Abstract
The evaluation of financial risks and the pricing of financial derivatives are based on statistical models trying to encompass the main features of underlying asset prices. From the seminal works of Bachelier (Ann Sci Ecole Norm Supér 17:21–86, 1900) based on Gaussian distributions, the random walk hypothesis for the returns or the log-returns has frequently been suggested. Its remarkable mathematical tractability, in particular in the multidimensional case, was the keystone of nice financial theories like Markowitz’s (Portfolio selection: efficient diversification of investments. Wiley, New York, 1959) portfolio management or Black and Scholes (J Polit Econ 81:637–659, 1973) option pricing model, among others. Nevertheless, during the last decades, the explosion of computational tools efficiency has allowed researchers to pay more attention to the analysis of financial datasets and the test of models assumptions. It is now well-documented that in spite of their huge heterogeneity concerning the nature of financial assets (stocks, commodities, interest rates, currencies…), the frequency of observations or the multiplication of financial centers, financial time series exhibit common statistical regularities (called stylized facts) that make satisfactory models difficult to obtain. A major attempt in this direction was done during the 1980s by Engle (Econometrica 50:987–1007, 1982) and Bollerslev (J Econ 31:307–327, 1986) through the ARCH/GARCH approach. After a brief reminder of the classical stylized facts observed for the daily log-returns of financial indices, the aim of the chapter is to present the main features of the GARCH modelling approach and its recent extensions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
When they exist, the skewness and the kurtosis of a random variable X are defined by \(\mathit{sk}[X] = \frac{E[(X-E[X])^{3}]} {\mathit{Var}(X)^{\frac{3} {2} }}\) and \(k[X] = \frac{E[(X-E[X])^{4}]} {\mathit{Var}(X)^{2}}\). For a Gaussian random variable they are respectively equal to 0 and 3. These indexes are commonly used to quantify asymmetry and fat tails of distributions.
- 2.
When (S 1, …, S T ) is a sample drawn from a distribution with density f, the Gaussian kernel estimator of f of bandwidth h is given by \(\hat{f}(x) = \frac{1} {\mathit{Th}}\sum \limits _{i=1}^{T}d\left (\frac{x-S_{i}} {h} \right )\) where d is the standard normal density function. In practice we take \(h = \left ( \frac{4\sigma ^{5}} {3T}\right )^{\frac{1} {5} }\) where σ is the standard deviation of the sample. This method is implemented in R via the density command of the stats package.
- 3.
If \((U_{n})_{n\in \mathbb{N}}\) is a sequence of nonnegative terms the associated series is convergent if \(\mathit{limsup}\ u_{n}^{ \frac{1} {n} } <\ 1.\)
- 4.
This property is not preserved if we include, for example, in the conditional mean ARMA terms (see Nelson 1991).
- 5.
For each experiment, the starting values of the parameters are obtained by perturbing the true values stated above in the following way:
$$\displaystyle{ \theta _{i}^{j,n} =\theta _{ i}^{0}\left (\frac{1} {2} + u\right ), }$$(2.55)for a given estimation approach j and the nth replication of the Monte Carlo experience. The number u is the realization of a random variable which follows a uniform distribution over [0, 1]. There are numbers of constraints required with these volatility structures and distributions. Once the parameters are perturbed, we check for their consistency with these constraints and discard those that do not match these requirements. What is more, we impose these constraints numerically within the optimization process. However, this is of little impact on the results as the starting point is selected to be close to the true value of the parameters. This would have a sharper influence on the results in the case of a real data set, involving the difficult step of the initialization of the parameters without knowing them.
- 6.
The volatility is initialized to its long term average as estimated from the sample using the method of moment estimator.
- 7.
Larger parameters are usually affected by larger estimation errors: in the criterion that we propose, the errors are weighted by the true value of the parameter, making the aggregation of the estimation errors coherent using relative quantities that are scale independent.
- 8.
This exercise has been done using a Intel Xeon E5420 PC (2.5 GHz, 1,333 MHz, 2X6 Mo).
- 9.
