domain: sdv phys

text:
Sophie Combet
email: sophie.combet@cea.fr

An introduction to neutrons for biology

), but rather to facilitate the reading of the present book by introducing a few neutron scattering notions that may be useful for the community of biologists eventually less familiar with this technique.

How can neutrons be a useful probe in biology?

Biology is a wide field whose subjects of study are as diverse as populations, organisms, cells, or macromolecules. Actually, almost only biologists interested by biological macromolecules (proteins, nucleic acids, or lipid membranes constituting the living cells) or by issues at the molecular scale can be concerned by neutron scattering techniques.

A crucial issue in biology remains the knowledge of how atoms and molecules organize themselves and move to accomplish a biological function. The long-term "dream" is to observe macromolecules at work in vivo and to describe not only the positions of atoms but also the atomic motions and conformational changes that occur.

To probe structure, interactions, or dynamics of biomacromolecules, the "classical" techniques currently used by biologists are X-ray crystallography, nuclear magnetic resonance (NMR) spectroscopy, or the increasingly used cryo-electron microscopy. However, neutron scattering, as a complementary or even alternative technique, can prove to be essential to answer specific but relevant questions of biology.

In practice, the success of any technical approach depends on its insertion in modern biology through several parameters: high throughput, automatization, facilities to label the samples, perform the experiment, treat the data, etc. Today, neutron scattering is still "underused" in biology, where it occupies a kind of "niche". Such situation reduces its potential importance and impact in the community of biologists. This is partly due to the a priori maladjustment between real space problems in biology and scattering methods, which are based on spatial and temporal correlations. More deeply, this under-utilization may be due to communication difficulties between physicists and biologists. For instance, the notion of "complexity" in biology drastically prevents or makes very difficult the practice of mathematical tools always present in physical models. There is therefore a "risk" for neutron users to move from biology issues to macromolecular physics.

This situation is also due to more "practical" reasons, such as difficulties of access, since neutron beams are produced only at nuclear reactors or spallation sources and thus are not accessible on a laboratory bench. The quantity and quality of sample necessary for a neutron experiment can sometimes be an obstacle in biology, although it depends on the beam intensity and the instrument used [5]. A special attention has indeed to be paid to the quality control of sample purity, monodispersity, and the monomeric state of the sample (e.g. by using SDS-PAGE, analytical NMR, SEC-DLS, or SEC-MALS) before but also all along an experiment, whose duration (from a few minutes to several days) can be tricky for most of biomolecules [5] (see also the book section written by Jeffries et al.). As such, the second generation of neutron spallation sources, like the European Spallation Source (ESS, Lund, Sweden) that should be operational in the next years, will open up new perspectives.

Like other neutron users, biologists can be driven by local contacts to perform a neutron experiment. As mentioned above, access to neutron experiments remains difficult since beam time is mainly accessible through experiment proposals evaluated by ad hoc scientific committees. Besides, there are many different home-made methods to reduce the data in neutron facilities that can make data treatment difficult but local contacts are usually available to provide help for this step.

Fortunately, recently, the rapid and huge development of neutron instruments, sample environments, tools for data treatment, and computer simulations [6,7] changed the situation to the point that nowadays biology represents a substantial part of the research activity at neutron facilities.

The neutron scattering technique

Properties of neutrons

Neutrons are nuclear particles discovered by J. Chadwick in 1932 (less than one century ago!) and were used the first time in 1946 by C. Shull as a tool for scattering experiments. Neutrons can be produced by two different ways (nuclear fission or spallation). Neutron scattering technique has since been developed to concern now all the aspects of condensed matter: physics, chemistry, materials, and so, as far as we are concerned, biology. Neutrons are an exceptional tool because they interact with atomic nuclei with very small energy, therefore without physical or chemical damage of the samples.

The energy of thermal neutrons is, by definition, that of room temperature, i.e. ~25 meV, since they are thermalized at room temperature. Such domain of energy is often that of molecular motions (diffusion, rotation) and of chemical reactions. From this point of view, biology is an ideal domain to apply neutron scattering, as many biological processes take place also at room temperature. However, the large size of biomolecules, even of their secondary and tertiary structures, implies motions too slow to be accessible to neutrons, limiting substantially the potential dynamical studies.

Indeed, with typical wavelengths between 1 and 10 Å, neutrons cover ideally the nm and the ps space-time domain. Modern instruments succeed to extend this domain to two or three orders of magnitude, in the domain of hundreds of nm or several ns, but with a lower quality of the data and longer data acquisition times.

Principles of neutron scattering

Quantum mechanics associates any moving particle to a wave of wavelength 𝜆 = ! " and energy 𝐸 = ℏ𝜔, where ℎ = 2𝜋ℏ is the Planck constant and 𝑝 = 𝑚𝑣 is the momentum of a particle of mass 𝑚 moving with the velocity 𝑣.

Like in other scattering experiments, a neutron scattering experiment consists of sending a monochromatic beam of particles to a sample. Incident neutrons are defined by the momentum, thus by the wave vector 𝑘 .⃗ # , and energy 𝐸 # :

0𝑘 .⃗ # 0 = 2𝜋 𝜆 # 𝐸 # = ℏ $ 𝑘 # $ 2𝑚 = 1 ℎ $ 2𝑚 2 1 1 𝜆 # $ 2 = 1 2 𝑚𝑣 # $
Neutrons interact with the nuclei of the sample through the nuclear interaction potential. They are scattered by the sample and measured in function of their wave vector 𝑘 .⃗ % and energy Es, providing information about the properties of the sample.

The scattering vector is defined by the wave vector 𝑄 .⃗ (Fig. 1) and the exchanged energy 𝐸: 𝑄 .⃗ = 𝑘 .⃗ # -𝑘 .⃗ % 𝐸 = ℏ𝜔 = 𝐸 # -𝐸 % Fig. 1. Principle of a scattering experiment in the reciprocal space. Due to interaction with atoms of the sample, neutrons undergo a modification of their wave vector and energy. The case 𝑄 "⃗ = 𝑄 "⃗ ! corresponds to elastic scattering, for which 0𝑘 .⃗ # 0 = 0𝑘 .⃗ % 0 and 𝜔 = 0.

In the case of inelastic scattering (0𝑘 .⃗ # 0 ≠ 0𝑘 .⃗ % 0), we can define the modulus of the vector 𝑄 .⃗ (Fig. 1) by:

𝑄 = 8𝑘 # $ + 𝑘 % $ -2𝑘 # 𝑘 % 𝑐𝑜𝑠𝜃 Eq. 1
In elastic scattering, i.e. without any energy exchange between the neutrons and the sample (𝐸 # = 𝐸 % , 0𝑘 .⃗ # 0 = 0𝑘 .⃗ % 0), we can define the modulus of the elastic scattering vector, noted 𝑄 .⃗ & (Fig. 1) by:

𝑄 & = 4𝜋 𝜆 sin (𝜃/2)
where 𝜃 is the scattering angle. Note that 𝑄 depends on the instrumental conditions (𝜆, 𝜃), not on the sample. Each incident neutron is described by a planar wave defined by a wave vector 𝑘 .⃗ that, when interacting with a single atom, is scattered isotropically in all directions, generating a spherical wave because the wavelength (10 -8 cm) is much larger than the size of an atomic nucleus (of the order of fm = 10 -13 cm) (Fig. 2). The anisotropic nature of the scattering is due to the interference of multiple spherical waves centered in the different nuclei that constitute the sample.

Note that neutron scattering does not measure distances like a microscope would do. It measures correlation lengths and correlation times in the reciprocal space.

