Table of Contents
Journal of Applied Mathematics and Decision Sciences
Volume 2008, Article ID 397028, 30 pages
http://dx.doi.org/10.1155/2008/397028
Research Article

Subordination, Self-Similarity, and Option Pricing

1Department of Statistics and Applied Probability, University of California, Santa Barbara, CA 93106-3110, USA
2Department of Mathematics, The Florida State University, Tallahassee, FL 32306-4510, USA

Received 17 February 2008; Accepted 27 October 2008

Academic Editor: Henry Schellhorn

Copyright © 2008 Mack L. Galloway and Craig A. Nolder. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 generalized 𝑤

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 the H-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.

tab001
Table 1

The 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 Section 2 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. An 𝑑-valued stochastic process {𝑋𝑡𝑡0} on (Ω,,) is an additive process if

(i)𝑋0=0 a.s.(ii)For any choice of 𝑛 and 0𝑡0<𝑡1<<𝑡𝑛, the random variables 𝑋𝑡0,𝑋𝑡1𝑋𝑡0,𝑋𝑡2𝑋𝑡1,,𝑋𝑡𝑛𝑋𝑡𝑛1 are independent.(iii){𝑋𝑡} is stochastically continuous.(iv)There exists a Ω0 with (Ω0)=1 such that, for every 𝜔Ω0,𝑋𝑡(𝜔) is right-continuous in 𝑡0 and has left limits in 𝑡>0.

Remark 2.2. A Lévy process is a time-homogeneous additive process—an additive process which satisfies 𝑋𝑡+𝛿𝑋𝑡𝑑=𝑋𝛿𝑡,𝛿0.(2.1)

Definition 2.3. A stochastic process {𝑋𝑡𝑡0} is self-similar if, for any 𝑎>0, there exists 𝑏>0, such that {𝑋𝑎𝑡𝑡0} and {𝑏𝑋𝑡𝑡0} are identical in law.

Furthermore, if {𝑋𝑡𝑡0} is an 𝑑-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. For 𝑑1 a probability measure 𝜇 on 𝑑 is self-decomposable if, for all 𝑏>1, there exists a probability measure 𝜌𝑏 on 𝑑 such that 𝑧𝑧𝜇=𝜇𝑏𝜌𝑏𝑧,(2.2) where 𝜇(𝑧) is the characteristic function of 𝜇.

Since a self-decomposable distribution 𝜇 is infinitely divisible, it generates a Lévy process {𝑋𝑡𝑡0}. In this case, the characteristic function of the distribution of 𝑋𝑡 is 𝜇𝑡(𝑧) for all 𝑡0 [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]). If 𝜇 is a nontrivial self-decomposable distribution on 𝑑, then, for any 𝐻>0, there exists, uniquely in law, a nontrivial 𝐻-self-similar additive process {𝑋𝑡𝑡0} such that 𝑋1=𝜇.

Remark 2.6 (Sato [5, Proof of Theorem 16.1, page 100]). In the previous theorem, the characteristic function of 𝑋𝑡 is given by Φ𝑋𝑡𝜉𝑡=𝜇𝐻𝜉,𝑡0,𝜉.(2.3)

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 measure 𝜇 on is self-decomposable if and only if 𝑧1𝜇=exp2𝐴𝑧2+𝑖𝛾𝑧+𝑒𝑖𝑧𝑥1𝑖𝑧𝑥11,1𝑥𝑘𝑥||𝑥||𝑑𝑥,(2.4) where 𝐴0,𝛾,𝑘(𝑥)0,(|𝑥|21)(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. A 𝑘-function of the self-decomposable distribution 𝜇 [5, page 403] is an +-valued function on {0} which is decreasing and left-continuous on + and which is increasing and right-continuous on with the Lévy density of 𝜇 given by 𝑘(𝑥)/|𝑥|,𝑥{0}.

The notation below will be used in the theorems and derivations which follow.

Notation 2.9. Given an -valued process 𝑋{𝑋𝑡𝑡0}, we define the following: Characteristicfunction:Φ𝑋𝑡𝜉𝑒=𝐸𝑖𝜉𝑋𝑡,𝜉Characteristicexponent:𝜂𝑋𝑡𝜉Φ=log𝑋𝑡𝜉,𝜉.(2.5)
If 𝑋 is +-valued, then we also have Laplaceexponent:Ψ𝑋𝑡𝜉𝐸𝑒=log𝜉𝑋𝑡𝜉,0.(2.6)

