nach oben

Soft Computing

Erschienen in:

23.07.2019 | Focus

Option valuation with IG-GARCH model and a U-shaped pricing kernel

verfasst von: Christophe Chorro, Rahantamialisoa H. Fanirisoa

Erschienen in: Soft Computing | Ausgabe 12/2020

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

Empirical and theoretical studies have attempted to establish the U-shape of the log-ratio of conditional risk-neutral and physical probability density functions. The main subject of this paper is to question the use of such a U-shaped pricing kernel to improve option pricing performances in a non-Gaussian setting. Starting from the so-called inverse Gaussian GARCH model (IG-GARCH), known to provide semi-closed-form formulas for classical European derivatives when an exponential-affine pricing kernel is used, we build a new pricing kernel that is non-monotonic and that still has this remarkable property. Using a daily dataset of call options written on the S&P500 index, we compare the pricing performances of these two IG-GARCH models proving, in this framework, that the new exponential U-shaped stochastic discount factor clearly outperforms the classical exponential-affine one. What is more, several estimation strategies including options or VIX information are evaluated taking advantage of the analytical tractability of these models. We prove that the parsimonious estimation approach using returns and VIX historical data remains competitive without having to work with the cross section of options.

Vorheriger Artikel Special issue on dynamics of socioeconomic systems: systemic risk—finance

Nächster Artikel A Quantile Regression approach for the analysis of the diversification in non-life premium risk

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Nur mit Berechtigung zugänglich

Concerning asymmetric volatility responses, refer to the EGARCH model introduced in Nelson (1991), the GJR GARCH model of Glosten et al. (1993), the APARCH model developed in Ding et al. (1993), as well as the TGARCH studied in Zakoian (1994).

See, among others, Christoffersen et al. (2006) for the inverse Gaussian distribution, Badescu et al. (2008) for the mixture of Gaussian distributions, Chorro et al. (2012) for the Generalized hyperbolic distribution.

The \(\mathrm {VIX}\) expresses the market expectations of the 30-day volatility implied in equity index options.

There exist in the literature different parameterizations of the inverse Gaussian distribution. In this paper, the definition and properties of the inverse Gaussian distribution are presented along the lines of Johnson et al. (1994) and Barndorff-Nielsen (1998). In particular, the associated density function is given by the one-parameter family: \( {\mathbf {1}}_{\{y>0\}} \dfrac{\delta }{\sqrt{2 \pi y^{3}}}\mathrm{e}^{-\left( \sqrt{y}-\delta /\sqrt{y}\right) ^{2} /2} \) where \(\delta \in {\mathbb {R}}^{*}_{+}\) and we have \({\mathbb {P}}(y_t=0)=0\).

This exponential-affine restriction of the stochastic discount factor is equivalent to Assumption (12) of Christoffersen et al. (2006).

The equations are derived by applying the pricing formula to the risk-free and risky assets.

Contrary to what happens for Gaussian GARCH models, the IG-GARCH framework is able to cope with the well-known stylized fact that the risk-neutral variance is in general greater than the historical one.

From (1), we obtain \(M^\mathrm{Ushp}_{t}=\mathrm{e}^{\theta _{t}\eta y_t+\frac{\rho _{t}}{y_{t}}+\varepsilon _{t}+\theta _{t} (r+\nu h_t)}\). In the empirical exercise performed in Sect. 4, we obtain, independently of the estimation process, \(\eta <0\), \(\theta _t<0\) and \(\rho _t>0\). Therefore, \(\underset{y_t\rightarrow 0^+}{\lim } M^\mathrm{Ushp}_{t}=\underset{y_t\rightarrow +\infty }{\lim } M^\mathrm{Ushp}_{t}=+\infty \) and \(M^\mathrm{Ushp}_{t}\) is a U-shaped function of \(y_t\). For more details on the shape of this new pricing kernel see “Appendix C”.

For the sake of brevity, we refer the reader to Chorro et al. (2015), p. 137, where a detailed algorithm is proposed with the associated R source code also used in the present paper.

In this section, we implicitly suppose that \({\mathbb {Q}}\) derives from the one-period stochastic discount factor processes defined in Sects. 2.2.1 and 2.2.2.