Interferences

Since the atomic nuclei are much smaller than the wavelengths used for neutron scattering, the nucleus acts as a point scatterer, resulting in isotropic scattering. For an incident plane wave described by the wave function Ψ, with unit amplitude, travelling in the positive z direction with wavenumber 𝑘 # = 0𝑘 .⃗ # 0, the scattered wave Ψ′ is then spherical (Fig. 2), with an amplitude ' ( specific of the atomic nucleus, where r is the distance from the nucleus (placed at r = 0):

Ψ = e #) ! * Ψ + = - 𝑏 𝑟 e #) " (
The positive constant b is the scattering length of a nucleus and is a measure of the "strength" of its interaction with the neutrons. The incident neutron flux is equal to:

Φ = 𝑣‖Ψ‖ $ = 𝑣
The number of neutrons scattered by time unit through dS, the surface element (on a detector for instance) defined by the solid angle 𝑑Ω = ,-( # , can be calculated by:

𝑣𝑑𝑆‖Ψ′‖ $ = 𝑣𝑑𝑆 𝑏 $ 𝑟 $ = 𝑣𝑏 $ 𝑑Ω
where v is the neutron velocity and ‖Ψ′‖ $ is the probability of finding neutrons in dS.

The number of neutrons per solid angle is the differential scattering cross section ,.

,/

and is calculated by the ratio of the scattered neutrons divided by the incident flux and the solid angle:

𝑑𝜎 𝑑Ω = 𝑣𝑏 $ 𝑑Ω Φ𝑑Ω = 𝑏 $
By integrating on the sphere around the sample (represented by a point scatterer), we find the differential section of a nucleus (the usual unit is barn, equal to 10 -24 cm 2 ): In a neutron scattering experiment, the spherical waves, generated by the sample nuclei from the planar wave of each incident neutron, interfere with each other (Fig. 3). Some interferences between i and j nuclei are, either constructive, or destructive, therefore the spherical symmetry is lost and we observe oscillations. The differential scattering cross section, that makes an average in the ensemble, is written:

𝜎 = 4𝜋𝑏 $
,. ,/ = 0 1 〈∑ 𝑏 # 𝑏 2 𝑒 (#4 5⃗ •( ⃗ !$ #82
〉 Eq. 2

Coherent/incoherent scattering

The term "coherence" we use for neutrons is not related to a property of the sample coherence (spatial correlation), nor to the neutron beam (which is rigorously incoherent).

In the interaction between a neutron and a nucleus, except the case of isotopes with null nuclear spin (e.g. C, O), there are two possibilities: the neutron couples with the nuclear spin of each atom (noted I), either parallel, or anti-parallel. Since the neutron has a spin equal to 0 $ , the spin of the ensemble, for each isotope, is: 𝐼 + Each interaction process corresponds to different scattering lengths, 𝑏 9 and 𝑏 ; , respectively. We can define the average and the standard deviation, that give, respectively, the "coherent" and "incoherent" scattering lengths:

𝑏 <=! = 𝑏 X = 𝑓 9 𝑏 9 + 𝑓 ; 𝑏 ; 𝑏 #>< = Y𝑏 $ XXX -(𝑏 X ) $
The "coherent" scattering gives information on the spatial and temporal coherences between positions and motions of pairs of atoms nuclei in the sample. The coherent part of the scattered intensity depends therefore on the interaction between pairs of atoms and on collective excitations.

The "incoherent" scattering does not give any information about pair correlations as it depends only on individual nuclei and so gives information about individual (or self) correlations. Self-correlation techniques are able to probe dynamics of individual atoms, even in a concentrated sample.

The "contrast variation" technique in biology

Neutrons interact with the nuclei of atoms, not with electrons like X-rays do. Because the size of an atomic nucleus is several orders of magnitude smaller than the neutron wavelength, the scattering is isotropic and the scattering length (bnucleus), which corresponds to the atomic "scattering power", does not depend on the scattering angle. Contrary to X-ray scattering, the nuclear scattering length of atoms is independent of their atomic numbers (Fig. 4). Therefore, light and heavy elements have comparable scattering lengths. Moreover, different isotopes of a given element scatter neutrons in different ways. Some can exhibit very different scattering lengths and even with opposite signs (the negative sign of b can be interpreted as the change of phase between the scattered and the incident waves).

In particular, this is the case for the coherent scattering length of hydrogen (bH = -3.74 fm) and deuterium, which is positive and very large (bD = 6.67 fm), and close to that of other atoms found in biological samples (Fig. 5A). The large difference between the coherent scattering length of these two isotopes is very useful for structural studies of biomolecules, namely in techniques such as neutron diffraction, small-angle neutron scattering (SANS), specular neutron reflectometry (SNR), or neutron spin-echo spectrometry (NSE).

The difference between the scattering lengths, or the scattering length densities (which are calculated by dividing the added scattering lengths of atoms that constitute parts of the scattering entities by the occupied volume), allows the definition of a "contrast".

As mentioned above, in biology, as well as in organic compounds, the remarkably large difference between the coherent scattering length of hydrogen and that of other common atoms (deuterium, carbon, oxygen, nitrogen, phosphorous) allows large contrasts between parts of the sample, if one replaces selectively, hydrogen by deuterium. We have also to take into account the exchangeable H nuclei from the sample with the buffer used, that are those bound to N and O atoms (present in NH, NH2, OH, etc.), and called the "labile" protons, but not bound to C atom (CH, CH2, CH3, etc.). See the book section of Jeffries et al. for more details.

Isotopic mixtures of H and D (for example, in hydrogenated and deuterated water) can therefore modulate the scattering length density of parts of a sample or of a solvent, erasing the contrast of such parts, thus enhancing the contrast of other parts. This technique, only possible with neutrons, is called "contrast matching" (Fig. 5). For instance, by mixing light and heavy water (or any other hydrogenous and deuterated solvents or buffers), the scattering length density of the solvent can be varied as desired. In the case of mixtures of compounds having different scattering lengths (hydrogenous and deuterated protein complexes, mixtures of hydrogenous and deuterated lipids, hydrogenous proteins in interaction with deuterated lipid membranes, hydrogenous proteins in the presence of a high concentration of deuterated crowding agents, etc.), it is therefore possible to mask either of the components of the compounds. This is particularly important in biology where hydrogen is very present (50% of all the atoms in most of biological macromolecules) and practically evenly distributed, which is a great advantage to make structural and dynamical studies using neutron scattering. Often, a neutron scattering measurement of a biological sample consists of measuring correlations between regions of the sample rich in hydrogen compared to other parts of the sample.

The notion of contrast must be applied differently at different scales, because a domain can be taken as homogeneous (thus, described by a single scattering length density) at a large scale, and heterogeneous at smaller scales, in which case the contrast must be defined with higher spatial resolution. Therefore, a large macromolecule can be defined by a single or by different scattering length densities at small or large Q-values, respectively. For example, amino acids constituting a protein can be distinguished or not, depending on the covered Qrange of the experiment. Some isotopes have also very different incoherent scattering cross sections. This is also the case of H and D (𝜎 #>< (𝐻) ≃ 80 barn compared to 𝜎 #>< (𝐷) ≃ 2 barn). Because of this large incoherent cross section, the motions of hydrogen atoms can be observed directly in quasielastic scattering, evaluated by time-of-flight spectroscopy (see 4.2). biology?

Several neutron scattering techniques are used in biology to study structure (crystallography [8], SANS (combined with simulations [9], SAXS [10,11], or NMR [12]), and reflectometry [13,14]) and dynamics [15,16], which are presented in the present book. The reader is invited to refer to the following book sections to focus on:

• Neutron diffraction ("Opportunities and challenges in neutron crystallography" by N. R. Zaccai and N. Coquelle),

• Small-angle neutron scattering theory ("The basics of small-angle neutron scattering" by C. M. Jeffries, Z. Pietras, and D. I. Svergun); applications ("The power of SANS, combined with deuteration and contrast variation, for structural studies of functional and dynamic biomacromolecular systems in solution" by E. Mahieu, Z. Ibrahim, M. Moulin, M. Härtlein, B. Franzetti, A. Martel, and F. Gabel), and simulations ("Solution structure of macromolecules using small-angle neutron scattering and molecular simulations" by J. Bhatt),

• Neutron reflectometry theory ("An introduction to neutron reflectometry" by F. Cousin and G. Fadda) and applications ("Applications of neutron reflectometry in biology" by Y. Gerelli),

• Quasi-elastic and inelastic neutron scattering ("Probing the dynamics of biological matter by elastic, quasi-elastic, and inelastic neutron scattering" by G. Schirò).