Remark 2.10. Given a Lévy process 𝑋 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 𝐸𝑒𝑖𝜉𝑋𝑍𝑡Ψ=exp𝑍𝑡𝜂𝑋1𝜉,𝜉.(2.7) 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. On the probability space (Ω,,) let 𝑋{𝑋𝑡𝑡0} be an -valued Brownian motion, where 𝑋1 is normal(𝜇,𝜎2) distributed. On the same probability space let 𝑍{𝑍𝑡𝑡0} be an +-valued, independent gamma process where, for any 𝑡0, 𝑍𝑡 is gamma(𝑡/𝜈,1/𝜈) distributed. The variance gamma process, {VG𝑡}, can be constructed as VG𝑡𝜔=𝑋𝑍𝑡𝜔𝜔,𝑡0,𝜔Ω,(2.8)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, Ψ𝑍𝑡1/𝜈,1/𝜈𝜉=log1𝜉𝜈𝑡/𝜈𝜉,for𝑡0,0(2.9) are used to obtain the characteristic function of the random variable VG𝑡 as follows: 𝐸𝑒𝑖𝜉VG𝑡Ψ=exp𝑍𝑡𝜂𝑋1𝜉=explog1𝜂𝑋1𝜉𝜈𝑡/𝜈=11+𝜈𝑖𝜇𝜉+2𝜎2𝜉2𝑡/𝜈.(2.10)

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. On the probability space (Ω,,) let 𝑋{𝑋𝑡𝑡0} be an -valued Brownian motion, where 𝑋1 is normal(𝛽𝛿2,𝛿2) distributed. On the same probability space let 𝑍{𝑍𝑡𝑡0} be an +-valued, independent inverse Gaussian process where, for any 𝑡0, 𝑍𝑡 is IG(𝑡,𝛿𝛾) distributed. The normal inverse Gaussian process, {NIG𝑡}, can be constructed as NIG𝑡𝜔=𝑋𝑍𝑡𝜔𝜔,𝑡0,𝜔Ω.(2.11) Since the Laplace exponent of 𝑍𝑡 is given by Ψ𝑍𝑡1,𝛿𝛾𝜉=𝑡{2𝜉+𝛿𝛾2𝜉𝛿𝛾},for𝑡0,0,(2.12) we have, similar to the previous case, the characteristic function of the distribution of NIG𝑡: 𝐸𝑒𝑖𝜉NIG𝑡Ψ=exp𝑍𝑡𝜂𝑋1𝜉=exp(𝑡{2𝜂𝑋1𝜉+𝛿𝛾2𝛿𝛾})=exp(𝛿𝑡{𝜉22𝑖𝛽𝜉+𝛾2𝛾}).(2.13)

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 Section 2.1, 𝑋1 is Normal(𝜇,𝜎2) distributed while 𝑍𝑡 is gamma(𝑡/𝜈,1/𝜈) distributed for all 𝑡0. 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 𝜚VG1𝑥=0,𝑑𝜇𝑋𝑠𝑥𝜚𝑑𝑥𝑍1𝑠=𝑑𝑠0,12𝜋𝜎2𝑠exp𝑥𝜇𝑠22𝜎2𝑠1𝑠𝜈𝑠exp𝜈=1𝑑𝑠2𝜋𝜎𝜈exp𝑥𝜇𝜎2[2(𝜇2/𝜎2+2/𝜈||𝑥||𝜎)1/2𝐾1/2(||𝑥||𝜎𝜇2𝜎2+2𝜈=1)]𝜈||𝑥||1exp(𝜎[𝜇2𝜎2+2𝜈||𝑥||𝜇𝜎𝑥]),(2.14) 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. 0,𝛼exp2𝑠2𝛽2𝑠2𝑠1+̃𝜈𝛼𝑑𝑠=2𝛽̃𝜈𝐾̃𝜈𝛼𝛽.(2.15) Identity (2.16) [29, page 109] is used in the final step. 𝐾1/2𝑧=𝜋𝑒2𝑧𝑧,𝑧>0.(2.16) The product given by 𝑥|𝑥|𝜚VG1(𝑥) is positive, decreasing on + and increasing on if 𝜈>0. By Theorem 2.7, the k-function given by 𝑘𝑥=1𝜈1exp(𝜎[𝜇2𝜎2+2𝜈||𝑥||𝜇𝜎0𝑥]),𝑥(2.17) 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].

tab1
Table 2: Lévy densities of subordinator distributions.