For the IG-GARCH model, the risk-neutral parameters are simple functions of the historical ones in the case of an exponential-affine stochastic discount factor, while they are functions of the historical parameters and \(\pi \) under \({\mathbb {Q}}^\mathrm{Ushp}\).

The implied volatility root-mean-square error (IVRMSE) will be used in the empirical study to evaluate and compare the pricing performances of the models.

This price is computed using the FFT methodology presented in Sect. 2.3 and depends on the risk-neutral conditional volatility at time t, \(h^*_t\), that is obtained from the log-returns and the risk-neutral GARCH updating rule initialized at its unconditional level.

In fact, when calibrating model parameters, all the attention is focused on the minimization of the in-sample error. Therefore, it is possible to overfit the options dataset and to produce poor out-of-sample pricing errors.

We have \( \theta ^{*}=\{ \nu , \omega , b, c,a,\eta \} \) in the case of the exponential-affine stochastic discount factor and \( \theta ^{*}=\{ \nu , \omega , b, c,a,\eta ,\pi \} \) in the case of the exponential U-shaped one.

Such a joint calibration of model parameters is also performed in Badescu et al. (2019) for the NGARCH model with non-Gaussian innovations when the risk-neutral dynamics is obtained using the so-called extended Girsanov principle of Elliott and Madan (1998) and in Papantonis (2016) for the Heston–Nandi model associated with the variance-dependent pricing kernel of Christoffersen et al. (2013). In the latter study a new bivariate normal model for log-returns and the \(\mathrm {VIX}\) is also introduced to take into account market correlations, but this approach is not a priori compatible with the conditional IG distribution of the log-returns in our setting.

We divide the option data into 18 categories according to either moneynesses or times to expiration. The moneyness is defined as the ratio between the forward price of the underlying asset and the option’s strike price.

We perform a similar analysis to test the moment condition for the returns on the options for different moneynesses and different times to maturities. The results are presented in Tables 6, 7, 8 and 9 with similar conclusions.

This conclusion was conjectured in Papantonis (2016): “This technique is expected to produce equivalent results to those obtained by using the whole cross section of options, while at the same time being straightforward and computationally more efficient”.

Having option pricing in mind, the existence and the simple expression of the moment generating of the inverse Gaussian distribution will be fundamental to using the so-called Esscher transform (and the variant presented in this paper) to specify the stochastic discount factors.

A priori, the parameter \(\eta ^*\) depends on time through \(\theta _{t+1}^{*}\), but as we are going to see below, \(\theta _{t+1}^{*}\) is time independent.

From the empirical values of the parameters obtained in Table 4, this condition is always fulfilled in our framework.

We obtain in particular from this expression that \(\eta ^*<0\).

We thank an anonymous referee for pointing out this question.

Using the fact that an IG random variable Z with degree of freedom \(\delta \) fulfills \(E[\frac{1}{Z}]=\frac{1}{\delta }+\frac{1}{\delta ^2}\).

Badescu A, Kulperger R, Lazar E (2008) Option valuation with normal mixture GARCH. Stud Nonlinear Dyn Econom 12(2):1–42MathSciNetMATH

Badescu A, Elliott RJ, Siu TK (2009) Esscher transforms and consumption-based models. Insur Math Econ 45(3):337–347MathSciNetMATHCrossRef

Badescu A, Elliott RJ, Kulperger R, Miettinen J, Siu TK (2011) A comparison of pricing kernels for GARCH option pricing with generalized hyperbolic distributions. Int J Theor Appl Finance 14(5):669–708MathSciNetMATHCrossRef

Badescu A, Cui Z, Ortega JP (2017) Non-affine GARCH option pricing models, variance dependent kernels, and diffusion limits. J Financ Econom 15(4):602–648CrossRef

Badescu A, Couch M, Cui Z (2019) A unified valuation framework for variance swaps under non-affine stochastic volatility models. Quant Finance 19(2):227–246MathSciNetMATHCrossRef

Bakshi G, Cao C, Chen Z (1997) Empirical performance of alternative option pricing models. J Finance 52:2003–2050CrossRef