We can also mention on this topic, the important result of Wang (2002) that shows that statistical inference for GARCH modelling and statistical inference for the diffusion limit are not equivalent in general.
- 10.
The factor \(\sqrt{\tau }\) (resp. τ) that appears in the conditional variance (resp. conditional mean) of the log-returns X (n) over a time period of length τ is introduced to respect classical formulas to convert volatility (resp. conditional mean) from one time period to another. Note that in the special case where τ = 1, the model (2.57) reduces to the classical GARCH in mean process (2.45).
- 11.
These parametric constraints are obtained if we take a 0(τ) = w 0 τ, \(a_{1}(\tau ) = \sqrt{w_{2}\tau }\) and b 1(τ) = 1 − a 1(τ) − w 1 τ.
- 12.
This diffusion limit has been extended in Badescu et al. (2013) for the AGARCH case. When the residuals are Gaussian (μ 3 = 0 and μ 4 = 3) several extensions for classical asymmetric GARCH specifications may be found in Duan (1997). Let us also mention that, in the spirit of Duan et al. (2006), it is possible to consider GARCH extensions (called GARCH-Jump processes) that include, as limiting cases, processes characterized by jumps in both prices and volatilities.
- 13.
One of the main features of the Heston and Nandi (2000) model is that both conditional expectation and variance of the volatility process are affine functions of the volatility at the preceding trading date (we speak about affine models). This is not true for the classical GARCH model (2.53) where the conditional variance of the volatility has a quadratic form. This major difference explains not only why we pass from two to one Brownian motions in (2.61) and (2.65) but also why the price of vanilla options in the Heston–Nandi model has a pseudo analytic form (see Sect. 3.7).
References
Abramowitz M, Stegun I (1964) Handbook of mathematical functions with formulas, graphs and mathematical tables. Dover, New York
Alexander C, Lazar E (2006) Normal mixture GARCH(1,1): applications to exchange rate modelling. J Appl Econ 21:307–336
Andersen TG, Bollerslev T, Diebold X (2010) Parametric and nonparametric volatility measurement, chapter 2. In: Hansen LP, Aït-Sahalia Y (eds) Handbook of financial econometrics. Elsevier, Amsterdam, pp 67–128
Bachelier L (1900) Théorie de la spéculation. Ann Sci Ecole Norm Supér 17:21–86
Badescu A, Kulperger R, Lazar E (2008) Option valuation with normal mixture GARCH. Stud Nonlinear Dyn Econ 12(2):1–42
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):669–708
Badescu A, Elliott RJ, Ortega JP (2013) On the extended Girsanov principle for GARCH models and their diffusion limits. Working Paper
Bai X, Russell JR, Tiao GC (2004) Kurtosis of garch and stochastic volatility models with non-normal innovations. J Econ 114:349–360
Baillie RT, Bollerslev T (1992) Prediction in dynamic models with time dependent conditional variances. J Econ 52:91–113
Barndorff-Nielsen OE (1977) Exponentially decreasing distributions for the logarithm of particle size. Proc R Soc Lond Ser A 353:401–419
Barndorff-Nielsen OE (1978) Hyperbolic distributions and distributions on hyperbolae. Scand J Stat 5:151–157
Barndorff-Nielsen OE (1995) Normal inverse Gaussian processes and the modelling of stock returns. Research report 300. Department Theoretical Statistics, Aarhus University, Aarhus
Barndorff-Nielsen OE, Blaesild P (1981) Hyperbolic distributions and ramifications: contributions to theory and applications. In: Taillie C, Patil GP, Baldessari BA (eds) Statistical distributions in scientific work, vol 4. Reidel, Dordrecht, pp 19–44
Barndorff-Nielsen OE, Halgreen O (1977) Infinite divisibility of the hyperbolic and generalized inverse Gaussian distributions. Z Wahrscheinlichkeit Verwandte Gebiete 38:309–312