Normal Inverse Gaussian Case
Using the same construction as in Section 2.1 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 𝜚NIG1𝑥=0,𝑑𝜇𝑋𝑠𝑥𝜚𝑑𝑥𝑍1𝑠=𝑑𝑠0,12𝜋𝛿2𝑠exp(𝑥𝛽𝛿2𝑠22𝛿2𝑠)1𝑠2𝜋3/21exp2𝛿2𝛾2𝑠=1𝑑𝑠2𝜋𝛿exp𝛽𝑥0,𝑥exp([2𝛿2+𝛽2𝑠2+𝛾2𝛿22𝑠])𝑠2𝑑𝑠.(2.18) Using (2.15) with ̃𝜈=1 yields 𝜚NIG1𝑥=1𝛿2𝜋𝛿exp𝛽𝑥[2(𝛽2+𝛾2||𝑥||/𝛿)𝐾1𝛿𝛽2+𝛾2||𝑥||𝛿]=𝛿𝛽2+𝛾2𝜋1||𝑥||𝐾exp𝛽𝑥1(𝛽2+𝛾2||𝑥||0.),𝑥(2.19) Since 𝑥|𝑥|𝜚NIG1(𝑥) is decreasing on + and increasing on if and only if 𝑥𝑒(𝛽/𝛽2+𝛾2)𝑥𝐾1(|𝑥|) is decreasing on + and increasing on , we make use of identity (2.20) [30, equation (9.6.24)]: 𝐾̃𝜈𝑡=0,𝑒𝑡cosh𝑧𝑡cosh̃𝜈𝑧𝑑𝑧,>0.(2.20) We have, in a manner similar to Laforgia [31], 𝑒(𝛽/𝛽2+𝛾2)𝑥𝐾1||𝑥||=0,𝑒||𝑥||cosh𝑧+𝑥(𝛽/𝛽2+𝛾2)𝑧=cosh𝑑𝑧0,𝑒||𝑥||cosh𝑧signum𝑥(𝛽/𝛽2+𝛾2)𝑧0.cosh𝑑𝑧,𝑥(2.21) It follows that the product of 𝜚NIG1 and |𝑥| is positive, decreasing on + and increasing on for all 𝛿>0,𝛽. By Theorem 2.7, the k-function given by 𝑘𝑥=𝛿𝛽2+𝛾2𝜋𝐾exp𝛽𝑥1(𝛽2+𝛾2||𝑥||0),𝑥(2.22) 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 of 𝐻-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, and 𝑚=VG1 be the self-decomposable variance gamma(𝜎,𝜇,𝜈) distribution on . By Theorem 2.5 and Remark 2.6, the characteristic function of VGSSD𝑡 is given by 𝐸𝑒𝑖𝜉VGSSD𝑡𝑡=𝑚𝐻=𝜉;𝜎,𝜇,𝜈1+𝜈𝑖𝜇𝑡𝐻1𝜉+2𝜎2𝑡2𝐻𝜉21/𝜈.(2.23)

For the characteristic function of the distribution of NIGSSD𝑡, let 𝛿,𝛾,𝐻>0,𝛽, and 𝑚=NIG1 be the self-decomposable normal inverse Gaussian(𝛿,𝛽,𝛾) distribution on . Similar to the previous case, the characteristic function of the distribution of NIGSSD𝑡 is given by 𝐸𝑒𝑖𝜉NIGSSD𝑡𝑡=𝑚𝐻𝜉;𝛽,𝛿,𝛾=exp(𝛿{𝑡2𝐻𝜉22𝑖𝛽𝑡𝐻𝜉+𝛾2𝛾}).(2.24) 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: 𝑘Γ𝑥𝑘=𝑎exp𝑏𝑥,𝑥>0,IG𝑥=12𝜋𝑎𝑥1/21exp2𝑏2𝑥,𝑥>0(2.25) verify self-decomposability of the time-one distributions of the gamma and inverse Gaussian processes, respectively.

The characteristic function of the distribution of VHG𝑡 is determined as follows. Let 𝜎,𝜈,𝐻>0, and 𝜇. On the probability space (Ω,,) let 𝑋{𝑋𝑡𝑡0} be an -valued Brownian motion where 𝑋1 is normal(𝜇,𝜎2) distributed. On the same probability space, using Theorem 2.5, define 𝑍{𝑍𝑡𝑡0} 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 VHG𝑡𝜔=𝑋𝑍𝑡𝜔𝜔,𝑡0,𝜔Ω.(2.26) Using (2.7) in Remark 2.10, Remark 2.6, and the Laplace exponent of 𝑍1, Ψ𝑍11/𝜈,1/𝜈𝜉=log1𝜉𝜈1/𝜈𝜉,0,(2.27) the characteristic function of the distribution of the random variable VHG𝑡 is given by 𝐸𝑒𝑖𝜉VHG𝑡Ψ=exp𝑍𝑡𝜂𝑋1𝜉Ψ=exp𝑍1𝑡𝐻𝜂𝑋1𝜉=explog1𝑡𝐻12𝜎2𝜉2𝜈+𝑖𝜇𝜉1/𝜈=1+𝑡𝐻𝜈1𝑖𝜇𝜉+2𝜎2𝜉21/𝜈.(2.28) Consequently, the distributions given by (2.10), (2.23), and (2.28) all agree at time one.

