξ2])−1/ν. Consequently, the distributions
given by (2.10), (2.23), and (2.28) all agree at time one.The characteristic function of the distribution ofNHIGt is similarly defined. Fix δ,γ,H>0,
and β∈ℝ.
On the probability space (Ω,ℱ,ℙ) define X≡{Xt:t≥0} to be an ℝ-valued Brownian motion where X1 is normal(βδ2,δ2) distributed. On the same probability space,
using Theorem 2.5, define Z≡{Zt:t≥0} to be an independent, increasing H-self-similar additive process such that Z1 is IG(1,δγ) distributed. The subordinated,
time-inhomogeneous additive analog of the NIG process, {NHIGt},
can be constructed as (2.29)NHIGt(ω)=XZt(ω)(ω),∀t≥0,ω∈Ω. The Laplace exponent of Z1 is given by (2.30)ΨZ1[1,δγ](ξ)=−{−2ξ+(δγ)2−δγ},ℜ(ξ)≤0. Similar to the previous case, the
characteristic function of the distribution of NHIGt is obtained as follows: (2.31)E[eiξ(NHIGt)]=exp(ΨZt(ηX1(ξ)))=exp(ΨZ1(tHηX1(ξ)))=exp(−{−2tHηX1(ξ)+(δγ)2−δγ})=exp(−δ{tH[ξ2−2iβξ]+γ2−γ}). Note that (2.31) agrees at time one
with (2.13) and (2.24).


## 3. Increment moments

Given anℝ-valued additive process Y≡{Yt} on (Ω,ℱ,ℙ),
let μYt−Ys denote the distribution of Yt−Ys where 0≤s<t.
By additivity of Y, the characteristic function of the increment
is given by (3.1)μ^Yt−Ys(ξ)=μ^Yt(ξ)μ^Ys(ξ),0≤s<t, [5, Theorem 9.7, page 51]. Consequently,
the moments of the increment are easily computed by taking derivatives of the
difference of characteristic exponents. The mean, variance, skewness, and
excess kurtosis of the increment Yt−Ys are calculated for each of the previously
defined additive processes in Tables 3–8.Table 3
Variance gamma
process: increment moments.
MomentVGMean(Yt−Ys)(t−s)μVar(Yt−Ys)(t−s)[σ2+μ2ν]Skew(Yt−Ys)(t−s)[3σ2+2μ2ν]μν[Var(Yt−Ys)]−3/2Ex.Kurt(Yt−Ys)(t−s)[σ4+4σ2μ2ν+2μ4ν2]3ν[Var(Yt−Ys)]−2Table 4
Normal inverse
Gaussian process: increment moments.
MomentNIGMean(Yt−Ys)(t−s)δγ−1βVar(Yt−Ys)(t−s)δγ−3(β2+γ2)Skew(Yt−Ys)(t−s)3δγ−5β(β2+γ2)[Var(Yt−Ys)]−3/2Ex.Kurt(Yt−Ys)(t−s)3δγ−7(5β2+γ2)(β2+γ2)[Var(Yt−Ys)]−2Table 5
Self-similar
variance gamma process: increment moments.
MomentVGSSDMean(Yt−Ys)(tH−sH)μVar(Yt−Ys)(t2H−s2H)[σ2+μ2ν]Skew(Yt−Ys)(t3H−s3H)[3σ2+2μ2ν]μν[Var(Yt−Ys)]−3/2Ex.Kurt(Yt−Ys)(t4H−s4H)[σ4+4σ2μ2ν+2μ4ν2]3ν[Var(Yt−Ys)]−2Table 6
Self-similar
normal inverse Gaussian process: increment moments.
MomentNIGSSDMean(Yt−Ys)(tH−sH)δγ−1βVar(Yt−Ys)(t2H−s2H)δγ−3(β2+γ2)Skew(Yt−Ys)(t3H−s3H)3δγ−5β(β2+γ2)[Var(Yt−Ys)]−3/2Ex.Kurt(Yt−Ys)(t4H−s4H)3δγ−7(5β2+γ2)(β2+γ2)[Var(Yt−Ys)]−2Table 7
VarianceH-gamma process: increment moments.
MomentVHGMean(Yt−Ys)(tH−sH)μVar(Yt−Ys)(t2H−s2H)μ2ν+(tH−sH)σ2Skew(Yt−Ys)[(t3H−s3H)2μ2ν+(t2H−s2H)3σ2]μν[Var(Yt−Ys)]−3/2Ex.Kurt(Yt−Ys)[(t4H−s4H)2μ4ν2+(t3H−s3H)4σ2μ2ν+(t2H−s2H)σ4]3ν⋅[Var(Yt−Ys)]−2Table 8
NormalH-inverse Gaussian process: increment moments.
MomentNHIGMean(Yt−Ys)(tH−sH)δγ−1βVar(Yt−Ys)[(t2H−s2H)β2+(tH−sH)γ2]δγ−3Skew(Yt−Ys)[(t3H−s3H)β2+(t2H−s2H)γ2]3δβγ−5⋅[Var(Yt−Ys)]−3/2Ex.Kurt(Yt−Ys)[(t4H−s4H)5β4+(t3H−s3H)6β2γ2+(t2H−s2H)γ4]3δγ−7⋅[Var(Yt−Ys)]−2An interesting property among the three classes of
additive processes is the dependence of the variance of the incrementYt+δ−Yt on time t (for any t≥0 and any fixed δ>0). The property of time homogeneity of a Lévy
process implies that the variance of a Lévy increment is constant with respect
to time. See Tables 3 and 4 for VG and NIG cases. Using Remark 2.6 and (3.1), the
variance of the time-t increment of any H-self-similar additive process {Xt} is given by (3.2)Var(Xt+δ−Xt)=((t+δ)2H−t2H)Var(X1), provided the variance of X1 exists. Consequently, these processes have an
increment variance which is constant (resp., decreasing, increasing) with
respect to time when H=1/2 (resp., H<1/2,H>1/2). See Tables 5 and 6 for self-similar VG and
NIG cases. Using (2.7) in Remark 2.10, Remark 2.6, and (3.1), the variance of the
time-t increment of a Brownian motion {Xt} time changed by an independent, increasing H-self-similar additive process {Zt} is given by (3.3)Var(XZt+δ−XZt)=((t+δ)2H−t2H)E[X1]2Var(Z1)+((t+δ)H−tH)E[Z1]Var(X1), provided the variance of Z1 exists. Consequently, this class of
subordinated additive processes has an increment variance which is decreasing
(resp., increasing) in time for H≤1/2 (resp., H≥1). For H∈(1/2,1) the signum of the rate of change of increment
variance with respect to time depends on the parameter values of the
distributions of X1 and Z1,
as well as time t.
See Tables 7 and 8 for the increment moments associated with the
time-inhomogeneous, subordinated analogs of the VG and NIG processes.


