## 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.

### Keywords

- Conditional Variance
- Stylize Fact
- Exponentially Weight Move Average
- Financial Time Series
- Leverage Effect

*These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.*

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## Buying options

## 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*n*th 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).

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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

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