Behboodian J (1970) On the modes of a mixture of two normal distributions. Technometrics 12(1):131–139
Bertholon H, Monfort A, Pegoraro F (2006) Pricing and inference with mixtures of conditionally normal processes. Working Paper. CREST, Paris
Black F (1976) Studies of stock prices volatility changes. In: Proceedings of the 1976 meetings of the American Statistical Association, Business and Economics Statistics Section, pp 177–189
Black F, Scholes M (1973) The pricing of options and corporate liabilities. J Polit Econ 81:637–659
Blaesild P (1999) Generalized hyperbolic and generalized inverse Gaussian distributions. Working Paper. University of Arhus, Arhus
Bollerslev T (1986) Generalized autoregressive conditional heteroscedasticity. J Econ 31:307–327
Bollerslev T (1987) A conditionally heteroskedastic time series model for speculative prices and rates of return. Rev Econ Stat 69(2):542–547
Bollerslev T (2011) Glossary to ARCH (GARCH). In: Bollerslev T, Russell R, Watson M (eds) Volatility and time series econometrics: essays in honor of Robert F. Engle. Oxford University Press, Oxford
Bollerslev T, Chou RY, Kroner KF (1992) ARCH modeling in finance: a review of the theory and empirical evidence. J Econ 52:5–59
Bouchaud JP, Potters M (2000) Theory of financial risks. Cambridge University Press, Cambridge
Bougerol P, Picard N (1992) Stationarity of GARCH processes and some nonnegative time series. J Econ 52:115–127
Box GEP, Jenkins GM, Reinsel GC (2008) Time series analysis: forecasting and control. Wiley, New York
Brockwell P, Davis RA (1996) Time series: theory and methods. Springer, New York
Chorro C, Guégan D, Ielpo F (2012) Option pricing for GARCH type models with generalized hyperbolic innovations. Quant Finance 12(7):1079–1094
Chorro C, Guégan D, Ielpo F, Lalaharison H (2014) Testing for leverage effect in financial returns. CES Working Papers, 2014.22
Christensen BJ, Dahl CM, Iglesias EM (2012) Semiparametric inference in a GARCH-in-mean model. J Econ 167(2):458–472
Christoffersen P, Jacobs K (2004) Which garch model for option valuation? Manag Sci 50(9):1204–1221
Cont R (2001) Empirical properties of asset returns: stylized facts and statistical issues. Quant Finance 1:223–236
Cont R, Tankov P (2003) Financial modelling with jump processes. Chapman & Hall, London
Corradi V (2000) Reconsidering the continuous time limit of the GARCH(1,1) process. J Econ 96:145–153
Dacorogna M, Gençay R, Müller U, Olsen R, Pictet O (2001) An introduction to high-frequency finance. Academic, San Diego
Davis RA, Mikosch T (1998) Limit theory for the sample ACF of stationary process with heavy tails with applications to ARCH. Ann Stat 26:2049–2080
Ding Z, Granger CWJ, Engle RF (1993) A long memory property of stock market returns and a new model. J Empir Finance 1:83–106
Duan JC (1995) The GARCH option pricing model. Math Finance 5:13–32
Duan JC (1997) Augmented GARCH(p,q) process and its diffusion limit. J Econ 79:97–127
Duan JC, Ritchken P, Sun Z (2006) Approximating GARCH-jump models, jump diffusion processes, and option pricing. Math Finance 16:21–52
Eberlein E, Hammerstein EAV (2003) Generalized hyperbolic and inverse Gaussian distributions: limiting cases and approximation of processes. In: Dalang RC, Dozzi M, Russo F (eds) Proceedings of the 4th Ascona conference. Birkhauser, Basel
Eberlein E, Keller U (1995) Hyperbolic distributions in finance. Bernoulli 1:281–299
Eberlein E, Prause K (2002) The generalized hyperbolic model: financial derivatives and risk measures. In: Geman H, Madan D, Pliska S, Vorst T (eds) Mathematical finance-bachelier congress 2000. Springer, New York, pp 245–267
Embrechts P, Frey R, McNeil AJ (2005) Quantitative risk management. Princeton University Press, Princeton