## 4. Definition of the price processes

### 4.1. Exponential Additive Models

LetX≡{Xt}t∈[0,T] be an additive process on the stochastic basis (Ω,ℱ,𝔽={ℱt},ℙ),
where X is adapted to the filtration 𝔽.
The underlying of a European-exercised option is modeled by the price process {St}t∈[0,T],
which is defined, path by path, in (4.1)(4.1)St=S0exp(Xt+[r−q]t)𝔼ℙ[eXt]for  eacht∈[0,T], where r and q are the continuously compounded values of both
interest rate and dividend yield, respectively. Note that the property of
independent increments of {Xt} implies that {e−(r−q)tSt} is an (𝔽−ℙ) martingale.The characteristic functions of the time-t distributions of the logarithm of the
underlying for the additive processes defined in Sections 2.1–2.3 are given
below:(i)
exponential variance gamma(4.2)ΦlnStVG(ξ)=exp{iξ(lnS0+t[r−q])−(t/ν)log(1+ν[−iμξ+(1/2)σ2ξ2])}[(1−ν[μ+(1/2)σ2])−t/ν]iξ;(ii)
exponentialH-self-similar variance gamma (4.3)ΦlnStVGSSD(ξ)=exp{iξ(lnS0+t[r−q])−(1/ν)log(1+ν[−iμtHξ+(1/2)σ2t2Hξ2])}[(1−ν[μtH+(1/2)σ2t2H])−1/ν]iξ;(iii)
exponential varianceH-gamma(4.4)ΦlnStVHG(ξ)=exp{iξ(lnS0+t[r−q])−(1/ν)log(1+tHν[−iμξ+(1/2)σ2ξ2])}[(1−tHν[μ+(1/2)σ2])−1/ν]iξ;(iv)
exponential normal inverse Gaussian(4.5)ΦlnStNIG(ξ)=exp{iξ(lnS0+t[r−q])−δt[ξ2−2iβξ+γ2−γ]}[exp(−δt[−1−2β+γ2−γ])]iξ;(v)
exponentialH-self-similar normal inverse
Gaussian(4.6)ΦlnStNIGSSD(ξ)=exp{iξ(lnS0+t[r−q])−δ[t2Hξ2−2iβtHξ+γ2−γ]}[exp(−δ[−t2H−2βtH+γ2−γ])]iξ;(vi)
exponential normalH-inverse Gaussian (4.7)ΦlnStNHIG(ξ)=exp{iξ(lnS0+t[r−q])−δ[tH(ξ2−2iβξ)+γ2−γ]}[exp(−δ[tH(−1−2β)+γ2−γ])]iξ.


### 4.2. Exponential Lévy Stochastic Volatility Models

We now define, according to CGMY [3] and Schoutens [4], two Lévy stochastic
volatility processes to be used for the creation of model benchmarks for the
previously defined models. These processes are formed by time changing a Lévy
process by the time integral of a process of Ornstein-Uhlenbeck type with
associated stationary gamma distribution.
Let {Zt} be defined by (4.8)Zt=∫0tysds, where, for λ>0, y0>0, and subordinator {Lt}, {yt} satisfies the Ornstein-Uhlenbeck type SDE
[5] (4.9)yt=y0+Lt−λ∫0tysdsfort≥0. The solution of (4.9) is given by (4.10)yt=y0e−λt+∫0te−λ(t−s)dLs. We fix a,b>0 and specify {Lt} to be a compound Poisson process with
exponentially distributed jump sizes where (4.11)ΨL1(ξ)=λaξb−ξ,ℜ(ξ)≤0. CGMY [3] and Cont and Tankov
[2, page 488] show that
such a choice for {Lt} yields the following expression for the
characteristic function of the distribution of Zt: (4.12)ΦZt(ξ;λ,a,b,y0)=exp{iy0λ−1[1−e−λt]ξ+λaiξ−λb[blog(bb−iλ−1(1−e−λt)ξ)−itξ]},ξ∈ℝ (see [4]).We now conclude the definition of Lévy stochastic
volatility processes via subordination. Let{Xt} be a variance gamma process or normal inverse
Gaussian process which is independent of the increasing process {Zt},
as determined by (4.8), (4.10), and (4.11).
The VG-OU-Γ and NIG-OU-Γ processes can be constructed such that, for
each t≥0 and ω in the sample space (4.13)VG-OU-Γt(ω)=XZt(ω)(ω),{Xt}is  a  VG  process,(4.14)NIG-OU-Γt(ω)=XZt(ω)(ω),{Xt}is  a  NIG  process.Let{XZt}t∈[0,T] be an ℝ-valued Lévy stochastic volatility process on (Ω,ℱ,𝔽,ℙ) as in (4.13) or (4.14).
The underlying price process at time t is defined, as in [3, 4], to satisfy (4.15)St=dS0exp(XZt+[r−q]t)𝔼ℙ[eXZt]for  eacht∈[0,T], where {e−t(r−q)St} is a (𝔽˜−ℙ˜) martingale on another filtered probability
space (Ω˜,ℱ˜,𝔽˜,ℙ˜).
The exponential pricing models corresponding to (4.13) and (4.14) are equivalent to
the “VGSG” and “NIGSG” models of CGMY [3] when the “VGSG” and “NIGSG” correlation parameters are
set to zero. Unlike the exponential additive pricing models, these do not
prohibit dynamic arbitrage. Rather, the intent of definition (4.15) is to
prohibit arbitrages which result from a limited form of dynamic trading, known
as “static arbitrages” [3, 33]. CGMY [3] demonstrated that this simple (as opposed to
stochastic) exponentiation of Lévy stochastic volatility processes yielded
models which priced out-of-the-money SPX and equity options quite well, and
better than their stochastically exponentiated counterparts. We refer to Cont
and Tankov [2, pages 492, 493] for an alternative formulation in which the discounted price process is
a martingale with respect to the enlarged filtration determined from the
time-changed Lévy process and the directing process. The characteristic
functions of the time-t marginal distributions are given below for the
exponential VG-OU-Γ and NIG-OU-Γ models.(i) Exponential Variance Gamma Ornstein-Uhlenbeck-Gamma
Forσ,ν>0,
and μ∈ℝ,
define the parameters C,G,M as in [3]:(4.16)C=1ν,G=(14μ2ν2+12σ2ν−12μν)−1,M=(14μ2ν2+12σ2ν+12μν)−1. The characteristic function of the
time-one marginal of the VG process is given by (4.17)ΦVG1(ξ;C,G,M)=(GMGM+i(M−G)ξ+ξ2)C. The number of parameters in (4.13)
is reduced from seven to six via (4.18)ΦVG-OU-Γ(ξ;C,G,M,λ,a,b,y0)=ΦVG-OU-Γ(ξ;Cy0,G,M,λ,a,by0,1) (see [4]). It follows that the characteristic
function of the time-t distribution of the logarithm of the
underlying is given by (4.19)ΦlnStVG-OU-Γ(ξ)=exp(iξ(lnS0+t[r−q]))⋅ΦZt(−ilogφ(ξ);λ,a,b,1)[ΦZt(−ilogφ(−i);λ,a,b,1)]iξ, where  (4.20)φ(ξ)=ΦVG1(ξ;C,G,M) (see [4]) and ΦZt is defined in (4.12).(ii) Exponential Normal Inverse Gaussian Ornstein-Uhlenbeck-Gamma
The characteristic function of the time-one marginal
of the NIG process is(4.21)ΦNIG1(ξ;β,δ,γ)=exp{−δ[ξ2−2iβξ+γ2−γ]}. The number of parameters in (4.14)
is reduced from seven to six via (4.22)ΦNIG-OU-Γ(ξ;β,δ,γ,λ,a,b,y0)=ΦNIG-OU-Γ(ξ;β,δy0,γ,λ,a,by0,1) (see [4]). It follows that the
characteristic function of the time-t distribution of the logarithm of the
underlying is given by (4.23)ΦlnStNIG-OU-Γ(ξ)=exp(iξ(lnS0+t[r−q]))⋅ΦZt(−ilogφ(ξ);λ,a,b,1)[ΦZt(−ilogφ(−i);λ,a,b,1)]iξ, where (4.24)φ(ξ)=ΦNIG1(ξ;β,δ,γ) (see [4]) and ΦZt is defined in (4.12).