Figure 6 compares the space-time domains covered by different neutron spectrometers compared to other techniques. It is worth noting that methods are complementary and cannot be compared in a simple way. This is because i) different probes interact with the sample in different ways and ii) measured correlations correspond to individual, pair, or collective quantities. ) regions associated with the main physical phenomena potentially observable are shown. One can note that neutron scattering has the capability to simultaneously probe a significant region of this spatial and temporal map. Also, the spatial dependence of the correlation time is directly accessible. From [START_REF] Berrod | Inelastic and quasi-elastic neutron scattering. Application to soft matter[END_REF] and the European Spallation Source (ESS) web site (https://europeanspallationsource.se/science-using-neutrons).

To give a flavor of the following book sections, we give below a succinct description of both structural and dynamical information neutron scattering can bring to biology, hoping that "appetite will come with eating".

Structural information

In structural studies (spatial correlations), incoherent scattering generates a constant background that adds to the coherent intensity, the only one containing information about spatial correlations in the sample. This is illustrated by the following equations. The general relation of neutron intensity can be written from Eq. 2 (see 2.3):

𝑆]𝑄 .⃗ , 𝑡_ = `𝑏# <=! 𝑏 2 <=! 〈𝑒 ?#4 5⃗ •( ⃗ ! (&)A 𝑒 ?;#4• 5555⃗ ( ⃗ $ (B)A 〉 + `(𝑏 # #>< ) $ 〈𝑒 ?#4 5⃗ •( ⃗ ! (&)A 𝑒 ?;#4• 5555⃗ ( ⃗ ! (B)A 〉 # #,2
When using the static approximation, we obtain:

𝑆]𝑄 .⃗ _ = `𝑏# <=! 𝑏 2 <=! 𝑒 ?#4 5⃗ •(( ⃗ ! ;( ⃗ $ )A + `(𝑏 # #>< ) $ # #,2
In a totally random sample, the coherent scattered intensity is null. However, in biological samples, like in condensed matter in general, correlations are always present, at least due to steric effects or to compressibility in the case of fluids.

The extension of spatial correlations goes from interatomic distances (e.g. atoms of a molecule) to infinity, in a perfect crystal. A particular neutron technique can measure correlations within a relatively small spatial range, reason why different instruments are necessary to evaluate coherences at different scales.

Elastic scattering gives access to the structural properties of matter, i.e. the way atoms, molecules or complex structures organize themselves and interact in space. Information is always derived from spatial correlations between sites or spatial domains. The size of such domains goes from 0.1 to 1,000 nm.

High spatial resolution is mostly used to determine crystalline structures by diffraction. Most of the time, neutron diffraction complements X-ray diffraction, in particular to determine positions of hydrogen atoms or protons (H + ), which interact too weakly with electromagnetic radiation. The basic principles of neutron diffraction are similar to those of X-rays. In certain directions, the periodicity of the lattice of a crystal induces strong constructive interferences resulting in the observation of Bragg peaks (for more details, especially on diffraction theory, refer to the book section written by Zaccai and Coquelle).

With a much lower spatial resolution, small-angle neutron scattering (SANS) probes global sizes and shapes of macromolecules in solution as well as their mutual interactions. Obtained results are very reliable because they correspond to statistical averages over an extremely large number of scattering entities, essentially, all those contained in a sample volume of the order of 100 mm 3 . Cryo-microscopy can give equivalent information but statistics refers only to a few molecules and the necessary preliminary preparation of the sample can influence the observations. On the other hand, random inhomogeneity is averaged and is therefore not observed in reciprocal space.

Measuring with high resolution the small Q-domain, SANS of biomolecules in solution can often be decomposed into two independent terms. One of them, dominating the large Qdomain, gives information about size, shape, and surface area of the molecules, while scattering at lower angles gives information about mutual interactions. In this case, scattering is the most precise technique allowing the evaluation of interactions, thus of the effective potentials that define large-scale interactions. Within simple, often valid, assumptions, the total scattered intensity is the product of two terms, form and structure factors describing molecular shape and interactions, respectively.

Whenever reliable, extrapolation to zero angle of scattered intensity yields information about thermodynamic quantities such as molecular mass or osmotic pressure. Even less accurate SANS data can give information about global quantities such as radius of gyration (Rg) of an aggregate or a complex, or about the area of interfaces. Indeed, at large Q-values, SANS probes correlations at small length scales, i.e. intra-particle correlations, while at the smallest Q-values, large correlation lengths characterizing inter-particle correlations are the main contribution to the scattered intensity (Fig. 7). In real space, the product of form and structure factors corresponds to the convolution of the self-correlation of the particles with the pair correlation function that describes the spatial arrangements of the particles in the solution.

More details on SANS theory, applications, and simulations can be found in the book sections written by Jeffries et al., Mahieu et al., and Bhatt.

Scattering principles of specular neutron reflectometry (SNR) are the same as for SANS but applied to surfaces, instead of volumes. SNR consists on sending a collimated neutron beam onto a surface with a certain incidence angle and measuring the reflectivity of the sample as a function of the vector scattering, noted Qz, in the direction normal to the surface, and with an amplitude equal to:

𝑄 * = 4𝜋 𝜆 sin (𝜃)
where λ is the neutron wavelength and θ the incidence angle (for more details, refer to the book section written by Cousin and Fadda). The reflectivity curve obtained makes it possible to determine the thickness and the scattering length density of each layer of the sample. This is called "specular" reflectivity (like a mirror) as opposed to the "grazing-incidence" (GI-SANS) SANS technique which provides information in the plane area.

Dynamical information

Neutron scattering uses "thermal neutron" particles that have been thermalized by inelastic exchanges with a thermal bath, either water at room temperature, or liquid hydrogen at 20 K. The resulting kinetic energy of neutrons that interact with matter is similar to that of intermolecular bonds, thermally activated processes, or collective excitations, which are associated to energies smaller than a few tens of meV. Therefore, the energy domain probed by neutrons is very different from that of X-rays or light spectroscopy, for which the characteristic energy of the photons is one million or one thousand times larger, respectively.

Additionally, the interaction with the only atomic nuclei (we exclude magnetic interactions) makes the theory relatively simple because describing only atomic displacements (structural information) and vibrational and collective motions (dynamical information).

As pointed out above, a specific advantage of neutron scattering is the existence of two contributions to the scattered intensity, coherent and incoherent, which can be tuned often by isotopic substitutions that, within good approximation, do not modify physical or chemical properties of the samples. In practice, this remark applies almost exclusively to hydrogen and deuterium atoms, often present in biomolecules and hydration water.

While, in structural studies, the incoherent scattering represents a simple Qindependent background without information about pair correlations, in dynamic studies, it gives information about individual motions of atoms. This property is almost unique in scattering techniques, for which information about individual motions can be obtained but only indirectly from measurements of mutual diffusion coupled with appropriate models (for example, in dynamic light scattering).

However, in practice, measurements of individual motions are possible only for hydrogen atoms. This is because the total coherent scattering intensity is proportional to N 2 while the incoherent part is proportional to N, where N is the number of nuclei in a sample. Only the very large incoherent cross section of hydrogen, typically two orders of magnitude larger than all other nuclei other nuclei found in biological samples (𝜎 #>< (𝐻) ≃ 80 barn compared to 𝜎 #>< (𝐶, 𝑂, 𝑁, 𝐷) ≃ 0-2 barn) can compensate for this difference.

Therefore, in practice, evaluations of individual dynamics are almost exclusively achieved through measurements of motions of hydrogen atoms.

In biology, atomic fluctuations on the ps-ns timescale are of particular interest because they result from and reflect the forces that structure biological macromolecules, including atomic motions and molecular flexibility associated with biological activity.