The characteristic function of the distribution of NHIG𝑡 is similarly defined. Fix 𝛿,𝛾,𝐻>0, and 𝛽. On the probability space (Ω,,) define 𝑋{𝑋𝑡𝑡0} to be an -valued Brownian motion where 𝑋1 is normal(𝛽𝛿2,𝛿2) distributed. On the same probability space, using Theorem 2.5, define 𝑍{𝑍𝑡𝑡0} 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 NHIG𝑡𝜔=𝑋𝑍𝑡𝜔𝜔,𝑡0,𝜔Ω.(2.29) The Laplace exponent of 𝑍1 is given by Ψ𝑍11,𝛿𝛾𝜉={2𝜉+𝛿𝛾2𝜉𝛿𝛾},0.(2.30) Similar to the previous case, the characteristic function of the distribution of NHIG𝑡 is obtained as follows: 𝐸𝑒𝑖𝜉NHIG𝑡Ψ=exp𝑍𝑡𝜂𝑋1𝜉Ψ=exp𝑍1𝑡𝐻𝜂𝑋1𝜉=exp({2𝑡𝐻𝜂𝑋1𝜉+𝛿𝛾2𝛿𝛾})=exp(𝛿{𝑡𝐻𝜉22𝑖𝛽𝜉+𝛾2𝛾}).(2.31) Note that (2.31) agrees at time one with (2.13) and (2.24).

3. Increment moments

Given an -valued additive process 𝑌{𝑌𝑡} on (Ω,,), let 𝜇𝑌𝑡𝑌𝑠 denote the distribution of 𝑌𝑡𝑌𝑠 where 0𝑠<𝑡. By additivity of 𝑌, the characteristic function of the increment is given by 𝜇𝑌𝑡𝑌𝑠𝜉=𝜇𝑌𝑡𝜉𝜇𝑌𝑠𝜉,0𝑠<𝑡,(3.1) [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 𝑌𝑡𝑌𝑠 are calculated for each of the previously defined additive processes in Tables 38.

tab2
Table 3: Variance gamma process: increment moments.
tab3
Table 4: Normal inverse Gaussian process: increment moments.
tab4
Table 5: Self-similar variance gamma process: increment moments.
tab5
Table 6: Self-similar normal inverse Gaussian process: increment moments.
tab6
Table 7: Variance𝐻-gamma process: increment moments.
tab7
Table 8: Normal𝐻-inverse Gaussian process: increment moments.

An interesting property among the three classes of additive processes is the dependence of the variance of the increment 𝑌𝑡+𝛿𝑌𝑡 on time 𝑡 (for any 𝑡0 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 𝑋Var𝑡+𝛿𝑋𝑡=𝑡+𝛿2𝐻𝑡2𝐻𝑋Var1,(3.2) 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 𝑋Var𝑍𝑡+𝛿𝑋𝑍𝑡=𝑡+𝛿2𝐻𝑡2𝐻𝐸𝑋12𝑍Var1+𝑡+𝛿𝐻𝑡𝐻𝐸𝑍1𝑋Var1,(3.3) 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 𝐻1/2 (resp., 𝐻1). 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

Let 𝑋{𝑋𝑡}𝑡[0,𝑇] be an additive process on the stochastic basis (Ω,,𝔽={𝑡},), 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)𝑆𝑡=𝑆0𝑋exp𝑡+𝑡𝑟𝑞𝔼𝑒𝑋𝑡foreach𝑡0,𝑇,(4.1) 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(𝑟𝑞)𝑡𝑆𝑡} is an (𝔽) martingale.