## 4.1. Exponential Additive Models

LetX≡{Xt}t∈[0,T] be an additive process on the stochastic basis (Ω,ℱ,𝔽={ℱt},ℙ),
where X is adapted to the filtration 𝔽.
The underlying of a European-exercised option is modeled by the price process {St}t∈[0,T],
which is defined, path by path, in (4.1)(4.1)St=S0exp(Xt+[r−q]t)𝔼ℙ[eXt]for  eacht∈[0,T], where r and q are the continuously compounded values of both
interest rate and dividend yield, respectively. Note that the property of
independent increments of {Xt} implies that {e−(r−q)tSt} is an (𝔽−ℙ) martingale.The characteristic functions of the time-t distributions of the logarithm of the
underlying for the additive processes defined in Sections 2.1–2.3 are given
below:(i)
exponential variance gamma(4.2)ΦlnStVG(ξ)=exp{iξ(lnS0+t[r−q])−(t/ν)log(1+ν[−iμξ+(1/2)σ2ξ2])}[(1−ν[μ+(1/2)σ2])−t/ν]iξ;(ii)
exponentialH-self-similar variance gamma (4.3)ΦlnStVGSSD(ξ)=exp{iξ(lnS0+t[r−q])−(1/ν)log(1+ν[−iμtHξ+(1/2)σ2t2Hξ2])}[(1−ν[μtH+(1/2)σ2t2H])−1/ν]iξ;(iii)
exponential varianceH-gamma(4.4)ΦlnStVHG(ξ)=exp{iξ(lnS0+t[r−q])−(1/ν)log(1+tHν[−iμξ+(1/2)σ2ξ2])}[(1−tHν[μ+(1/2)σ2])−1/ν]iξ;(iv)
exponential normal inverse Gaussian(4.5)ΦlnStNIG(ξ)=exp{iξ(lnS0+t[r−q])−δt[ξ2−2iβξ+γ2−γ]}[exp(−δt[−1−2β+γ2−γ])]iξ;(v)
exponentialH-self-similar normal inverse
Gaussian(4.6)ΦlnStNIGSSD(ξ)=exp{iξ(lnS0+t[r−q])−δ[t2Hξ2−2iβtHξ+γ2−γ]}[exp(−δ[−t2H−2βtH+γ2−γ])]iξ;(vi)
exponential normalH-inverse Gaussian (4.7)ΦlnStNHIG(ξ)=exp{iξ(lnS0+t[r−q])−δ[tH(ξ2−2iβξ)+γ2−γ]}[exp(−δ[tH(−1−2β)+γ2−γ])]iξ.


## 4.2. Exponential Lévy Stochastic Volatility Models

We now define, according to CGMY [3] and Schoutens [4], two Lévy stochastic
volatility processes to be used for the creation of model benchmarks for the
previously defined models. These processes are formed by time changing a Lévy
process by the time integral of a process of Ornstein-Uhlenbeck type with
associated stationary gamma distribution.
Let {Zt} be defined by (4.8)Zt=∫0tysds, where, for λ>0, y0>0, and subordinator {Lt}, {yt} satisfies the Ornstein-Uhlenbeck type SDE
[5] (4.9)yt=y0+Lt−λ∫0tysdsfort≥0. The solution of (4.9) is given by (4.10)yt=y0e−λt+∫0te−λ(t−s)dLs. We fix a,b>0 and specify {Lt} to be a compound Poisson process with
exponentially distributed jump sizes where (4.11)ΨL1(ξ)=λaξb−ξ,ℜ(ξ)≤0. CGMY [3] and Cont and Tankov
[2, page 488] show that
such a choice for {Lt} yields the following expression for the
characteristic function of the distribution of Zt: (4.12)ΦZt(ξ;λ,a,b,y0)=exp{iy0λ−1[1−e−λt]ξ+λaiξ−λb[blog(bb−iλ−1(1−e−λt)ξ)−itξ]},ξ∈ℝ (see [4]).We now conclude the definition of Lévy stochastic
volatility processes via subordination. Let{Xt} be a variance gamma process or normal inverse
Gaussian process which is independent of the increasing process {Zt},
as determined by (4.8), (4.10), and (4.11).
The VG-OU-Γ and NIG-OU-Γ processes can be constructed such that, for
each t≥0 and ω in the sample space (4.13)VG-OU-Γt(ω)=XZt(ω)(ω),{Xt}is  a  VG  process,(4.14)NIG-OU-Γt(ω)=XZt(ω)(ω),{Xt}is  a  NIG  process.Let{XZt}t∈[0,T] be an ℝ-valued Lévy stochastic volatility process on (Ω,ℱ,𝔽,ℙ) as in (4.13) or (4.14).
The underlying price process at time t is defined, as in [3, 4], to satisfy (4.15)St=dS0exp(XZt+[r−q]t)𝔼ℙ[eXZt]for  eacht∈[0,T], where {e−t(r−q)St} is a (𝔽˜−ℙ˜) martingale on another filtered probability
space (Ω˜,ℱ˜,𝔽˜,ℙ˜).
The exponential pricing models corresponding to (4.13) and (4.14) are equivalent to
the “VGSG” and “NIGSG” models of CGMY [3] when the “VGSG” and “NIGSG” correlation parameters are
set to zero. Unlike the exponential additive pricing models, these do not
prohibit dynamic arbitrage. Rather, the intent of definition (4.15) is to
prohibit arbitrages which result from a limited form of dynamic trading, known
as “static arbitrages” [3, 33]. CGMY [3] demonstrated that this simple (as opposed to
stochastic) exponentiation of Lévy stochastic volatility processes yielded
models which priced out-of-the-money SPX and equity options quite well, and
better than their stochastically exponentiated counterparts. We refer to Cont
and Tankov [2, pages 492, 493] for an alternative formulation in which the discounted price process is
a martingale with respect to the enlarged filtration determined from the
time-changed Lévy process and the directing process. The characteristic
functions of the time-t marginal distributions are given below for the
exponential VG-OU-Γ and NIG-OU-Γ models.(i) Exponential Variance Gamma Ornstein-Uhlenbeck-Gamma
Forσ,ν>0,
and μ∈ℝ,
define the parameters C,G,M as in [3]:(4.16)C=1ν,G=(14μ2ν2+12σ2ν−12μν)−1,M=(14μ2ν2+12σ2ν+12μν)−1. The characteristic function of the
time-one marginal of the VG process is given by (4.17)ΦVG1(ξ;C,G,M)=(GMGM+i(M−G)ξ+ξ2)C. The number of parameters in (4.13)
is reduced from seven to six via (4.18)ΦVG-OU-Γ(ξ;C,G,M,λ,a,b,y0)=ΦVG-OU-Γ(ξ;Cy0,G,M,λ,a,by0,1) (see [4]). It follows that the characteristic
function of the time-t distribution of the logarithm of the
underlying is given by (4.19)ΦlnStVG-OU-Γ(ξ)=exp(iξ(lnS0+t[r−q]))⋅ΦZt(−ilogφ(ξ);λ,a,b,1)[ΦZt(−ilogφ(−i);λ,a,b,1)]iξ, where  (4.20)φ(ξ)=ΦVG1(ξ;C,G,M) (see [4]) and ΦZt is defined in (4.12).(ii) Exponential Normal Inverse Gaussian Ornstein-Uhlenbeck-Gamma
The characteristic function of the time-one marginal
of the NIG process is(4.21)ΦNIG1(ξ;β,δ,γ)=exp{−δ[ξ2−2iβξ+γ2−γ]}. The number of parameters in (4.14)
is reduced from seven to six via (4.22)ΦNIG-OU-Γ(ξ;β,δ,γ,λ,a,b,y0)=ΦNIG-OU-Γ(ξ;β,δy0,γ,λ,a,by0,1) (see [4]). It follows that the
characteristic function of the time-t distribution of the logarithm of the
underlying is given by (4.23)ΦlnStNIG-OU-Γ(ξ)=exp(iξ(lnS0+t[r−q]))⋅ΦZt(−ilogφ(ξ);λ,a,b,1)[ΦZt(−ilogφ(−i);λ,a,b,1)]iξ, where (4.24)φ(ξ)=ΦNIG1(ξ;β,δ,γ) (see [4]) and ΦZt is defined in (4.12).