Studies of dynamics are however not as numerous as structural investigations. Beyond the fact that there are fewer problems related to microscopic dynamics, neutron scattering has several limitations both at the level of the samples and of the covered dynamical range.

Without going into details of each technique (for that, the reader can refer to the book section written by Schirò), we can briefly describe the main classes of techniques and instruments currently used.

Quasi-elastic neutron scattering

Quasi-elastic neutron scattering (QENS), performed in time-of-flight (TOF) spectrometers, measures dynamical processes in the domain 1-100 ps with good accuracy. Through this technique, it is possible to discriminate between rotational and translational motions. It has been used in many studies of dynamics of water in hydrated molecules giving information about self-diffusion, rotational motions, jumps between sites, hydrogen bond lifetime, etc. It probes similar motions in biomolecules, as far as associated relaxation times are not too long as compared to the instrumental resolution, which is of the order of tens of μeV.

Given the characteristic space-time domain better covered by neutron scattering, the majority of QENS studies concerns motions at molecular scale, meaning that collective motions (cf. 4.2.3) often associated to biological functions only indirectly can be taken into account. This is the case of the kinetics of protein folding, which is too long to fit in the experimental window. Instead, dynamical properties of stable refolding intermediates can be measured in order to be compared with those of the native state.

This technique allows a direct measurement of the van Hove function Gself(r,t) of the hydrogen atoms present in the sample (the following being related to isotropic systems, we substitute 𝑟 to 𝑟 ⃗). Gself(r,t) is the probability of finding a hydrogen atom at point r and at time t assuming that the same atom was at the origin at time t = 0. To illustrate the meaning of Gself(r,t) with an image, R. Scherm and B. Fåk [START_REF] Scherm | Neutron Synchrotron Condens. Matter Stud[END_REF] suggested the image of a "fly correloscope", constituted by two viewfinders A and B connected by a graduated ruler and a chronometer, to illustrate how to measure the probability of having a fly at (𝑟 ⃗, 𝑡) if we have seen the same fly at (0,0) (Fig. 8).

Figure 9 depicts schematically the shape of Gself(r,t) in the case of a simple diffusion process with spherical symmetry. At t = 0, the diffusing particle is at the origin. With increasing time, it can be found in a larger region and, at infinite time, it can be anywhere in the sample. Obviously, all these bell-shaped functions have the same integral, equal to 1.

In practice, the measurement can evaluate diffusive motions of large particles containing hydrogen atoms, as far as internal motions of hydrogen atoms take place at a different time scale. In case of rotational motions, the particle remains confined in a limited region of the space, meaning that Gself(r,∞) = constant ≠ 0.

As we have seen above, about the structure factor, neutron scattering analyses correlations in reciprocal space, for which the variable reciprocal of the real space r is the momentum exchange Q. Similarly, concerning dynamics, many instruments measure the energy exchange, ΔE = ℏω, between neutrons and nuclei, ω being the conjugate variable of time t. We illustrate below the relationships between the van Hove function and the spectra measured by neutron scattering:

The index self refers to individual dynamics, which, most of the time (but not always) is measured by the incoherent part of the scattered intensity, Sinc(Q,ω). Arrows represent space and time Fourier transforms. I(Q,t) and S(Q,ω) are the intermediate and the dynamical scattering functions, respectively: Majority of spectrometers measure S(Q,ω). This is the case of different types of TOF and backscattering instruments. Typical energy resolutions are 50 and 0.1 μeV (FWHM), respectively. Based on a different principle, spin-echo instruments measure with a high energy resolution I(Q,t), i.e. relaxation times up to 10 ns (we remind that t [ps] » 0.658/ℏ𝜔 [meV]), typical of the dynamics of large molecules or aggregates.

Inelastic neutron scattering

Inelastic neutron scattering (INS) probes atomic and molecular vibrational modes, namely the vibrational density of states related to thermodynamic properties and phase transitions. They are mostly in the intermolecular energy domain below 100 meV.

They are measured in TOF spectrometers, often with low energy resolution in order to reach high energy exchange at moderate Q values.

Actually, as it can be seen in Figure 10, there is a strong interdependence between Q and ω, because of the small energy of neutrons. The neutron dispersion curve, that follows Eq. 1 (see 2.2), relates Q and ω for given incident neutron wavelength λ = 2π/k1 and scattering angle θ. On the same scale, the equivalent curve for X-rays or light scattering is reduced to a vertical line because the incident energy is much higher than the energy exchanges probed by neutrons.

One consequence of the neutron dispersion is the small extension of the Stokes side (neutron energy loss) of the spectra. Therefore, analysis is done on the anti-Stokes side (neutron energy gain) despite the lower intensity, particularly at low temperature.

Still, INS is a very useful probe that, in biology, often complements light scattering in the difficult domain of far infra-red absorption (typically up to a few THz, with 1 THz = 32 cm -1 ) where intermolecular contributions dominate.

Collective motions

More rarely in biology, INS can probe collective excitations, namely propagative excitations and acoustic modes (sound waves). Equivalent of Brillouin light scattering in a domain of frequency three orders of magnitude larger (THz instead of GHz) the experiments evaluate the energy dependence of the coherent part of the scattering function, Scoh(Q,ω) in the hydrodynamic limit, i.e. at small Q-values. Technically, experiments are difficult because only a narrow domain of Q and wavelength is available and the scattering intensity is always weak. This is because typical values of sound velocity imply using short wavelength neutrons, hardly compatible with small-angle scattering.

Today, high resolution X-ray inelastic scattering is the main method in this domain of research.

Elastic neutron scattering

Strictly speaking, elastic scattering is solely due to immobile atoms or to local motions taking place at scales relatively small. Examples of the latter are intramolecular rotations of methyl groups or diffusion under high confinement.

The eventual Q-dependence of the intensity of the elastic line gives information about the geometry of the confined volume where motions, namely rotations, take place. An important quantity often evaluated from the elastic component is the elastic incoherent structure factor (EISF) that gives information about the size of a domain where confined motions take place. Therefore, although indirectly, it gives structural information about the sample.

However, in case of slow diffusion, more generally in case of long relaxation times, the instrumental resolution is not good enough to discriminate slow motions from real immobility. In practice, what is measured like "elastic" scattering, does include motions associated to long relaxation times. In elastic scans, one measures the intensity scattered inside a dynamic window generally centered at ω = 0. In general, its width is the instrumental resolution, eventually larger. Measurements are performed in function of an external parameter, normally the temperature.

In the case of a sample in which all atoms are immobile, one will observe a regular decrease of the "elastic" intensity due to the Debye-Waller factor, which describes atomic delocalizations. With increasing temperature, part of the intensity is scattered inelastically, reducing the intensity scattered elastically. Actually, beyond this trivial effect, one looks for kinks or discontinuities of the elastically scattered intensity plotted against temperature. They correspond to dynamic changes, such as onset of dynamic processes, glass or phase transitions.

It is worth noting that the most frequent dynamics of biomolecules is associated to relaxation times larger than 1 ns, which are too large to be studied in TOF spectrometers. Therefore, spin echo instruments and elastic scans, often performed in backscattering instruments, are currently used.

Many other techniques are possible with neutrons in biology that are not described in the present book (e.g. imaging). We did not give also technical details on neutron sources and instruments (for that purpose, refer to [START_REF] Scherm | Neutron Synchrotron Condens. Matter Stud[END_REF][START_REF] Ederth | Neutrons for scattering: What they are, where to get them, and how to deal with them[END_REF]).

Nevertheless, we hope that we convinced the reader that neutron techniques are useful probes in many cases in biology and that this introduction encourages to go further, which is also the propose of the following chapters of this book, that we invite to discover.

Fig. 2 .

 2 Fig. 2. Representation of the incident planar wave, with the wave number 𝑘 "⃗ " , scattered at a point scatterer into a spherical wave with the wave number 𝑘 "⃗ # .

Fig. 3 .

 3 Fig. 3. Principle of a neutron scattering experiment showing interferences between three spherical waves generated by the interaction of the neutron planar wave with three point scatterers representing three nuclei in the sample.