The characteristic functions of the time-𝑡 distributions of the logarithm of the underlying for the additive processes defined in Sections 2.12.3 are given below:

(i)exponential variance gamma ΦVGln𝑆𝑡𝜉=exp𝑖𝜉ln𝑆0+𝑡𝑟𝑞(𝑡/𝜈)log1+𝜈𝑖𝜇𝜉+(1/2)𝜎2𝜉21𝜈𝜇+(1/2)𝜎2𝑡/𝜈𝑖𝜉;(4.2)(ii)exponential 𝐻-self-similar variance gamma ΦVGSSDln𝑆𝑡𝜉=exp𝑖𝜉ln𝑆0+𝑡𝑟𝑞(1/𝜈)log1+𝜈𝑖𝜇𝑡𝐻𝜉+(1/2)𝜎2𝑡2𝐻𝜉21𝜈𝜇𝑡𝐻+(1/2)𝜎2𝑡2𝐻1/𝜈𝑖𝜉;(4.3)(iii)exponential variance 𝐻-gammaΦVHGln𝑆𝑡𝜉=exp𝑖𝜉ln𝑆0+𝑡𝑟𝑞(1/𝜈)log1+𝑡𝐻𝜈𝑖𝜇𝜉+(1/2)𝜎2𝜉21𝑡𝐻𝜈𝜇+(1/2)𝜎21/𝜈𝑖𝜉;(4.4)(iv)exponential normal inverse Gaussian ΦNIGln𝑆𝑡𝜉=exp{𝑖𝜉ln𝑆0+𝑡𝑟𝑞𝛿𝑡[𝜉22𝑖𝛽𝜉+𝛾2𝛾]}[exp(𝛿𝑡[12𝛽+𝛾2𝛾])]𝑖𝜉;(4.5)(v)exponential 𝐻-self-similar normal inverse GaussianΦNIGSSDln𝑆𝑡𝜉=exp{𝑖𝜉ln𝑆0+𝑡𝑟𝑞𝛿[𝑡2𝐻𝜉22𝑖𝛽𝑡𝐻𝜉+𝛾2𝛾]}[exp(𝛿[𝑡2𝐻2𝛽𝑡𝐻+𝛾2𝛾])]𝑖𝜉;(4.6)(vi)exponential normal 𝐻-inverse Gaussian ΦNHIGln𝑆𝑡𝜉=exp{𝑖𝜉ln𝑆0+𝑡𝑟𝑞𝛿[𝑡𝐻𝜉22𝑖𝛽𝜉+𝛾2𝛾]}[exp(𝛿[𝑡𝐻12𝛽+𝛾2𝛾])]𝑖𝜉.(4.7)

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 𝑍𝑡=𝑡0𝑦𝑠𝑑𝑠,(4.8) where, for 𝜆>0, 𝑦0>0, and subordinator {𝐿𝑡}, {𝑦𝑡} satisfies the Ornstein-Uhlenbeck type SDE [5] 𝑦𝑡=𝑦0+𝐿𝑡𝜆𝑡0𝑦𝑠𝑑𝑠for𝑡0.(4.9) The solution of (4.9) is given by 𝑦𝑡=𝑦0𝑒𝜆𝑡+𝑡0𝑒𝜆𝑡𝑠𝑑𝐿𝑠.(4.10) We fix 𝑎,𝑏>0 and specify {𝐿𝑡} to be a compound Poisson process with exponentially distributed jump sizes where Ψ𝐿1𝜉=𝜆𝑎𝜉𝜉𝑏𝜉,0.(4.11) 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: Φ𝑍𝑡𝜉;𝜆,𝑎,𝑏,𝑦0=exp𝑖𝑦0𝜆11𝑒𝜆𝑡𝜉+𝜆𝑎𝑏𝑖𝜉𝜆𝑏𝑏log𝑏𝑖𝜆11𝑒𝜆𝑡𝜉𝑖𝑡𝜉,𝜉(4.12) (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 𝑡0 and 𝜔 in the sample space VG-OU-Γ𝑡𝜔=𝑋𝑍𝑡𝜔𝜔,𝑋𝑡isaVGprocess,(4.13)NIG-OU-Γ𝑡𝜔=𝑋𝑍𝑡𝜔𝜔,𝑋𝑡isaNIGprocess.(4.14)