## 5. Model implementation and data specifications

In order to calculate European option prices, the
“modified call” method of Carr and Madan [34] is used. This requires an analytical expression for
the inverse Fourier transform of a damped call price function in terms of the
characteristic function of the distribution of the logarithm of the risk-neutral
underlying. Put prices are calculated via the call price using put-call parity.
Let Θ denote the set of model parameter vectors such
that for all t, both θ∈Θ and t determine the time-t distribution of
the logarithm of the risk-neutral underlying, lnSt.
Given a set of N market option prices {Pi}i=1⋯N,
the set of corresponding model option prices is denoted by {πiθ}i=1⋯N for any θ∈Θ.
In order to estimate θ,
the quadratic pricing error, ℰ:Θ→ℝ+,
is minimized with (5.1)ℰ(θ)=∑i=1Nwi[πiθ(r,q,S0,Ti,Ki)−Pi(Ti,Ki)]2,θ∈Θ, where wi,Ti,Ki are the weighting factor, maturity, and strike
price of option i,
respectively. Following Cont and Tankov [2], the weighting factor of each option is given by the
reciprocal of the square of the Black-Scholes vega evaluated at the associated Black-Scholes
market implied volatility. Consequently, the sum of squared differences between
market and model implied volatilities is approximated by the sum of weighted
squared differences between model and market prices. Cont and Tankov [35] discuss some of the
computational difficulties pertaining to the calibration of a given model, for example, multiple local minima. For further discussion on sources of error in
calibration, we refer to [36].The option data used to calibrate the risk-neutral
models in this study were 4 p.m. EST time-stamped bid and ask quotes of
out-of-the-money Standard and Poor's 500 index options. The selected quote
dates coincided with the second Tuesday of the month for the time period
spanning January 2005 through December 2005. Market option prices were
calculated as the average of bid and ask, while the closing index value was
used as the underlying price. The interest rate and dividend yield for each
quote date were calculated using the daily updated rate on the three-month
Treasury bill [37] and
the quarterly updated SPX dividend yield [38], respectively.Several filters were applied to the data. First, the
nearest-term SPX options were excluded from this study, leaving options of five
maturities ranging from approximately 1 month to 1 year. Second, any option
with an ask or bid quote equal to zero was discarded. Third, options were
excluded which violated the requirement of monotonicity of price with respect
to strike. Most of these deep out-of-the-money options were eliminated, as in
[8], by requiring the
ratio of option price to underlying price to be bounded below by a sufficiently
large positive number. Here, an empirically determined value of 9.5×10−4 was used to filter out such options. After all
of the filters were applied, the moneyness (strike/underlying) values for all
options in this study ranged from 0.5893 to 1.2680, with an average of 117
options per quote date.


## 6. Calibration results

### 6.1. Performance Measures

In order to measure the pricing performance of each of
the eight previously defined models, the average percentage error(APE) is now defined. Denote by N the number of options used on a particular
quote date, {Pi} the set of observed bid-ask averages, and {πi} the set of corresponding calculated model
prices. The average percentage error is defined in [4] as (6.1)APE=∑i=1N|Pi−πi|∑i=1NPi.Since the market option prices are taken as the set of
averages of the bid and ask quotes on a given quote date, each model price may
deviate from the market price by at most(ask−bid)/2 if it is to lie between the bid and ask
quotes. Setting the price difference to be (ask−bid)/2 and the market price to be (ask+bid)/2 in (6.1) yields the measure of uncertainty in
market prices given below: (6.2)APEmarket=∑i=1N(aski−bidi)∑i=1N(aski+bidi).We define the in-sample APE using (6.1) with bid-ask
quotes and parameter estimates corresponding to the same quote date. The one-day-ahead (resp., one-week-ahead) out-of-sample APE is defined using (6.1) with
in-sample parameter estimates and bid-ask quotes obtained one day (resp., one
week) later than the in-sample quote date. The in-sample (resp., one-day-ahead
out-of-sample, one-week-ahead out-of-sample) market APE is calculated using
(6.2) with bid-ask quotes obtained zero days (resp., one day, one week) later
than the in-sample quote date.


### 6.2. Individual Model Performance

In Figures1, 2, 3 are the bar graphs of the in-sample,
one-day-ahead, and one-week-ahead out-of-sample APE for each quote date and
model. The time averages of APE, taken over all twelve-quote dates, are
reported in Table 9. The worst
performing model was the VG Lévy with 11.9% (resp., 12.7%, 14.6%) mean in-sample
(resp., one-day-ahead, one-week-ahead) APE. The best performing model was the
self-similar NIG (NIGSSD) with respective mean APEs of 4.06%, 5.13%, and 8.32%.
Furthermore, this model was the only one with a mean in-sample pricing error
below that of the market.Table 9
Mean of average percentage error: twelve quote date time average (January–December 2005).
Standard deviation is given in parentheses.
Process nameMean APE (%)In-sample1-day-ahead1-week-aheadVG11.9 (2.08)12.7 (1.95)14.6 (3.63)NIG10.2 (1.93)11.3 (1.85)12.9 (4.18)VHG7.24 (1.37)7.87 (1.75)10.5 (3.76)NHIG5.76 (1.10)6.61 (1.64)9.38 (4.16)VGSSD6.04 (1.42)6.63 (1.55)9.23 (4.26)NIGSSD4.06 (1.01)5.13 (1.51)8.32 (4.49)VG-OU-Γ5.25 (1.12)6.24 (1.61)8.78 (4.19)NIG-OU-Γ4.81 (0.966)5.97 (1.57)8.59 (4.33)Market4.63 (0.502)4.97 (1.28)4.52 (0.569)Average percentage error (in-sample, out-of-the-money options).
(a)(b)Average percentage error (1-day-ahead out-of-sample, out-of-the-money options).
(a)(b)Average percentage error (1-week-ahead out-of-sample, out-of-the-money options).
(a)(b)Table10 contains values of the mean ratio of APE,
taken over all twelve-quote dates, for a model of one class to that of another
with a common associated distribution (VG or NIG). As shown in
the sixth row of Table 10, the NIGSSD model outperformed its model benchmark,
with APE values which were, on average, 84% and 86% of those obtained by the
NIG-OU-Γ model for in-sample and one-day-ahead
out-of-sample cases, respectively. In contrast, the VGSSD model, on average,
had pricing errors which were 115% and 107% of those obtained by the VG-OU-Γ model for in-sample and one-day-ahead
out-of-sample cases, respectively. Below is a summary of the NIGSSD pricing
performance statistics taken from Figures 1–3.Table 10
Model class comparison: twelve quote date time-averaged ratio of average percentage errors
(January–December 2005). Standard deviation is given in parentheses.
Model 1: model 2〈APE1APE2〉In-sample1-day-ahead1-week-aheadVHG: VG0.622 (0.139)0.625
(0.126)0.728
(0.219)NHIG: NIG0.576 (0.128)0.585
(0.105)0.745
(0.255)VGSSD: VHG0.829 (0.0894)0.841
(0.0525)0.847
(0.125)NIGSSD: NHIG0.704 (0.124)0.776
(0.0900)0.847
(0.134)VGSSD: VG-OU-Γ1.15 (0.102)1.07
(0.0902)1.05
(0.129)NIGSSD: NIG-OU-Γ0.838 (0.109)0.858
(0.0635)0.947
(0.133)(i) The proportions of quote dates in which the
NIGSSD model had a lower APE than that of the market for the in-sample, one-day-ahead, and one-week-ahead out-of-sample cases were9/12, 6/12,
and 2/12,
respectively.(ii) The proportions of quote dates in which the
NIGSSD model had a lower APE than those of both stochastic volatility models
for the in-sample, one-day-ahead, and one-week-ahead out-of-sample cases were11/12, 12/12,
and 8/12,
respectively.