0$

  (parallel spins) or V𝐼 -0 $ V (anti-parallel spins) with the respective probabilities (weighting factors): 𝑓 9 = :90 $:90 and 𝑓 ; = : $:90 .

Fig. 4 .

 4 Fig. 4. Coherent scattering lengths of isotopes. Note that there is no relationship with their atomic numbers, in contrast to X-ray scattering lengths.

Fig. 5 .

 5 Fig. 5. (A)Coherent scattering lengths, in fm (1 fm = 10 -13 cm) of the main atoms present in biological molecules. Note that the value of the hydrogen atom ( 1 H) is largely different from those of other atoms, in particular of the deuterium atom ( 2 H = D), which allows high contrasts and is the basis of the "contrast variation" technique. (B) Scattering length densities of the natural and of a few fully deuterated biological molecules as a function of the isotopic composition of water. Lines are not horizontal because of labile hydrogen atoms, i.e. atoms of the solute that exchange with atoms of the solvent. This graph applies to water and, within good accuracy, to common buffers.

Fig. 6 .

 6 Fig. 6. Complementarity of inelastic neutron scattering, time-of-flight, backscattering, and neutron spin-echo with other spectroscopic techniques. The spatial (r [Å] » 2π/Q [Å -1 ]) and temporal (t [ps] » 0.658/ℏ𝜔 [meV]) regions associated with the main physical phenomena potentially observable are shown. One can note that neutron scattering has the capability to simultaneously probe a significant region of this spatial and temporal map. Also, the spatial dependence of the correlation time is directly accessible. From[START_REF] Berrod | Inelastic and quasi-elastic neutron scattering. Application to soft matter[END_REF] and the European Spallation Source (ESS) web site (https://europeanspallationsource.se/science-using-neutrons).

Fig. 7 .

 7 Fig. 7. (A) Schematic representation of a two-dimensional sample, showing different degrees of spatial resolution as a function of Q and (B) the corresponding regions of the form factor (I(Q) = P(Q)) curve. From [1].

Fig. 8 .

 8 Fig. 8. A "fly correloscope". From [2].

Fig. 9 .

 9 Fig. 9. Shape of Gself(r,t) as a function of time.

Fig. 10 .

 10 Fig. 10. The neutron dispersion curve relying the momentum Q and the energy ω, for a given wavelength λ and a scattering angle θ.


Option Pricing under GARCH models with Generalized Hyperbolic innovations (II): Data and results

Keywords:

In this paper, we provide a new dynamic asset pricing model for plain vanilla options and we discuss its ability to produce minimum mispricing errors on equity option books. The data set is the daily log returns of the French CAC 40 index, on the period January 2, 1988, October 26, 2007. Under the historical measure, we estimate an EGARCH model with Generalized Hyperbolic innovations, using this dataset. We showed in Chorro, Guégan and Ielpo (2008) that when the pricing kernel is an exponential affine function of the state variables, the risk neutral distribution is unique and implies again a Generalized Hyperbolic dynamic, with changed parameters. Thus, using this theoretical result associated to Monte Carlo simulations, we compare our approach to natural competitors in order to test its efficiency. More generally, our empirical investigations analyze the ability of specific parametric innovations to reproduce market prices in the context of the exponential affine specification of the stochastic discount factor.

Introduction

In this paper, we provide a new dynamic asset pricing model for plain vanilla options and we discuss its ability to produce minimum mispricing errors on equity option books. The data set is the daily log returns of the French CAC 40 index, on the period January 2, 1988, October 26, 2007. Under the historical measure, we estimate an EGARCH model with Generalized Hyperbolic innovations, using this dataset. We showed in Chorro, [START_REF] Chorro | Option pricing under GARCH models with generalized Hyperbolic Innovations (I): Methodology[END_REF] that when the pricing kernel is an exponential affine function of the state variables, the risk neutral distribution is unique and implies again a Generalized Hyperbolic dynamic, with changed parameters. Thus, using this theoretical result associated to Monte Carlo simulations, we compare our approach to natural competitors in order to test its efficiency. More generally, our empirical investigations analyze the ability of specific parametric innovations to reproduce market prices in the context of the exponential affine specification of the stochastic discount factor.

As soon as the assumption of market completeness is questionable in real-world equity markets, we need to work in an incomplete setting which is more problematic. Therefore, in an incomplete market with no arbitrage opportunities, there is more than one equivalent martingale measure and hence a range of no-arbitrage prices for a contingent claim. One crucial issue is to identify an equivalent martingale measure which gives an economically consistent and justifiable price for contingent claims compatible with market data.

To settle the latter problem, we proceed as follows:

• We use an EGARCH [START_REF] Nelson | Conditional Heteroskedasticity in Asset Returns[END_REF]) discrete time modeling for the underlying asset (under the historical distribution P) with an appropriated distribution for the innovations to take into account the most important features which characterize financial time series (skewness, kurtosis).

• In order to price options we need to move from the historical to the risk neutral world. Under the assumption of an exponential affine Stochastic Discount Factor, we exhibit the exact representation of this model under the risk neutral probability measure Q, proving the stability of the innovation distribution, with changed parameters.

• We show, throughout a deep empirical study based on Monte Carlo simulations on the CAC 40 French index that our model behaves fairly well to price options for a wide range of moneyness and maturities comparing to its natural competitors.

The first point is the choice of the model under the historical measure P. Until recently, classical well known methods have been considered like the [START_REF] Black | The pricing of Options and corporate liabilities[END_REF] model in continuous time, or the [START_REF] Duan | The GARCH Option Pricing Model[END_REF] and [START_REF] Heston | A Closed-Form GARCH Option Valuation[END_REF] models in discrete time. These latter models are known to fail to capture the short term behavior of equity options smiles. In this paper we propose an EGARCH model with Generalized Hyperbolic innovations. This distribution introduced by Barndorff-Nielsen (1977) is known to fit financial datasets remarkably especially to handle the particular tail behavior found in equity indexes returns (see section 2). It has already been used with empirical successes to model the dynamic of several stock markets in discrete or continuous time (see e.g [START_REF] Eberlein | The generalized hyperbolic model: financial derivatives and risk measures[END_REF] or [START_REF] Guégan | Pricing bivariate option under GARCH-GH model with dynamic copula[END_REF]).

For the second point we have to select a particular risk neutral probability with an interesting economic interpretation. One of the major attempt in this topic was proposed by Gerber and Siu (1994) that provide an elegant way to choose an equivalent martingale measure using the Esscher transformation. This tool has been introduced in actuarial science by [START_REF] Esscher | On the probability function in the collective theory of risk[END_REF], in an incomplete market setting. This latter approach permits to choose a wide variety of parametric models within the class of infinite divisible distributions (see Chorro, Guégan and Ielpo, 2008 proposition 4 for a general result in the GARCH setting). This method has been used to price options in discrete financial models (Bühlmann, Delbaen, Embretchs and Shiryaev (1996), Siu, Tong and Yang (2004) or Christoffersen, Heston and Jacobs ( 2006)) and even in continuous time Lévy type ones [START_REF] Eberlein | The generalized hyperbolic model: financial derivatives and risk measures[END_REF]). An equivalent formulation of the work of Gerber and Siu (1994) consists in using for the classical Stochastic Discount Factor an exponential affine parameterization. We have shown in Chorro, [START_REF] Chorro | Option pricing under GARCH models with generalized Hyperbolic Innovations (I): Methodology[END_REF] that in the framework of an EGARCH model with Generalized Hyperbolic innovations, this particular change of probability implies again a Generalized Hyperbolic dynamic with explicit parameters allowing for Monte Carlo simulations.