Let {𝑋𝑍𝑡}𝑡[0,𝑇] be an -valued Lévy stochastic volatility process on (Ω,,𝔽,) as in (4.13) or (4.14). The underlying price process at time 𝑡 is defined, as in [3, 4], to satisfy 𝑆𝑡𝑑=𝑆0𝑋exp𝑍𝑡+𝑡𝑟𝑞𝔼𝑒𝑋𝑍𝑡foreach𝑡0,𝑇,(4.15) where {𝑒𝑡(𝑟𝑞)𝑆𝑡} is a (𝔽) martingale on another filtered probability space (Ω,,𝔽,). 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
For 𝜎,𝜈>0, and 𝜇, define the parameters 𝐶,𝐺,𝑀 as in [3]:1𝐶=𝜈,𝐺=14𝜇2𝜈2+12𝜎21𝜈2𝜇𝜈1,𝑀=14𝜇2𝜈2+12𝜎21𝜈+2𝜇𝜈1.(4.16) The characteristic function of the time-one marginal of the VG process is given by ΦVG1=𝜉;𝐶,𝐺,𝑀𝐺𝑀𝐺𝑀+𝑖𝑀𝐺𝜉+𝜉2𝐶.(4.17) The number of parameters in (4.13) is reduced from seven to six via ΦVG-OU-Γ𝜉;𝐶,𝐺,𝑀,𝜆,𝑎,𝑏,𝑦0=ΦVG-OU-Γ𝜉;𝐶𝑦0,𝐺,𝑀,𝜆,𝑎,𝑏𝑦0,1(4.18) (see [4]). It follows that the characteristic function of the time-𝑡 distribution of the logarithm of the underlying is given by ΦVG-OU-Γln𝑆𝑡𝜉=exp𝑖𝜉ln𝑆0Φ+𝑡𝑟𝑞𝑍𝑡𝜉𝑖log𝜑;𝜆,𝑎,𝑏,1Φ𝑍𝑡𝑖log𝜑𝑖;𝜆,𝑎,𝑏,1𝑖𝜉,(4.19) where 𝜑𝜉=ΦVG1𝜉;𝐶,𝐺,𝑀(4.20) (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 ΦNIG1𝜉;𝛽,𝛿,𝛾=exp{𝛿[𝜉22𝑖𝛽𝜉+𝛾2𝛾]}.(4.21) The number of parameters in (4.14) is reduced from seven to six via ΦNIG-OU-Γ𝜉;𝛽,𝛿,𝛾,𝜆,𝑎,𝑏,𝑦0=ΦNIG-OU-Γ𝜉;𝛽,𝛿𝑦0,𝛾,𝜆,𝑎,𝑏𝑦0,1(4.22) (see [4]). It follows that the characteristic function of the time-𝑡 distribution of the logarithm of the underlying is given by ΦNIG-OU-Γln𝑆𝑡𝜉=exp𝑖𝜉ln𝑆0Φ+𝑡𝑟𝑞𝑍𝑡𝜉𝑖log𝜑;𝜆,𝑎,𝑏,1Φ𝑍𝑡𝑖log𝜑𝑖;𝜆,𝑎,𝑏,1𝑖𝜉,(4.23) where 𝜑𝜉=ΦNIG1𝜉;𝛽,𝛿,𝛾(4.24) (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 𝜃Θ 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=1𝑁, the set of corresponding model option prices is denoted by {𝜋𝜃𝑖}𝑖=1𝑁 for any 𝜃Θ. In order to estimate 𝜃, the quadratic pricing error, Θ+, is minimized with 𝜃=𝑁𝑖=1𝑤𝑖𝜋𝜃𝑖𝑟,𝑞,𝑆0,𝑇𝑖,𝐾𝑖𝑃𝑖𝑇𝑖,𝐾𝑖2,𝜃Θ,(5.1) 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 9.5×104 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 APE=𝑁𝑖=1|||𝑃𝑖𝜋𝑖|||𝑁𝑖=1𝑃𝑖.(6.1)

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 (askbid)/2 if it is to lie between the bid and ask quotes. Setting the price difference to be (askbid)/2 and the market price to be (ask+bid)/2 in (6.1) yields the measure of uncertainty in market prices given below: APEmarket=𝑁𝑖=1ask𝑖bid𝑖𝑁𝑖=1ask𝑖+bid𝑖.(6.2)

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 Figures 1, 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.

tab8
Table 9: Mean of average percentage error: twelve quote date time average (January–December 2005). Standard deviation is given in parentheses.
fig1
Figure 1: Average percentage error (in-sample, out-of-the-money options).
fig2
Figure 2: Average percentage error (1-day-ahead out-of-sample, out-of-the-money options).
fig3
Figure 3: Average percentage error (1-week-ahead out-of-sample, out-of-the-money options).

Table 10 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 13.

tab9
Table 10: Model class comparison: twelve quote date time-averaged ratio of average percentage errors (January–December 2005). Standard deviation is given in parentheses.

(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 were 9/12, 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 were 11/12, 12/12, and 8/12, respectively.