### 6.3. Performance Comparison of Additive Classes

Rows one and
two of Table10 indicate the benefits of time inhomogeneity, in that the
in-sample pricing error of the subordinated additive class was approximately
60% of that corresponding to the Lévy class. The benefit of self-similarity in
the log-return process over self-similarity of the directing process is
observed via rows three and four of Table 10. For the in-sample case, the NIGSSD
model (mean ratio of 0.70) performed considerably better against its
subordinated additive counterpart than did the VGSSD model (mean ratio of
0.83). Such differences in mean error ratio decreased with time lag, such that,
regardless of time-one distribution, the one-week-ahead values were
approximately 73% and 85% for the [subordinated additive]:[Lévy] and
[self-similar]:[subordinated additive] comparisons, respectively.


### 6.4. Performance Comparison of Nig and Vg Models

In each row of Table11 a model class is fixed, and
the mean ratio (taken over all twelve-quote dates) of APE of the NIG model to
that of the corresponding VG model is given. The mean ratio
of in-sample APE of the NIG (resp., NHIG, NIGSSD) model to that of the VG (resp.,
VHG, VGSSD) model was approximately 86% (resp., 80%, 67%). Furthermore, the
variation in mean APE ratio across additive model class decreased with time
lag. As shown in rows one through three of the third column of Table 11, the
one-week-ahead out-of-sample mean ratios were approximately 87%, regardless of
the additive model class under consideration. Finally, the consistency with
which the NIG distributions outperformed those of the VG family may be observed
in the following summary statistics taken from Figures 1–3. The proportions of quote dates in which each
of the four models of the NIG family had a lower APE than that of the
corresponding member of the VG family for in-sample, one-day-ahead, and one-week-ahead out-of-sample cases were 12/12, 11/12,
and 8/12,
respectively.Table 11
Variance gamma and normal inverse
Gaussian distribution comparison: twelve quote date time-averaged ratio of
average percentage errors (January–December 2005). Standard deviation is given in parentheses.
Model 1: model 2〈APE1APE2〉In-sample1-day-ahead1-week-aheadNIG: VG0.857
(0.0179)0.890
(0.0528)0.867
(0.0782)NHIG: VHG0.796
(0.0575)0.837
(0.0473)0.873
(0.0926)NIGSSD: VGSSD0.670
(0.0590)0.773
(0.0984)0.873
(0.109)NIG-OU-Γ:
VG-OU-Γ0.921
(0.0585)0.957
(0.0449)0.969
(0.0532)


### 6.5. Model Price Behavior across Strikes and Maturities

In Figures4–7 the market and model prices are
plotted versus moneyness (strike/underlying) for the set of options available on April 12, 2005.
Results for this month are displayed in order to provide a typical observation
of option pricing errors for the self-similar NIG model. Figures 4 and 5 consist
of the plots for the VG family of models, while Figures 6 and 7 consist of the
plots for the NIG family. We define the relative model price difference for two
models corresponding to a given option as the absolute value of the difference
between model prices divided by the market option price. The relative model
price differences between the Lévy and subordinated additive models were the
greatest for deep out-of-the-money puts belonging to the two nearest term
maturities (see Figures 4 and 6). The relative model price differences between
the subordinated additive and self-similar additive models were the greatest
among the out-of-the-money calls of the nearest two and longest two maturities,
as shown in Figures 4, 5 and 6, 7. This same observation is true for relative price
differences between the self-similar and stochastic volatility models, as shown
in Figures 5 and 7.Market and model option prices versus moneyness (strike/underlying)
for VG and VHG models: April 2005 quote date.
(a)(b)Market and model option prices versus moneyness (strike/underlying)
for VGSSD and VG-OU-gamma models: April 2005 quote date.
(a)(b)Market and model option prices versus moneyness (strike/underlying)
for NIG and NHIG models: April 2005 quote date.
(a)(b)Market and model option prices versus moneyness (strike/underlying)
for NIGSSD and NIG-OU-gamma models: April 2005 quote date.
(a)(b)


### 6.6. Term Structure of Moments

The first four moments of the risk-neutral
distributions determined from the options on the April 12, 2005 quote date are
plotted versus maturity in Figure8. For the set of parameters chosen by the
market in the subordinated additive models, the skewness (resp., kurtosis) is
decreasing (resp., increasing) with maturity. This is contrary to the case of
the Lévy models in which the skewness (resp., kurtosis) is increasing (resp.,
decreasing) with maturity. Self-similar processes have one-dimensional marginal
distributions with a rather striking constant skewness and kurtosis. Invariance
with respect to maturity is due to the fact that the characteristic function at
time t is obtained by composition of the
characteristic function at time one with the map ξ↦tHξ.
When the chain rule is applied to the composition, a factor of tH is produced. Consequently, the moment number
is matched by the same number of factors of tH, thereby allowing cancelation of maturity in
the calculation of skewness and excess kurtosis.Risk-neutral mean, variance, skewness,
and kurtosis term structures for April 12, 2005 quote date (VG models—top row) (NIG models—bottom row).
(a)(b)(c)(d)(e)(f)(g)(h)