For the last point, we compare the pricing performances of our EGARCH model Generalized Hyperbolic innovations to the [START_REF] Black | The pricing of Options and corporate liabilities[END_REF], [START_REF] Heston | A Closed-Form GARCH Option Valuation[END_REF] and Bertolon, Monfort and Pegoraro (2003) ones. The choice of this competitors is natural because, for each of them, the exponential affine form of the pricing kernel is assumed (implicitly or explicitly) to move from the historical distribution to the risk neutral one. Especially, for the Bertolon, Monfort and Pegoraro (2003) model, the use of a mixture of two normal distributions to model the innovations may be interestingly compared to our choice of the Generalized Hyperbolic distribution because these two families have exactly the same number of parameters. With this study we want to highlight the importance of the choice of the Generalized Hyperbolic distribution to model stock price processes.

The paper is organized as follows. In section two, we describe the dataset on which we work. Section three is a summary of the methodology used in this paper. Mathe-matical details being provided in a previous paper [START_REF] Chorro | Option pricing under GARCH models with generalized Hyperbolic Innovations (I): Methodology[END_REF], we especially focus on the explicit description of the carrying out of the theoretical method. The section four is devoted to the analysis of empirical results. Section five concludes.

The dataset

The dataset that we use contains the following time series. Following this preliminary study we test various probability distributions in order to find the best one. The results are provided in Table 2. In this paper we are particularly interested in the Generalized Hyperbolic (GH) distributions introduced by Barndorff-Nielsen (1977). Let us remind briefly its definition.

For (λ, α, β, δ, µ) ∈ R 5 with δ > 0 and α >| β |> 0, the one dimensional GH(λ, α, β, δ, µ) distribution is defined by the following density function

d GH (x, λ, α, β, δ, µ) = ( α 2 -β 2 /δ) λ √ 2πK λ (δ α 2 -β 2 ) e β(x-µ) K λ-1/2 α δ 2 + (x -µ) 2 δ 2 + (x -µ) 2 /α 1/2-λ , (1)
where K λ is the modified Bessel function of the third kind. For λ ∈ 1 2 Z, the basic properties of the Bessel function allow to find simpler forms for the density. In particular, for λ = 1, we get the Hyperbolic distributions (HYP) whose log-density is a hyperbola. For λ = -1 2 , we obtain the Normal Inverse Gaussian distribution (NIG) which is closed under convolution. This family has already been suggested as a model for financial price processes because its exponentially decreasing tails seem to fit the statistical behavior of asset returns remarkably (Barndorff-Nielsen (1995), [START_REF] Eberlein | The generalized hyperbolic model: financial derivatives and risk measures[END_REF]).

From the results provided in Table 2 we observe that the Gaussian distribution is strongly rejected as a model for the conditional distribution of the CAC 40 log returns. On the contrary, the distributions belonging to the Generalized Hyperbolic family give rise to satisfactory results. We provide several figures that confirm these results. On Figure 1, we present the empirical log-density (plain black line) vs. the estimated log density obtained with the NIG (red), Hyperbolic (green), Generalized Hyperbolic (dark blue) and Gaussian (light blue) distributions for the daily log returns of the French CAC 40. In Figure 2, we present the qq-plots comparing the empirical quantiles of the CAC 40 returns vs. the estimated quantiles obtained with the NIG (red), Hyperbolic (green), Generalized Hyperbolic (dark blue) and Gaussian (light blue) distributions.

Based on the graphics and the tables previously mentioned, it is natural that a dynamic asset pricing model is based on a distribution that belongs to the Generalized Hyperbolic family.

The financial theory states that the price of any asset is equal to the present value of the expected pay-off, under a well chosen distribution. Thus, to compute option prices, we need to know the risk-free rate that can be used to compute the necessary discount factors. The risk free rate used in this paper is the zero rate obtained from the European interest rates swap (IRS). We use closing swap rates whose maturities range from 1 month to 3 years. Intermediate maturities required for option pricing are computed using the Svensson (1994) model 1 .

Beyond this initial dataset, we use the available option contracts written on the French CAC 40. The most liquid contracts available are the quarterly ones, that is the con- tracts maturing on March, June, September and December for every available years. We focused on these maturities, neglecting the intermediate monthly maturities that are far less liquid on average. The option dataset starts on January 2, 2006 and ends on October 26, 2007. The strikes are chosen so that the moneynesses used here range from .8 to 1.2, which is the standard moneyness window used in the literature. The table 3 presents key statistics regarding the option dataset across moneynesses and maturities.

3 Methodology

Theoretical approach

In this section, we recall briefly the methodology we are going to use. Since we developed it in Chorro, Guégan and Ielpo (2008), we only focus on the main points here :

• Under the historical measure P, the discrete time economy we consider is character- ized by the time series dynamic of the stock price process (S t ) t∈{0,1,...,T } given by

Y t = log S t S t-1 = r + m t + h t z t , S 0 = s ∈ R + , (2)
where r is the risk free rate expressed on a daily basis and supposed to be constant and where z t is a sequence of independent and identically distributed centered random variables with variance 1. In this model we allow for non Gaussian innovations in order to model extreme returns behavior. Following our previous study, we assume that the random variables z t follow a centered and reduced Generalized Hyperbolic (GH) distribution GH(λ, α, β, δ, µ) where (λ, α, β, δ, µ) ∈ R 5 with δ > 0 and α >| β |> 0. This distribution may be characterized by its moment generating function

G GH (u) = e µu α 2 -β 2 α 2 -(β + u) 2 λ 2 K λ (δ α 2 -(β + u) 2 ) K λ (δ α 2 -β 2 ) , | β + u |< α, (3)
where K λ is the modified Bessel function of the third kind. In equation ( 2), we consider a general time varying excess of return m t that depends on the constant risk premium λ 0 . In practice, it will be fixed for the empirical study and we retain the following classical form:

m t = λ 0 h t - 1 2 h t . (4)
In order to capture asymmetry phenomena (as the leverage effect) observed on our dataset, we choose for the conditional variance of the returns h t an EGARCH model [START_REF] Nelson | Conditional Heteroskedasticity in Asset Returns[END_REF]) that ensures positivity without restrictions on the coefficients

log(h t ) = a 0 + a 1 (| z t-1 | +γ z t-1 ) + b 1 log(h t-1 ). (5)
According to Blaesild (1981), the conditional distribution of Y t may be now deduced from the z t one. In fact, given

F t-1 = σ(z u ; 0 ≤ u ≤ t -1), Y t → GH λ, α √ h t , β √ h t , δ h t , M t + µ h t , (6)
where

M t = r + m t .
• Now, our model is entirely specified under P. Since we want to use it to price contingent claims, we need to postulate an explicit risk premium to perform the change in distribution. Classically, in a discrete time dynamic equilibrium model (or in an arbitrage free continuous one), the price of any asset equals the expected present value of its future payoffs under an equivalent martingale measure Q. For example, the price P t at time t of an European asset paying Φ T at T (Φ T being F T measurable) is given by

P t = E Q [Φ T e -r(T -t) | F t ] (7)
or equivalently

P t = E P [Φ T M t,T | F t ]. (8)
The F t+1 measurable random variable M t,t+1 is the so called stochastic discount factor (SDF) (the quantity M t,t+1 e r is also known as the pricing kernel).

An important point of our approach concerns the assumption on the stochastic discount factor M t,t+1 that is constrainted to be an exponential affine function of the log returns:

∀t ∈ {0, ..., T -1} M t,t+1 = e θ t+1 Y t+1 +ξ t+1 , (9)
where θ t+1 and ξ t+1 are F t measurable random variables. Thus, we need to compute explicitly (θ t+1 , ξ t+1 ).

• Under our previous assumptions, we may show that there exists a unique risk neutral probability Q. This result comes from the following facts. For all t ∈ {0, ..., T -1}, we denote by G t the conditional moment generating function of Y t+1 given F t defined on a non empty convex set. Applying the pricing equation ( 8) to the risky and the non risky assets we just have to solve the following system, to get θ t+1 and ξ t+1 :

G t (θ t+1 ) = e -(r+ξ t+1 ) G t (θ t+1 + 1) = e -ξ t+1 . (10)
In the case where the innovations of the GARCH process (2) follow a GH distribution, the following proposition ensures under mild conditions the existence of a solution for [START_REF] Duan | The GARCH Option Pricing Model[END_REF] (see Chorro, Guégan and Ielpo, 2008 proposition 5 for the proof).

