Risk Management

Modelling Conditional Mean and Volatility

If the variable depends on one or more other variables, a conditional mean can be estimated by using a least squares regression. In this case, the information available in the previous period is taken into account. The conditional volatility, on the other hand, is estimated by considering the linear relationship between the current level of uncertainty and the value of a certain variable, unless there is a noise factor. The estimates reflect the current level of uncertainty generated by past shocks.

Conversely, autoregressive refers to the method used to model conditional heteroskedasticity, which is based on a variance "self" regression. This means that past volatility levels influence future levels. If the variable is found to linearly depend on its past value, the conditional mean estimation model would change accordingly.

The key difference between unconditional mean and conditional mean is that the former is a constant, while the latter takes into account the relationship with other variables.

The latter is time-Variance in t is estimated based on 2 components: variance in t-1 and a marketvarying, and is estimated based upon a given model. In this sense, the different versions shock (surprise) in t-1. of GARCH models represent different specifications of the model "explaining" conditional

To start with, let us consider the original version of the ARCH model proposed by Engle. It considers variance as a function of the prediction errors ε made in p past periods. In symbols

2 2 2 σt = α0 + α1εt-1 + α2εt-2 + ... + αpεt-p (6.12)

In practice, if a sudden shock of the variable occurs, this will cause a prediction

error,0 1 0 1p pt t−1 t−pwhich, in turn, if its coefficient is positive, will generate an increase in the predictedα174 Risk Management and Shareholders’ Value in Banking

So, the model estimates variance as a moving average of squared past predictionvolatility for future periods (the fact that squared prediction errors are considered resultsppSo, the model estimates variance as a moving average of p squared past13in the increase occurring regardless of the error’s sign). So, a first marked change in the

In practice, if a sudden shock of the variable occurs, this will cause a prediction error,errors; for this reason, it is referred to as “p delay model”, or simply ARCH(p).variable is thought to be likely to be followed by others; this is consistent with theprediction errors; for this reason, it is referred to as “p delay model”, or simplywhich, in turn, if its coefficient is positive, will generate an increase in the

Predicted α empirical evidence that a significant price change tends to be followed by just as many volatility for future periods (the fact that squared prediction errors are considered results ARCH(p). p significant changes. In this sense, the ARCH model explicitly recognizes the difference 9 When estimating parameters and of the model, the common hypotheses relating to errors (zero mean, α β ε t in the increase occurring regardless of the error’s sign). So, a first marked change in the between unconditional volatility and conditional volatility, acknowledging that the latter serial independence, identical distribution) are usually introduced. The main limitation of the ARCH model is that its empirical applications often variable is thought to be likely to be followed by others; this is consistent with the 10 The majority of historical series econometric models are aimed at predicting the mean of a certain variable varies in time in relation to past prediction

The main limitation of the ARCH model is that its empirical applications often require errors; it also adequately models the fact that required a large number of delays, making the model not very flexible and empirical evidence that a significant price change tends to be followed by just as many by using an equation to be estimated as (6.10). On the contrary, the primary purpose of GARCH models is to a marked change in a certain market factor tends to persist in time, generating a volatility model and predict the variance of a random variable. significant changes. In this sense, the ARCH model explicitly recognizes the difference clustering phenomenon as described above. burdensome. The generalization introduced by Bollersev (GARCH), conversely, 11 The construction of an autoregressive conditional heteroskedasticity model would, in fact, involve two distinct specifications: one for the mean, and one for variance.

For the model's stability to be ensured (i.e., in order to prevent estimated variance from "exploding"), the main limitation of the ARCH model is that its empirical applications often require a large number of delays, making the model not very flexible and burdensome. The generalization introduced by Bollersev (GARCH), conversely, has made it more flexible and capable of handling a large number of delays, making the model more suitable for empirical applications.

As will be seen better below, if the variable considered is given (as in our case) by market factor returns:

R_t = α₀ + α₁R_t-1 + α₂R_t-2 + ... + α_pR_t-p + β₁σ_t-1 + β₂σ_t-2 + ... + β_qσ_t-q

where σ_t² = α₀ + α₁ε_t-1² + α₂ε_t-2² + ... + α_pε_t-p² + β₁σ_t-1² + β₂σ_t-2² + ... + β_qσ_t-q²

which are assumed to have a zero mean, then the error term ε_t will coincide with R_t - α₀ - α₁R_t-1 - α₂R_t-2 - ... - α_pR_t-p.

The sum of error coefficients (α₁ + α₂ + ... + α_p) also needs to be less than one for the model's stability to be ensured.

obtaining the same degree of accuracy using fewer delays. Its analytical formulation is as follows:

In (6.13), conditional variance is modelled by inserting not only prediction error delays, but also delays relating to past values of variance, hence the name The GARCH(p,q).

In (6.13), conditional variance is modelled by inserting not only p prediction q2 2 2 2 2 2= + + · · · + + + + · · · +σ α α ε α ε β σ β σ β σformer are aimed at capturing the short/very short-term effects related to the trend of 0 1 1 2p qt t−1 t−p t−1 t−2 t−qerror delays, but also q delays relating to past values of variance, hence the name the variable considered, the latter, conversely, are aimed at capturing long-term effects;!with: 0α 0 (6.13)α > , . . . , α , β , . . . , β0 1 1p qconditional variance therefore depends on its historical values,

as well.GARCH(p,q). The former are aimed at capturing the short/very short-term effects

The majority of applications of the GARCH model are based upon the GARCH(1,1)

related to the trend of the variable considered, the latter, conversely, are aimed at

version, which considers only one prediction error (the last one), and the value of variance

In (6.13), conditional variance is modelled by inserting not only prediction error delays,

pin the previous period. Analytically:

capturing long-term effects; conditional variance therefore depends on its

but also delays relating to past values of variance, hence the name TheGARCH(p,q).q

former are aimed at capturing the short/very short-term effects related to the trend of

historical values, as well. 2 2 2= + +

the variable considered, the latter, conversely, are aimed at capturing long-term effects;

σ α α ε β σ (6.14)0 1 1t t−1 t−1

The majority of applications of the GARCH model are based upon the

GARCH(1,1) conditional variance therefore depends on its historical values, as well. So, (6.14) specifies conditional variance at time as a function of three factors: (i) aversion, which considers only one prediction error (the last one), and the value of variance. The majority of applications of the GARCH model are based upon the GARCH(1,1) t^2 (ii) the variance prediction made in the previous period (σ), and (iii) the constant (α), version, which considers only one prediction error (the last one), and the value of variance 0 of variance in the previous period. Analytically: t−12 i.e., what was learnt about the trend of the variable. prediction error (ε).