### 6.7. Hurst Exponent of Additive Models

Tables12 and 13 contain the time-averaged parameter
estimates, taken over the twelve-quote dates, for each additive model. The
sample means of H were approximately 0.6 and 0.9 for the self-similar and subordinated additive
classes, respectively. Furthermore, the relative difference between sample
means of H in the two self-similar models was a mere 0.091%,
compared with 8.9% in the subordinated additive case. In the next
section, we will see that the modest variation of Hurst exponent in self-similar
models during the year 2005 is not maintained over the course of the first half
of the following year.Table 12
Variance gamma
family: twelve quote date time-averaged parameter estimates (January–December
2005). Standard deviation is given in parentheses.
ModelParameterVGVHGVGSSDμ−0.1143 (0.03232)−0.1320 (0.02859)−0.09432 (0.01381)σ0.1227  (0.005445)0.1105  (0.009046)0.1330  (0.006274)ν0.3818  (0.1008)0.9515 (0.1997)0.9713 (0.1405)H0.9566  (0.08177)0.6251  (0.02720)Table 13
Normal
inverse Gaussian family: twelve quote date time-averaged parameter estimates
(January–December 2005). Standard deviation is given in parentheses.
ModelParameterNIGNHIGNIGSSDβ−6.788 (2.134)−30.47  (23.33)−6.919 (1.681)δ0.1732 (0.03245)0.1134  (0.01260)0.1413  (0.01566)γ11.28 (1.764)15.81  (7.157)8.701  (1.005)H0.8754  (0.1236)0.6245  (0.02648)


### 6.8. Comparison with Other Studies

Recall in Table9 the mean in-sample APEs for the
self-similar VG and NIG models were 6.04% and 4.06%, respectively. These values
are in contrast to the respective mean values of 3.19% and 3.02% obtained by
CGMY  [8] and
3.31% and 3.38%, obtained by Madan and Eberlein [28]. In both of these studies,
self-similar models were calibrated using out-of-the-money SPX options. In the
CGMY study, options were used with quotes occurring during the years 2000 and
2001, and with maturities ranging from approximately one month to 15 months.
The data in the study by Madan and Eberlein consisted of options quoted on
May 17, 2006, with maturities ranging from one month to two years.In order to investigate the influence of the choice of
weighting scheme and maturity range on pricing error, we calibrate the VGSSD
and NIGSSD models using out-of-the-money SPX options quoted on May 17, 2006.
The same data filters are used as in Section5, but with varying maturity
ranges and weighting methods. The maturity ranges are one month to one year in
the first case, and one month to two years in the second. The weighting methods
consist of the reciprocal vega2 method, described in Section 5, and a uniform
weighting across both strike and maturity. In each calibration the sum of
weighted squared differences between model and market prices is minimized in
order to obtain the parameter estimates. In Tables 14–17, an asterisk is used
to identify a value which is determined using the same weighting scheme and
maturity range as described in Section 5.Table 14
Self-similar
variance gamma model: percentage error comparison with respect to maturity
range and weighting method for May 17, 2006 quote date.
VGSSD: APE [%]Weighting MethodRange of maturityτ[yrs]1/vega2uniformτ∈(112,1)4.47*3.74τ∈(112,2)5.183.58Table 15
Self-similar
normal inverse Gaussian model: percentage error comparison with respect to
maturity range and weighting method for May 17, 2006 quote date.
NIGSSD: APE [%]Weighting Methodrange of maturityτ[yrs]1/vega2uniformτ∈(112,1)4.78*3.91τ∈(112,2)5.163.57Table 16
Self-similar
variance gamma model: Hurst exponent estimate comparison with respect to
maturity range and weighting method for May 17, 2006 quote date.
VGSSD: Hurst exponentH:Weighting Methodrange of maturityτ[yrs]1/vega2uniformτ∈(112,1)0.4916*0.5201τ∈(112,2)0.50850.5440Table 17
Self-similar
normal inverse Gaussian model: Hurst exponent estimate comparison with respect
to maturity range and weighting method for May 17, 2006 quote date.
NIGSSD: Hurst exponentH:Weighting Methodrange of maturityτ[yrs]1/vega2uniformτ∈(112,1)0.4921*0.5214τ∈(112,2)0.50930.5451The pricing errors for the VGSSD and NIGSSD models are
reported in Tables14 and 15,
respectively. Interestingly, the differences in weighting method and maturity
range may account for as much as a 1.2% deviation in APE (NIGSSD case) from the pricing error obtained with the conventions of Section 5. Furthermore,
an increase in maturity range resulted in a decrease in APE under the uniform
weighting method and an increase in APE under the reciprocal vega2 weighting method. The behavior of pricing
error with maturity range, for a fixed reciprocal vega2 weighting scheme, is consistent with the fact
that this method places greater weight on nearer-term options.The estimate of the Hurst exponent in the VGSSD model,
obtained with options available on May 17, 2006, is reported in Madan and
Eberlein [28] as 0.5326.
In Tables 16 and 17 are the estimates of the Hurst exponent, H, for the VGSSD
and NIGSSD models, respectively. The estimate for H, denoted by (*) in Table 16, was 0.492.
This is in contrast to the value of 0.660 obtained for the VGSSD model on the last quote
date of this study (December 2005), as well as the time-averaged value of 0.625 in Table 12. The use of uniform weights and a
two-year maturity range resulted in an estimate of 0.544 for H,
as opposed to the estimate of 0.492,
obtained using the conventions of Section 5. Since parameter estimates for the
NIGSSD model were not presented in [28], only the VGSSD values are compared.


## 6.1. Performance Measures

In order to measure the pricing performance of each of
the eight previously defined models, the average percentage error(APE) is now defined. Denote by N the number of options used on a particular
quote date, {Pi} the set of observed bid-ask averages, and {πi} the set of corresponding calculated model
prices. The average percentage error is defined in [4] as (6.1)APE=∑i=1N|Pi−πi|∑i=1NPi.Since the market option prices are taken as the set of
averages of the bid and ask quotes on a given quote date, each model price may
deviate from the market price by at most(ask−bid)/2 if it is to lie between the bid and ask
quotes. Setting the price difference to be (ask−bid)/2 and the market price to be (ask+bid)/2 in (6.1) yields the measure of uncertainty in
market prices given below: (6.2)APEmarket=∑i=1N(aski−bidi)∑i=1N(aski+bidi).We define the in-sample APE using (6.1) with bid-ask
quotes and parameter estimates corresponding to the same quote date. The one-day-ahead (resp., one-week-ahead) out-of-sample APE is defined using (6.1) with
in-sample parameter estimates and bid-ask quotes obtained one day (resp., one
week) later than the in-sample quote date. The in-sample (resp., one-day-ahead
out-of-sample, one-week-ahead out-of-sample) market APE is calculated using
(6.2) with bid-ask quotes obtained zero days (resp., one day, one week) later
than the in-sample quote date.