Proposition 1. For a GH(λ, α, β, δ, µ) distribution with α > 1 2 , then,

1. If λ ≥ 0, the equation log G GH (1+θ) G GH (θ)
= r has a unique solution, 2. If λ < 0, the equation log G GH (1+θ)

G GH (θ)
= r has a unique solution if and only if µ -C < r < µ + C where

C = log Γ[-λ] 2 λ+1 -log K λ (δ α 2 -(α -1) 2 ) δ α 2 -(α -1) 2 . ( 11
)
The constant C is strictly positive because d dx K λ (x)

x λ = -K λ+1 (x)
x λ < 0.

For the unicity we just have to apply the general result of Gerber and Siu (1994) and we denote by (θ q t+1 , ξ q t+1 ) the unique solution of [START_REF] Duan | The GARCH Option Pricing Model[END_REF]. Then, the Stochastic discount factor M t,t+1 = e θ q t+1 Y t+1 +ξ q t+1 [START_REF] Eberlein | The generalized hyperbolic model: financial derivatives and risk measures[END_REF] being explicitly known, we may deduce easily the form of the associated equivalent martingale measure Q and the dynamic of the log returns under Q (see Chorro, Guégan and Ielpo, 2008 propositions 3 and 4).

Proposition 2. Under Q, the distribution of Y t given F t-1 is a GH λ, α √ h t , β √ h t + θ q t , δ h t , M t + µ h t ( 13
)
where M t = r + m t .

We deduce from the preceding result that, under Q,

Y t = r + m t + h t z t εt , S 0 = s, (14)
where the z t are F t measurable random variables such that, conditionally to F t-1 ,

z t → GH(λ, α, β + h t θ q t , δ, µ). ( 15
)
In particular, we can observe that the GH distribution is stable under the change of measure, allowing us to simulate easily the sample paths of the risky asset and price contingent claims by Monte Carlo simulations.

We can also remark that under Q, conditionally to F t-1 , the random variable ε t is no more centered and its variance is not h t but

var(ε t ) = h t δK λ+1 (δγ t ) γ t K λ (δγ t ) + (β + √ h t θ q t ) 2 δ 2 γ 2 t K λ+2 (δγ t ) K λ (δγ t ) - K 2 λ+1 (δγ t ) K 2 λ (δγ t )
where

γ t = α 2 -(β + √ h t θ q t ) 2 .
Thus the GARCH structure of the volatility is modified in a nonlinear way when we move from P to Q.

Carrying out

Now, we apply the preceding methodology to price european vanilla options on the CAC 40 index.

The strategy to price all available call options at a given date t between January 2, 2006 and October 26, 2007 unfolds as follows.

1. We first select a subsample containing 4000 working days2 and ending on the date t.

Using this subsample, we estimate the EGARCH model with GH innovations

(2) favoring a two-stages estimation procedure explained in details below. This estimation step also yields the conditional variance of returns for the date t + 1, h t+1 .

3. Now, starting from the date t + 1, we then simulate sampled paths, under the risk neutral distribution:

(a) Start from the estimated conditional variance h t+1 .

(b) Compute θ q t+1 and ξ q t+1 by solving equation ( 10). (c) Sample z t+1 from a distribution GH(α, β+ h t+1 θ q t+1 , δ, µ) using the methodology of [START_REF] Bibby | Hyperbolic processes in Finance, In: Handbook of Heavy Tailed Distributions in Finance[END_REF].

(d) Compute the log return Y t+1 and the conditional variance h t+2 .

(e) Then go back to step (a), replacing t by t + 1, until t = T -1, where T is the maturity of the option we need to price.

This simulation scheme gives the sample returns under the properly chosen risk neutral distribution, from time t to time T . The final price for the underlying asset at time T is given by:

S T = S t T k=t+1 e Y k . ( 16
)
4. Finally, to price a vanilla call option with time to maturity T -t and strike price K, we simulate N paths for the underlying future price S T . The i th sampled final price for the underlying is denoted by S T,i . Then using the Monte Carlo option pricing standard approach, we get the approximated option price Ĉ(.) as the sample average of the simulated final prices:

Ĉ(t, T, K) = e -r(T -t) 1 N N i=1 (S T,i -K) + . (17)
In practice, the number of sampled paths N is equal to 10 0003 .

The previous option pricing strategy requires a few remarks. At step (2), a Quasi Maximum Likelihood Estimation (QMLE) is first used to determine the parameters (λ 0 , a 0 , a 1 , b 1 , γ) of the EGARCH model. At the second stage, since we exactly know the form of the density function of a GH distribution (1) we adopt a classical maximum likelihood approach to estimate the unknown remaining parameters (λ, α, β, δ, µ) using the residuals obtained at the previous stage. The maximization of the log-likelihood of the EGARCH process is initialized using the unconditional variance of the process.

The option pricing methodology used here focuses on out-of-sample option pricing errors: for a given current date, we estimate the time series parameters using a dataset of constant size ending on the current date. We used each time 4000 observations. The key point in our approach is to maintain as much outliers in the dataset as possible: these extremal events are essential to fit the GH parameters and to control the tail behaviors. It is essential to remark that the Monte Carlo simulation we used is indeed pathdependent: for any date t, we need to solve equation ( 10) to obtain θ q t+1 and ξ q t+1 , given the simulated volatility h t+1 and the GH parameters. Finally, so as to reduce the option prices errors linked to the use of Monte Carlo methods, we follow the Duan and Simonato (1998) method that imposes martingality within the sampled processes. This approach makes it possible to reduce significantly the variance of the estimator of the option price.

a 0 b 1 a 1 γ λ 0 Average -5.
For any available date t between January 2, 2006 and October 26, 2007, we reproduce the previous option pricing methodology. Thus we use in fact a rolling window estimator for both the EGARCH and GH parameters. So, we are able to check for the stability of the estimations across our dataset. Table 4 and Table 5 provide respectively descriptive statistics for the estimated parameters of the EGARCH model and the GH distribution. The estimated parameters for the EGARCH model seem to be quite stable over the different rolling windows. This is not exactly the case for the GH parameters : the standard deviation associated to each estimated parameters can be large. This is not a real problem for our option pricing framework, given the good pricing performances of our approach, when compared to market quotes.

Finally, to compare various option pricing models, we use the criterion introduced in Heston and Nandi (2000): it corresponds to the average absolute relative pricing errors criterion for the working days t between January 2, 2006 (τ 1 ) and October 26, 2007 (τ 2 ). Let Ĉ(t, T j , K i ) be the estimated call option price with a time to maturity equal to T j -t and a strike price worth K i . Let C(t, T j , K i ) be the corresponding quoted market option price. Then the criterion we use here is

AARP E = τ 2 t=τ 1 Jt j=1 G t,j i=1 Ĉ(t, T j , K i ) -C(t, T j , K i ) C(t, T j , K i ) , (18)
where J t is the number of call option maturities T j available at time t and G t,j the number of strikes K i available at time t for this particular maturity T j .

We chose this criterion for two reasons. First, it is the usual criterion selected in the empirical literature previously mentioned. We will thus be able to compare the range of errors found in the dataset used with our model with the remaining major models of the literature. Second, this criterion is robust to the well known fact that option pricing errors are proportional to the moneyness: out of the money call option prices are very low, and so are the usual errors found. The converse is true for deep in the money option prices. This criterion rescales the errors using the level of the market option price: it is therefore robust to this effect and the analysis is largely eased.

Results

Here, we present the pricing errors results obtained with our methodology and the dataset presented previously. A deep empirical study have been done. We classically compare our EGARCH-GH model with natural competitors for which the stochastic discount factor is also constrainted to be an exponential affine function of the log returns. Through this study we want to highlight the importance of the choice of the GH distribution to model stock price processes.