6.3. Performance Comparison of Additive Classes

Rows one and two of Table 10 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 Table 11 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 13. 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.

tab10
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.
6.5. Model Price Behavior across Strikes and Maturities

In Figures 47 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.

fig4
Figure 4: Market and model option prices versus moneyness (strike/underlying) for VG and VHG models: April 2005 quote date.
fig5
Figure 5: Market and model option prices versus moneyness (strike/underlying) for VGSSD and VG-OU-gamma models: April 2005 quote date.
fig6
Figure 6: Market and model option prices versus moneyness (strike/underlying) for NIG and NHIG models: April 2005 quote date.
fig7
Figure 7: Market and model option prices versus moneyness (strike/underlying) for NIGSSD and NIG-OU-gamma models: April 2005 quote date.
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 Figure 8. 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 𝜉𝑡𝐻𝜉. 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.

fig10
Figure 8: Risk-neutral mean, variance, skewness, and kurtosis term structures for April 12, 2005 quote date (VG models—top row) (NIG models—bottom row).
6.7. Hurst Exponent of Additive Models

Tables 12 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.

tab11
Table 12: Variance gamma family: twelve quote date time-averaged parameter estimates (January–December 2005). Standard deviation is given in parentheses.
tab12
Table 13: Normal inverse Gaussian family: twelve quote date time-averaged parameter estimates (January–December 2005). Standard deviation is given in parentheses.
6.8. Comparison with Other Studies

Recall in Table 9 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 Section 5, 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 1417, an asterisk is used to identify a value which is determined using the same weighting scheme and maturity range as described in Section 5.

tab13
Table 14: Self-similar variance gamma model: percentage error comparison with respect to maturity range and weighting method for May 17, 2006 quote date.
tab14
Table 15: Self-similar normal inverse Gaussian model: percentage error comparison with respect to maturity range and weighting method for May 17, 2006 quote date.
tab15
Table 16: Self-similar variance gamma model: Hurst exponent estimate comparison with respect to maturity range and weighting method for May 17, 2006 quote date.
tab16
Table 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.

The pricing errors for the VGSSD and NIGSSD models are reported in Tables 14 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 of H-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.

Acknowledgments

The authors thank Warren Beaves, as well as Rick Fortier of deltaneutral.com for providing the option data for this study. The authors also thank the referees of JAMDS, Jean-Pierre Fouque, José Figueroa-López, and Janos Englander for their constructive comments and suggestions.