## 6.2. Individual Model Performance

In Figures1, 2, 3 are the bar graphs of the in-sample,
one-day-ahead, and one-week-ahead out-of-sample APE for each quote date and
model. The time averages of APE, taken over all twelve-quote dates, are
reported in Table 9. The worst
performing model was the VG Lévy with 11.9% (resp., 12.7%, 14.6%) mean in-sample
(resp., one-day-ahead, one-week-ahead) APE. The best performing model was the
self-similar NIG (NIGSSD) with respective mean APEs of 4.06%, 5.13%, and 8.32%.
Furthermore, this model was the only one with a mean in-sample pricing error
below that of the market.Table 9
Mean of average percentage error: twelve quote date time average (January–December 2005).
Standard deviation is given in parentheses.
Process nameMean APE (%)In-sample1-day-ahead1-week-aheadVG11.9 (2.08)12.7 (1.95)14.6 (3.63)NIG10.2 (1.93)11.3 (1.85)12.9 (4.18)VHG7.24 (1.37)7.87 (1.75)10.5 (3.76)NHIG5.76 (1.10)6.61 (1.64)9.38 (4.16)VGSSD6.04 (1.42)6.63 (1.55)9.23 (4.26)NIGSSD4.06 (1.01)5.13 (1.51)8.32 (4.49)VG-OU-Γ5.25 (1.12)6.24 (1.61)8.78 (4.19)NIG-OU-Γ4.81 (0.966)5.97 (1.57)8.59 (4.33)Market4.63 (0.502)4.97 (1.28)4.52 (0.569)Average percentage error (in-sample, out-of-the-money options).
(a)(b)Average percentage error (1-day-ahead out-of-sample, out-of-the-money options).
(a)(b)Average percentage error (1-week-ahead out-of-sample, out-of-the-money options).
(a)(b)Table10 contains values of the mean ratio of APE,
taken over all twelve-quote dates, for a model of one class to that of another
with a common associated distribution (VG or NIG). As shown in
the sixth row of Table 10, the NIGSSD model outperformed its model benchmark,
with APE values which were, on average, 84% and 86% of those obtained by the
NIG-OU-Γ model for in-sample and one-day-ahead
out-of-sample cases, respectively. In contrast, the VGSSD model, on average,
had pricing errors which were 115% and 107% of those obtained by the VG-OU-Γ model for in-sample and one-day-ahead
out-of-sample cases, respectively. Below is a summary of the NIGSSD pricing
performance statistics taken from Figures 1–3.Table 10
Model class comparison: twelve quote date time-averaged ratio of average percentage errors
(January–December 2005). Standard deviation is given in parentheses.
Model 1: model 2〈APE1APE2〉In-sample1-day-ahead1-week-aheadVHG: VG0.622 (0.139)0.625
(0.126)0.728
(0.219)NHIG: NIG0.576 (0.128)0.585
(0.105)0.745
(0.255)VGSSD: VHG0.829 (0.0894)0.841
(0.0525)0.847
(0.125)NIGSSD: NHIG0.704 (0.124)0.776
(0.0900)0.847
(0.134)VGSSD: VG-OU-Γ1.15 (0.102)1.07
(0.0902)1.05
(0.129)NIGSSD: NIG-OU-Γ0.838 (0.109)0.858
(0.0635)0.947
(0.133)(i) The proportions of quote dates in which the
NIGSSD model had a lower APE than that of the market for the in-sample, one-day-ahead, and one-week-ahead out-of-sample cases were9/12, 6/12,
and 2/12,
respectively.(ii) The proportions of quote dates in which the
NIGSSD model had a lower APE than those of both stochastic volatility models
for the in-sample, one-day-ahead, and one-week-ahead out-of-sample cases were11/12, 12/12,
and 8/12,
respectively.


## 6.3. Performance Comparison of Additive Classes

Rows one and
two of Table10 indicate the benefits of time inhomogeneity, in that the
in-sample pricing error of the subordinated additive class was approximately
60% of that corresponding to the Lévy class. The benefit of self-similarity in
the log-return process over self-similarity of the directing process is
observed via rows three and four of Table 10. For the in-sample case, the NIGSSD
model (mean ratio of 0.70) performed considerably better against its
subordinated additive counterpart than did the VGSSD model (mean ratio of
0.83). Such differences in mean error ratio decreased with time lag, such that,
regardless of time-one distribution, the one-week-ahead values were
approximately 73% and 85% for the [subordinated additive]:[Lévy] and
[self-similar]:[subordinated additive] comparisons, respectively.


## 6.4. Performance Comparison of Nig and Vg Models

In each row of Table11 a model class is fixed, and
the mean ratio (taken over all twelve-quote dates) of APE of the NIG model to
that of the corresponding VG model is given. The mean ratio
of in-sample APE of the NIG (resp., NHIG, NIGSSD) model to that of the VG (resp.,
VHG, VGSSD) model was approximately 86% (resp., 80%, 67%). Furthermore, the
variation in mean APE ratio across additive model class decreased with time
lag. As shown in rows one through three of the third column of Table 11, the
one-week-ahead out-of-sample mean ratios were approximately 87%, regardless of
the additive model class under consideration. Finally, the consistency with
which the NIG distributions outperformed those of the VG family may be observed
in the following summary statistics taken from Figures 1–3. The proportions of quote dates in which each
of the four models of the NIG family had a lower APE than that of the
corresponding member of the VG family for in-sample, one-day-ahead, and one-week-ahead out-of-sample cases were 12/12, 11/12,
and 8/12,
respectively.Table 11
Variance gamma and normal inverse
Gaussian distribution comparison: twelve quote date time-averaged ratio of
average percentage errors (January–December 2005). Standard deviation is given in parentheses.
Model 1: model 2〈APE1APE2〉In-sample1-day-ahead1-week-aheadNIG: VG0.857
(0.0179)0.890
(0.0528)0.867
(0.0782)NHIG: VHG0.796
(0.0575)0.837
(0.0473)0.873
(0.0926)NIGSSD: VGSSD0.670
(0.0590)0.773
(0.0984)0.873
(0.109)NIG-OU-Γ:
VG-OU-Γ0.921
(0.0585)0.957
(0.0449)0.969
(0.0532)


## 6.5. Model Price Behavior across Strikes and Maturities

In Figures4–7 the market and model prices are
plotted versus moneyness (strike/underlying) for the set of options available on April 12, 2005.
Results for this month are displayed in order to provide a typical observation
of option pricing errors for the self-similar NIG model. Figures 4 and 5 consist
of the plots for the VG family of models, while Figures 6 and 7 consist of the
plots for the NIG family. We define the relative model price difference for two
models corresponding to a given option as the absolute value of the difference
between model prices divided by the market option price. The relative model
price differences between the Lévy and subordinated additive models were the
greatest for deep out-of-the-money puts belonging to the two nearest term
maturities (see Figures 4 and 6). The relative model price differences between
the subordinated additive and self-similar additive models were the greatest
among the out-of-the-money calls of the nearest two and longest two maturities,
as shown in Figures 4, 5 and 6, 7. This same observation is true for relative price
differences between the self-similar and stochastic volatility models, as shown
in Figures 5 and 7.Market and model option prices versus moneyness (strike/underlying)
for VG and VHG models: April 2005 quote date.
(a)(b)Market and model option prices versus moneyness (strike/underlying)
for VGSSD and VG-OU-gamma models: April 2005 quote date.
(a)(b)Market and model option prices versus moneyness (strike/underlying)
for NIG and NHIG models: April 2005 quote date.
(a)(b)Market and model option prices versus moneyness (strike/underlying)
for NIGSSD and NIG-OU-gamma models: April 2005 quote date.
(a)(b)


## 6.6. Term Structure of Moments

The first four moments of the risk-neutral
distributions determined from the options on the April 12, 2005 quote date are
plotted versus maturity in Figure8. For the set of parameters chosen by the
market in the subordinated additive models, the skewness (resp., kurtosis) is
decreasing (resp., increasing) with maturity. This is contrary to the case of
the Lévy models in which the skewness (resp., kurtosis) is increasing (resp.,
decreasing) with maturity. Self-similar processes have one-dimensional marginal
distributions with a rather striking constant skewness and kurtosis. Invariance
with respect to maturity is due to the fact that the characteristic function at
time t is obtained by composition of the
characteristic function at time one with the map ξ↦tHξ.
When the chain rule is applied to the composition, a factor of tH is produced. Consequently, the moment number
is matched by the same number of factors of tH, thereby allowing cancelation of maturity in
the calculation of skewness and excess kurtosis.Risk-neutral mean, variance, skewness,
and kurtosis term structures for April 12, 2005 quote date (VG models—top row) (NIG models—bottom row).
(a)(b)(c)(d)(e)(f)(g)(h)