Bakshi G, Madan D, Panayotov G (2015) Heterogeneity in beliefs and volatility tail behavior. J Financ Quant Anal 50(6):1389–1414CrossRef

Bates DS (1996) Testing option pricing models. In: Maddale GS, Rao CR (eds) Handbook of statistics: statistical methods in finance, vol 14. Elsevier, Amsterdam, pp 567–611CrossRef

Barndorff-Nielsen OE (1998) Processes of normal inverse Gaussian type. Finance Stoch 2(1):41–68MathSciNetMATHCrossRef

Black F, Scholes M (1973) The pricing of options and corporate liabilities. J Polit Econ 81:637–659MathSciNetMATHCrossRef

Bollerslev T (1986) Generalized autoregressive conditional heteroskedasticity. J Econom 31:307–327MathSciNetMATHCrossRef

Brennan M (1979) The pricing of contingent claims in discrete-time models. J Finance 34:53–68MathSciNetCrossRef

Bühlmann H, Delbaen F, Embrechts P, Shiryaev AN (1996) No arbitrage, change of measure and conditional esscher transforms in a semi-martingale model of stock processes. CWI Econ Stat 9:291–317MATH

Campbell J, Lo AW, MacKinlay AC (1997) The econometrics of financial markets princeton. Princeton University Press, New JerseyMATHCrossRef

Carr P, Madan DB (1999) Option valuation using the fast Fourier transform. J Comput Finance 2(4):61–73CrossRef

Chabi-Yo F, Garcia R, Renault E (2008) State dependence can explain the risk aversion puzzle. Rev Financ Stud 21:973–1011CrossRef

Chorro C, Guégan D, Ielpo F (2012) Option pricing for GARCH-type models with generalized hyperbolic innovations. Quant Finance 12(7):1079–1094MathSciNetMATHCrossRef

Chorro C, Guégan D, Ielpo F (2015) A time series approach to option pricing: models, methods and empirical performances, vol (7). Springer, Berlin, XVI, p 188MATH

Christoffersen P, Heston S, Jacobs K (2006) Option valuation with conditional Skewness. J Econom 131:253–284MathSciNetMATHCrossRef

Christoffersen P, Heston S, Jacobs K (2010) Option anomalies and the pricing kernel. Working paper, University of Maryland

Christoffersen P, Jacobs K, Ornthanalai C (2012) Dynamic jump intensities and risk premiums: evidence from S&P500 returns and options. J Financ Econ 106:447–472CrossRef

Christoffersen P, Heston S, Jacobs K (2013) Capturing option anomalies with a variance-dependent pricing kernel. Rev Financ Stud 26(8):1963–2006CrossRef

Cochrane JH (2001) Asset pricing. Princeton University Press, Princeton

Ding Z, Granger CWJ, Engle RF (1993) A long memory property of stock market returns and a new model. J Empir Finance 1:83–106CrossRef

Duan JC (1995) The GARCH option pricing model. Math Finance 5:13–32MathSciNetMATHCrossRef

Duan JC (1999) Conditionally fat-tailed distributions and the volatility smile in options. Working paper

Duan JC, Ritchken P, Sun Z (2005) Jump starting GARCH: pricing and hedging options with jumps in returns and volatilities. Working Paper, University of Toronto

Duan JC, Ritchken P, Sun Z (2006) Approximating GARCH-jump models, jump diffusion processes, and option pricing. Math Finance 16:21–52MathSciNetMATHCrossRef

Engle RF (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50(4):987–1007MathSciNetMATHCrossRef

Elliott RJ, Madan DB (1998) A discrete time equivalent martingale measure. Math Finance 8:127–152MathSciNetMATHCrossRef

Esscher F (1932) On the probability function in the collective theory of risk. Skandinavisk Aktuarietidskrift 15:175–195MATH

Gerber H, Shiu E (1994) Option pricing by Esscher transforms. Trans Soc Actuar XLVI:99–191

Glosten LR, Jagannathan R, Runkle DE (1993) On the relation between the expected value and the volatility if the nominal excess return on stocks. J Finance 48(5):1779–1801CrossRef