References

  1. P. Carr and L. Wu, “The finite moment log stable process and option pricing,” The Journal of Finance, vol. 58, no. 2, pp. 753–778, 2003. View at Publisher · View at Google Scholar
  2. R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, Fla, USA, 1st edition, 2004. View at Zentralblatt MATH · View at MathSciNet
  3. P. Carr, H. Geman, D. B. Madan, and M. Yor, “Stochastic volatility for Lévy processes,” Mathematical Finance, vol. 13, no. 3, pp. 345–382, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. W. Schoutens, Lévy Processes in Finance: Pricing Financial Derivatives, John Wiley & Sons, West Sussex, UK, 1st edition, 2003.
  5. K. Sato., Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, UK, 1st edition, 2002.
  6. O. Barndorff-Nielsen, “Processes of normal inverse Gaussian type,” Finance and Stochastics, vol. 2, no. 1, pp. 41–68, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. C. A. Nolder, “Selfsimilar additive processes and financial modelling,” in Proceedings of the Financial Engineering and Applications Conference (FEA '03), Banff, Canada, July 2003.
  8. P. Carr, H. Geman, D. B. Madan, and M. Yor, “Self-decomposability and option pricing,” Mathematical Finance, vol. 17, no. 1, pp. 31–57, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  9. J.-M. Courtault, Y. Kabanov, B. Bru, P. Crépel, I. Lebon, and A. Le Marchand, “Louis Bachelier on the centenary of Théorie de la spéculation,” Mathematical Finance, vol. 10, no. 3, pp. 341–353, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. B. Mandelbrot, “The variation of certain speculative prices,” Journal of Business, vol. 36, no. 4, pp. 394–419, 1963. View at Publisher · View at Google Scholar
  11. P. Samuelson, “Rational theory of warrant pricing,” Industrial Management Review, vol. 6, no. 2, pp. 13–39, 1965. View at Google Scholar
  12. C. Sprenkle, “Warrant prices as indicators of expectations and preferences,” Yale Economic Essays, vol. 1, pp. 178–231, 1961. View at Google Scholar
  13. F. Black and M. Scholes, “The pricing of options and corporate liabilities,” The Journal of Political Economy, vol. 81, no. 3, pp. 637–654, 1973. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. R. C. Merton, “Theory of rational option pricing,” The Bell Journal of Economics and Management Science, vol. 4, pp. 141–183, 1973. View at Google Scholar · View at MathSciNet
  15. R. C. Merton, “Option pricing when underlying stock returns are discontinuous,” Journal of Financial Economics, vol. 3, pp. 125–144, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. M. Rubinstein, “Nonparametric tests of alternative option pricing models using all reported trades and quotes on the 30 most active CBOE option classes from August 23, 1976 through August 31, 1978,” The Journal of Finance, vol. 40, no. 2, pp. 455–480, 1985. View at Publisher · View at Google Scholar
  17. D. Bates, “The crash of ‘87: Was it expected? The evidence from options markets,” The Journal of Finance, vol. 46, no. 3, pp. 1009–1044, 1991. View at Publisher · View at Google Scholar
  18. D. B. Madan and E. Seneta, “The variance gamma (V.G) model for share market returns,” The Journal of Business, vol. 63, no. 4, pp. 511–524, 1990. View at Publisher · View at Google Scholar
  19. D. B. Madan, P. Carr, and E. Chang, “The variance gamma process and option pricing,” European Finance Review, vol. 2, no. 1, pp. 79–105, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. W. Schoutens and J. L. Teugels, “Lévy processes, polynomials and martingales,” Stochastic Models, vol. 14, no. 1-2, pp. 335–349, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. S. Boyarchenko and S. Levendorsii, “Option pricing for truncated Lévy processes,” International Journal of Theoretical and Applied Finance, vol. 3, no. 3, pp. 549–552, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. S. Boyarchenko and S. Levendorskiĭ, “Barrier options and touch-and-out options under regular Lévy processes of exponential type,” The Annals of Applied Probability, vol. 12, no. 4, pp. 1261–1298, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. P. Carr, H. Geman, D. B. Madan, and M. Yor, “The fine structure of asset returns: an empirical investigation,” Journal of Business, vol. 75, no. 2, pp. 305–332, 2002. View at Publisher · View at Google Scholar
  24. E. Eberlein and U. Keller, “Hyperbolic distributions in finance,” Bernoulli, vol. 1, no. 3, pp. 281–299, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. O. Barndorff-Nielsen, “Exponentially decreasing distributions for the logarithm of particle size,” Proceedings of the Royal Society of London. Series A, vol. 353, no. 1674, pp. 401–419, 1977. View at Publisher · View at Google Scholar
  26. E. Eberlein and K. Prause, “The generalized hyperbolic model: financial derivatives and risk measures,” in Mathematical Finance—Bachelier Congress, 2000 (Paris), Springer Finance, pp. 245–267, Springer, Berlin, Germany, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. R. Cont and P. Tankov, “Non-parametric calibration of jump-diffusion option pricing models,” Journal of Computational Finance, vol. 7, no. 3, pp. 1–49, 2004. View at Google Scholar
  28. C. Madan and E. Eberlein, “Sato processes and the valuation of structured products,” September 2008, http://www.ssrn.com/.
  29. F. E. Relton, Applied Bessel Functions, Dover, New York, NY, USA, 1st edition, 1965. View at Zentralblatt MATH · View at MathSciNet
  30. M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, M. Abramowitz and I. A. Stegun, Eds., Dover, New York, NY, USA, 9th edition, 1972.
  31. A. Laforgia, “Bounds for modified Bessel functions,” Journal of Computational and Applied Mathematics, vol. 34, no. 3, pp. 263–267, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. D. B. Madan and M. Yor, “Making Markov martingales meet marginals: with explicit constructions,” Bernoulli, vol. 8, no. 4, pp. 509–536, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. P. Carr and D. B. Madan, “A note on sufficient conditions for no arbitrage,” Finance Research Letters, vol. 2, no. 3, pp. 125–130, 2005. View at Publisher · View at Google Scholar
  34. P. Carr and D. B. Madan, “Option pricing and the fast Fourier transform,” Journal of Computational Finance, vol. 2, no. 4, pp. 61–73, 1999. View at Google Scholar
  35. R. Cont and P. Tankov, “Retrieving Lévy processes from option prices: regularization of an Ill-posed inverse problem,” SIAM Journal of Control and Optimization, vol. 45, no. 1, pp. 1–25, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. P. Tankov, Lévy processes in finance: inverse problems and dependence modelling, dissertation, L'Ecole Polytechnique, Palaiseau Cedex, France, 2004.
  37. Board of Governors of the Federal Reserve System. U.S. government securities/Treasury constant maturities/Nominal, http://www.federalreserve.gov/releases/h15/data/Business_day/H15_TCMNOM_M3.txt.
  38. Standard and Poor's, “Standard and Poor's index services: S and P 500 CASH DIVIDENDS,” September 2008, http://www2.standardandpoors.com/spf/xls/index/SP500EPSEST.XLS.