## 6.7. Hurst Exponent of Additive Models

Tables12 and 13 contain the time-averaged parameter
estimates, taken over the twelve-quote dates, for each additive model. The
sample means of H were approximately 0.6 and 0.9 for the self-similar and subordinated additive
classes, respectively. Furthermore, the relative difference between sample
means of H in the two self-similar models was a mere 0.091%,
compared with 8.9% in the subordinated additive case. In the next
section, we will see that the modest variation of Hurst exponent in self-similar
models during the year 2005 is not maintained over the course of the first half
of the following year.Table 12
Variance gamma
family: twelve quote date time-averaged parameter estimates (January–December
2005). Standard deviation is given in parentheses.
ModelParameterVGVHGVGSSDμ−0.1143 (0.03232)−0.1320 (0.02859)−0.09432 (0.01381)σ0.1227  (0.005445)0.1105  (0.009046)0.1330  (0.006274)ν0.3818  (0.1008)0.9515 (0.1997)0.9713 (0.1405)H0.9566  (0.08177)0.6251  (0.02720)Table 13
Normal
inverse Gaussian family: twelve quote date time-averaged parameter estimates
(January–December 2005). Standard deviation is given in parentheses.
ModelParameterNIGNHIGNIGSSDβ−6.788 (2.134)−30.47  (23.33)−6.919 (1.681)δ0.1732 (0.03245)0.1134  (0.01260)0.1413  (0.01566)γ11.28 (1.764)15.81  (7.157)8.701  (1.005)H0.8754  (0.1236)0.6245  (0.02648)


## 6.8. Comparison with Other Studies

Recall in Table9 the mean in-sample APEs for the
self-similar VG and NIG models were 6.04% and 4.06%, respectively. These values
are in contrast to the respective mean values of 3.19% and 3.02% obtained by
CGMY  [8] and
3.31% and 3.38%, obtained by Madan and Eberlein [28]. In both of these studies,
self-similar models were calibrated using out-of-the-money SPX options. In the
CGMY study, options were used with quotes occurring during the years 2000 and
2001, and with maturities ranging from approximately one month to 15 months.
The data in the study by Madan and Eberlein consisted of options quoted on
May 17, 2006, with maturities ranging from one month to two years.In order to investigate the influence of the choice of
weighting scheme and maturity range on pricing error, we calibrate the VGSSD
and NIGSSD models using out-of-the-money SPX options quoted on May 17, 2006.
The same data filters are used as in Section5, but with varying maturity
ranges and weighting methods. The maturity ranges are one month to one year in
the first case, and one month to two years in the second. The weighting methods
consist of the reciprocal vega2 method, described in Section 5, and a uniform
weighting across both strike and maturity. In each calibration the sum of
weighted squared differences between model and market prices is minimized in
order to obtain the parameter estimates. In Tables 14–17, an asterisk is used
to identify a value which is determined using the same weighting scheme and
maturity range as described in Section 5.Table 14
Self-similar
variance gamma model: percentage error comparison with respect to maturity
range and weighting method for May 17, 2006 quote date.
VGSSD: APE [%]Weighting MethodRange of maturityτ[yrs]1/vega2uniformτ∈(112,1)4.47*3.74τ∈(112,2)5.183.58Table 15
Self-similar
normal inverse Gaussian model: percentage error comparison with respect to
maturity range and weighting method for May 17, 2006 quote date.
NIGSSD: APE [%]Weighting Methodrange of maturityτ[yrs]1/vega2uniformτ∈(112,1)4.78*3.91τ∈(112,2)5.163.57Table 16
Self-similar
variance gamma model: Hurst exponent estimate comparison with respect to
maturity range and weighting method for May 17, 2006 quote date.
VGSSD: Hurst exponentH:Weighting Methodrange of maturityτ[yrs]1/vega2uniformτ∈(112,1)0.4916*0.5201τ∈(112,2)0.50850.5440Table 17
Self-similar
normal inverse Gaussian model: Hurst exponent estimate comparison with respect
to maturity range and weighting method for May 17, 2006 quote date.
NIGSSD: Hurst exponentH:Weighting Methodrange of maturityτ[yrs]1/vega2uniformτ∈(112,1)0.4921*0.5214τ∈(112,2)0.50930.5451The pricing errors for the VGSSD and NIGSSD models are
reported in Tables14 and 15,
respectively. Interestingly, the differences in weighting method and maturity
range may account for as much as a 1.2% deviation in APE (NIGSSD case) from the pricing error obtained with the conventions of Section 5. Furthermore,
an increase in maturity range resulted in a decrease in APE under the uniform
weighting method and an increase in APE under the reciprocal vega2 weighting method. The behavior of pricing
error with maturity range, for a fixed reciprocal vega2 weighting scheme, is consistent with the fact
that this method places greater weight on nearer-term options.The estimate of the Hurst exponent in the VGSSD model,
obtained with options available on May 17, 2006, is reported in Madan and
Eberlein [28] as 0.5326.
In Tables 16 and 17 are the estimates of the Hurst exponent, H, for the VGSSD
and NIGSSD models, respectively. The estimate for H, denoted by (*) in Table 16, was 0.492.
This is in contrast to the value of 0.660 obtained for the VGSSD model on the last quote
date of this study (December 2005), as well as the time-averaged value of 0.625 in Table 12. The use of uniform weights and a
two-year maturity range resulted in an estimate of 0.544 for H,
as opposed to the estimate of 0.492,
obtained using the conventions of Section 5. Since parameter estimates for the
NIGSSD model were not presented in [28], only the VGSSD values are compared.


## 7. Conclusion

In this paper, we considered additive processes
defined via subordination, Sato's construction ofH-self-similar additive
processes from self-decomposable distributions, and a combination of the two
methods. For each of these three methods, or model classes, two processes were
defined: one with a time-one marginal variance gamma (VG) distribution, and the
other, a time-one marginal normal inverse Gaussian (NIG) distribution. The
corresponding models for the price process were calibrated, along with those
constructed from two benchmark Lévy stochastic volatility processes, using
out-of-the-money Standard and Poor's 500 index options from the year 2005. The
four-parameter subordinated additive models had in-sample pricing errors which
were approximately 60% (pooled across time and time-one distribution) of those
obtained by the corresponding three-parameter Lévy models. The four-parameter
self-similar additive models had in-sample pricing errors which were
approximately 77% (pooled across time and time-one distribution) of those
obtained by the corresponding subordinated additive models. The results for the
comparisons between self-similar and stochastic volatility models were mixed.
The in-sample pricing errors of the self-similar VG and NIG models were 115%
and 84% of those obtained by the VG and NIG Lévy stochastic volatility models,
respectively. For a given model class, the normal inverse Gaussian model
consistently outperformed its variance gamma counterpart for in-sample and one-day-ahead out-of-sample cases. The Hurst exponent estimates associated with the
self-similar and subordinated classes were approximately 0.6 and 0.9,
respectively. Although the estimates of Hurst exponent of self-similar models
had a modest range during the year 2005, Hurst exponent estimates can vary
greatly over the course of a year, as demonstrated in Section 6.8.



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*Source: 397028-2009-03-25.xml*