Engle RF (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50:987–1007
Engle RF (1990) Discussion: stock market volatility and the crash of 87. Rev Financ Stud 3:103–106
Engle RF (2001) GARCH 101: an introduction to the use of arch/garch models in applied econometrics. NYU Working Paper No. FIN-01-030
Engle RF, Bollerslev T (1986) Modelling the persistence of conditional variances. Econ Rev 94:405–420
Engle RF, Ng VK (1993) Measuring and testing the impact of news on volatility. J Finance 48:1749–1777
Engle RF, Lilien DM, Robins RP (1987) Estimating time varying risk premia in the term structure: the ARCH-M model. Econometrica 55(2):391–407
Fama E (1965) The behavior of stock markets prices. J Bus 38:34–105
Fornari F, Mele A (1996) Modeling the changing asymmetry of conditional variances. Econ Lett 50:197–203
Francq C, Zakoian JM (2006) Mixing properties of a general class of GARCH(1,1) models without moment assumptions. Econ Theory 22:815–834
Francq C, Zakoian JM (2010) GARCH models: structure, statistical inference and financial applications. Wiley, New York
Francq C, Wintenberger O, Zakoian JM (2013) GARCH models without positivity constraints: exponential or Log GARCH? J Econ 177:34–46
Glosten LR, Jagannathan R, Runkle DE (1993) On the relation between the expected value and the volatility of the nominal excess return on stocks. J Finance 48(5):1779–1801
Gonzalez-Rivera G (1998) Smooth transition GARCH models. Stud Nonlinear Dyn Econ 3:161–178
Gouriéroux C (1997) ARCH models and financial applications. Springer, New York
Granger CWJ (2005) The past and future of empirical finance: some personal comments. J Econ 129:35–40
Granger CWJ, Andersen AP (1978) An introduction to bilinear time series analysis. Vandenhoeck and Ruprecht, Gottingen
Haas M, Mittnik S, Paolella M (2004) Mixed normal conditional heteroskedasticity. J Financ Econ 2:211–250
Hansen BE (1994) Autoregressive conditional density estimation. Int Econ Rev 35(3):705–730
He C, Terasvirta T (1999a) Fourth-moment structure of the GARCH(p,q) process. Econ Theory 15:824–846
He C, Terasvirta T (1999b) Properties of moments of a family of GARCH processes. J Econ 92:173–192
He C, Terasvirta T (1999c) Statistical properties of the asymmetric power ARCH process. In: Engle RF, White H (eds) Cointegration, causality and forecasting. A festschrift in honour of Clive W.J. Granger. Oxford University Press, New York
He C, Terasvirta T, Malmsten H (2002) Moment structure of a family of first-order exponential GARCH models. Econ Theory 18:868–885
Hentschel L (1995) All in the family. Nesting symmetric and asymmetric GARCH models. J Financ Econ 39:71–104
Heston SL (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev Financ Stud 6(2):327–343
Heston SL, Nandi S (2000) A closed-form GARCH option valuation. Rev Financ Stud 13:585–625
Hull J, White A (1987) The pricing of options on assets with stochastic volatilities. J Finance 42:281–300
Jensen MB, Lunde A (2001) The NIG-ARCH model: a fat tailed stochastic and autoregressive heteroscedastic volatility model. Econ J 4:319–342
Karanasos M, Kim J (2006) A re-examination of the asymmetric power ARCH model. J Empir Finance 13:113–128
Kelker D (1971) Infinite divisibility and variance mixtures of the normal distribution. Ann Math Stat 42(2):802–808
Kluppelberg C, Lindner A, Maller R (2004) A continuous-time GARCH process driven by a Lévy process: stationarity and second-order behavior. J Appl Probab 41(3):601–622
Kon SJ (1984) Models of stock returns: a comparison. J Finance 39:147–165
Ling S, McAleer M (2002a) Necessary and sufficient moment conditions for the GARCH(r,s) and the asymmetric power GARCH(r,s) models. Econ Theory 18:722–729
Ling S, McAleer M (2002b) Stationarity and the existence of moments of a family of GARCH processes. J Econ 106:109–117