Gourieroux C, Monfort A (2007) Econometric specifications of stochastic discount factor models. J Econom 136:509–530MathSciNetMATHCrossRef

Guégan D, Ielpo F, Lalaharison H (2013) Option pricing with discrete time jump processes. J Econ Dyn Control 37:2417–2445MathSciNetMATHCrossRef

Hansen LP, Singleton KJ (1982) Generalized instrumental variables estimation of nonlinear rational expectations models. Econometrica 50:1269–1286MathSciNetMATHCrossRef

Hansen LP, Singleton KJ (1983) Stochastic consumption, risk aversion, and the temporal behavior of asset returns. J Polit Econ 91:249–265CrossRef

Hao J, Zhang JE (2013) GARCH option pricing models, the CBOE VIX, and variance risk premium. J Financ Econom 11(3):556–580CrossRef

Harrison JM, Kreps DM (1979) Martingales and arbitrage in multiperiod securities markets. J Econ Theory 20:381–408MathSciNetMATHCrossRef

Heston SL (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev Financ Stud 6(2):327–343MathSciNetMATHCrossRef

Heston S, Nandi S (2000) A closed-form GARCH option valuation model. Rev Financ Stud 13:585–625CrossRef

Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions, vol 1, 2nd edn. Wiley, HobokenMATH

Kanniainen J, Binghuan L, Hanxue Y (2014) Estimating and using GARCH models with VIX data for option valuation. J Bank Finance 43(C):200–211

Monfort A, Pégoraro F (2012) Asset pricing with second order Esscher transforms. J Bank Finance 36(6):1678–1687CrossRef

Nelson DB (1991) Conditional heteroscedasticity in asset returns. Econometrica 59:347–370MathSciNetMATHCrossRef

Papantonis I (2016) Volatility risk premium implications of GARCH option pricing models. Econ Model 58:104–115CrossRef

Qiang L, Gaoxiu Q, Guo S (2015) VIX forecasting and variance risk premium: a new GARCH approach. North American J Econ Finance 34:314–322CrossRef

Ross SA (1978) A simple approach to the valuation of risky streams. J Bus 51:453–475CrossRef

Rubinstein M (1976) The valuation of uncertain income streams and the pricing of options. Bell J Econ 7:407–425MathSciNetCrossRef

Simonato JG, Stentoft L (2015) Which pricing approach for options under GARCH with non-normal innovations? Working paper

Siu TK, Tong H, Yang H (2004) On pricing derivatives under GARCH models: a dynamic Gerber-Shiu approach. North American Actuar J 8:17–31MathSciNetMATH

Stentoft L (2008) American option pricing using GARCH models and the normal inverse Gaussian distribution. J Financ Econom 6(4):540–582CrossRef

Terasvirta T (2009) An Introduction to univariate GARCH models. In: Andersen TG, Davis RA, Kreiss JP, Mikosch T (eds) Handbook of financial time series. Springer, Berlin, pp 17–42CrossRef

Zakoian JM (1994) Threshold heteroskedastic models. J Econ Dyn Control 18(5):931–955MATHCrossRef

Ziegler A (2007) Why does implied risk aversion smile? Rev Financ Stud 20:859–904CrossRef

Titel: Option valuation with IG-GARCH model and a U-shaped pricing kernel
verfasst von: Christophe Chorro
Rahantamialisoa H. Fanirisoa
Publikationsdatum: 23.07.2019
Verlag: Springer Berlin Heidelberg
Erschienen in: Soft Computing / Ausgabe 12/2020
Print ISSN: 1432-7643
Elektronische ISSN: 1433-7479
DOI: https://doi.org/10.1007/s00500-019-04236-4

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Weitere Artikel der Ausgabe 12/2020

A vertical ranking technique for linguistic hesitant fuzzy sets

Performance analysis of six meta-heuristic algorithms over automated test suite generation for path coverage-based optimization

An approach for solving fully fuzzy multi-objective linear fractional optimization problems

A fuzzy generalized power series method under generalized Hukuhara differentiability for solving fuzzy Legendre differential equation

Automating fake news detection system using multi-level voting model

MV-algebras as sheaves of -groups on fuzzy topological spaces