More precisely, we compare the pricing performances of our EGARCH-GH model to the Black and Scholes (1973) (BS), Heston and Nandi (2000) (HN) and Bertolon, Monfort and Pegoraro (2003) (MN) ones. Each model is estimated on an equal basis to make the inter-model comparison easier.

First, the choice of these competitors is natural because, in each of them, the exponential affine form of the pricing kernel is assumed to move from the historical distribution to the risk neutral one. Moreover, it is well known that the historical Black and Scholes (1973) model always appears as a benchmark in the financial literature and that the exactly the same number of parameters.

Table 6 presents the values of the AARPE criterion given in [START_REF] Nelson | Parsimonious modeling of yield curves[END_REF] for each model. These results are graphically presented in figure 3.

Looking at these results, wo observe that

• The worst competitors are the Black and Scholes and Heston and Nandi models, yielding errors beyond 100 percent in this out-of-sample testing strategy.

• The EGARCH-GH and EGARCH-MN models behaved fairly well.

Thus, it appears that

• The models based on a fine understanding of the tail behaviors (EGARCH-GH and EGARCH-MN models) yield the best results.

• Our model outperforms the EGARCH-MN one for a wide range of moneynesses and maturities, or is equivalent to it. • The EGARCH-GH model clearly outperforms the EGARCH-MN one for deep out-of-the-money option prices.

What is more, for the EGARCH-GH model, the size of the errors obtained is close to what is obtained during a calibration exercise, see for example Barone-Adesi, Engle and Mancini (2008). This result is very interesting because, in our approach, we do not perform any optimization exercise to call options: we especially focus on the direct modeling approach. In fact, starting from the historical distribution, we impose no arbitrage restrictions based on a specific choice on the shape of risk aversion and we compute after option prices, using an enhanced Monte Carlo method. We must underline that our approach is much less time consuming since it avoids the Monte Carlo based optimization. The key idea here is that a successful option pricing approach should focus on the tail behavior of the returns time series, under the historical distribution. The best competing models are the one based on the choice of a realistic distribution for the stock market returns especially the GH one.

Conclusion

In this paper, we presented a new model based on two assumptions. First, the pricing kernel is an exponential affine function of the log of the future value of the underlying asset. Second, returns in this economy are conditionally Generalized Hyperbolic distributed. Using these assumptions, we show how to price options and compare the empirical performances of our model with the one found in the existing literature. The performances of the model are found to be close to those found when performing a calibration exercise. Finally, we find that what seems to really matter for option pricing is the degree of fit in the tails of the historical distribution, which is a very interesting message for the econometrics of asset pricing framework.

Figure 1 :

 1 Figure 1: Log density This figure presents the empirical log-density (plain black line) vs. the estimated log density obtained with the NIG (red), Hyperbolic (green), Generalized Hyperbolic (dark blue) and Gaussian (light blue) distributions using the CAC 40 returns dataset. Time varying variance has been filtered out using an EGARCH process. The sample starts on January 2, 1988 and ends on October 26, 2007.

Figure 2 :

 2 Figure 2: QQ-Plot This figure presents the qq-plots comparing the empirical quantiles of the CAC 40 returns vs. the estimated quantiles obtained with the NIG (red), Hyperbolic (green), Generalized Hyperbolic (dark blue) and Gaussian (light blue) distributions. Time varying variance has been filtered out using an EGARCH process. The sample starts on January 2, 1988 and ends on October 26, 2007.

Figure 3 :

 3 Figure 3: Absolute average pricing errors expressed in percentage for CAC 40 option prices using the exponential affine pricing kernel for different kinds of maturities.

Table 1 :

 1 We consider daily log returns of the French CAC 40 whose value at time t is denoted S t . The sample starts on January 2, 1988 and ends on October 26, 2007. Descriptive statistics for the CAC 40 dataset.

	Number of observations	Mean	Minimum Maximum
	4159	0.0003	-0.0768	0.07
	Median	Annualized Volatility	Skewness	Kurtosis
	0.0005	0.0021	-0.15	2.87

Table 2 :

 2 Kolmogorov-Smirnov (KS) and Andersen-Darling (AD) adequation tests

	CAC

This table presents the

Kolmogorov-Smirnov and Andersen-Darling adequation tests, testing the adequation of the NIG, Hyperbolic, Generalized Hyperbolic and Gaussian distributions to a dataset of the daily log returns of the French CAC 40. Time varying variance has been filtered out using an EGARCH process. The sample starts on January 2, 1988 and ends on October 26, 2007.

Table 3 :

 3 Descriptive statistics for the CAC 40 option dataset used in the paper.

		Number of available option contracts
		<.8	[.8-.9]	[.9-1]	[1-1.1]	[1.1-1.2]	>1.2
	.25<Maturity<.5	903	2579	2879	2835	1519	100
	.5<Maturity<1	1294	4577	5530	5511	4065	1859
	Maturity>1	1026	7262	15067	16010	14823	14688
			Average option price
	Maturity/Moneyness	<.8	[.8-.9]	[.9-1]	[1-1.1]	[1.1-1.2]	>1.2
	.25<Maturity<.5	1435.2837	949.245273 471.477398 139.330296	20.6086241	2.4202
	.5<Maturity<1	1547.91826	1051.23124 621.277436	287.889459 94.3190627	19.5322431
	Maturity>1	1651.09504	1158.02038 780.151686	483.018575	261.69954	124.69807

Table 4 :

 4 51 0.47 0.01 0.91 0.08 Std. Dev. 0.33 0.04 0.02 0.02 0.03 Descriptive statistics on the average estimated parameters for the EGARCH model.

	α	β	δ	µ	λ
	Average 308.42 -2.77 335.64 3.04 -78.46
	Std. Dev. 139.90 5.08 152.02 5.51 76.68

Table 5 :

 5 Descriptive statistics on the average estimated parameters for the GH distribution.

Table 6 :

 6 [START_REF] Heston | A Closed-Form GARCH Option Valuation[END_REF] model is the discrete time counterpart of the main model used in the banking industry for option pricing: the Heston (1993) stochastic volatility model. Finally, for the Bertolon, Monfort and Pegoraro (2003) model, the use of a mixture of two normal distributions (MN) to model the innovations may be interestingly compared to our choice of the GH distribution because these two families have Absolute average pricing errors for CAC 40 option prices.

	.25<Maturity<.5
	Moneyness <.8 [.8-.9] [.9-1] [1-1.1] [1.1-1.2] >1.2
	BS 0.049 0.073 0.234 1.694	20.504	56.189
	HN 0.043 0.069 0.219 1.560	25.422	73.120
	EGARCH-MN 0.055 0.077 0.174 0.656	6.604	55.893
	EGARCH-GH 0.046 0.058 0.118 0.294	0.989	1.492
	.5<Maturity<1
	Moneyness <.8 [.8-.9] [.9-1] [1-1.1] [1.1-1.2] >1.2
	BS 0.067 0.122 0.252 0.749	2.818	9.852
	HN 0.057 0.120 0.248 0.766	3.143	9.615
	EGARCH-MN 0.069 0.101 0.152 0.384	1.245	4.542
	EGARCH-GH 0.055 0.071 0.080 0.181	0.343	0.699
	Maturity>1
	Moneyness <.8 [.8-.9] [.9-1] [1-1.1] [1.1-1.2] >1.2
	BS 0.132 0.215 0.361 0.664	1.379	3.064
	HN 0.139 0.220 0.387 0.727	1.525	3.549
	EGARCH-MN 0.125 0.150 0.177 0.272	0.534	1.287
	EGARCH-GH 0.106 0.111 0.088 0.104	0.169	0.307

We compared the results obtained with the Nelson and Siegel (1987) approach and the performances of the models presented here are globally unchanged.

We compared the empirical results for samples of size 4000 and 4500, and the results work broadly the same.

We compared the convergence of the estimator of the option price using 10 000, 15 000 and 20 000 simulations, and the results are globally the same. For the sake of numerical feasibility, we favored 10 000 simulations.