Madan DB, Carr PP, Chang EC (1998) The variance gamma process and option pricing. Eur Finan Rev 2:79–105
Malmsten H, Terasvirta T (2004) Stylized facts of financial time series and three popular models of volatility. SSE/EFI Working Paper series in economics and finance, vol 563. Stockholm School of Economics, Stockholm
Mandelbrot B (1963) The variation of certain speculative prices. J Bus 36:394–419
Markowitz H (1959) Portfolio selection: efficient diversification of investments. Wiley, New York
Meitz M, Saikkonen P (2012) Parameter estimation in nonlinear AR.GARCH models. Econ Theory 27(6):1236–1278
Merton RC (1973) An intertemporal capital asset pricing model. Econometrica 41:867–888
Mikosch T, Rezapur M (2012) Stochastic volatility models with possible extremal clustering. Bernoulli (to appear)
Nelson DB (1990a) Stationary and persistence in the GARCH(1,1) model. Econ Theory 6:318–334
Nelson DB (1990b) ARCH models as diffusion approximations. J Econ 45:7–38
Nelson DB (1991) Conditional heteroskedasticity in asset returns: a new approach. Econometrica 59:347–370
Nelson DB, Cao CQ (1992) Inequality constraints in the univariate GARCH model. J Bus Econ Stat 10:229–235
Newey W, Steigerwald D (1997) Asymptotic bias for quasi-maximum-likelihood estimators in conditional heteroscedasticity models. Econometrica 65:587–599
Nocedal J, Wright SJ (1999) Numerical optimization. Springer, New York
Pantula SG (1986) Modeling the persistence of conditional variances: a comment. Econ Rev 5:71–74
Poon SH (2005) A practical guide to forecasting financial market volatility. Wiley, New York
Rodriguez MJ, Ruiz M (2012) Revisiting several popular GARCH models with leverage effect: differences and similarities. J Financ Econ 10(4):637–668
Rogers LCG, Satchell SE (1991) Estimating variance from high, low and closing prices. Ann Appl Probab 1(4):504–512
Sentana E (1995) Quadratic ARCH models. Rev Econ Stud 62:639–661
Straumann D (2005) Estimation in conditionally heteroscedastic time series models. Lecture notes in statistics, vol 181. Springer, Berlin
Stroock DW, Varadhan SRS (1979) Multidimensional diffusion processes. Springer, New York
Taylor SJ (1986) Modelling financial time series. Wiley, New York
Taylor SJ (2005) Asset price dynamics, volatility, and prediction. Princeton University Press, Princeton
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, New York, pp 17–42
Terasvirta T, Zhao Z (2011) Stylized facts of return series, robust estimates and three popular models of volatility. Appl Financ Econ 21(1–2):67–94
Venkataraman S (1997) Value at risk for a mixture of normal distributions: the use of quasi-Bayesian estimation techniques. Econ Perspect Federal Reserve Bank of Chicago 21:2–13
Wang Y (2002) Asymptotic nonequivalence of GARCH models and diffusions. Ann Stat 30:754–783
Wintenberger O (2013) Continuous invertibility and stable QML estimation of EGARCH(1,1) model. Scand J Stat 40(4): 846–867
Wold H (1938) A study in the analysis of stationary time series. Almqvist and Wiksell, Uppsala
Yang M (2011) Volatility feedback and risk premium in GARCH models with generalized hyperbolic distributions. Stud Nonlinear Dyn Econ 15(3):1–21
Zakoian JM (1994) Threshold heteroskedastic models. J Econ Dyn Control 18:931–955
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Chorro, C., Guégan, D., Ielpo, F. (2015). The Time Series Toolbox for Financial Returns. In: A Time Series Approach to Option Pricing. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45037-6_2
Download citation
DOI: https://doi.org/10.1007/978-3-662-45037-6_2
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-45036-9
Online ISBN: 978-3-662-45037-6
eBook Packages: Business and EconomicsEconomics and Finance (R0)