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4

These are 2 completely different ways of estimating volatility. GARCH models are calibrated on historical time series i.e. information provided under the real-world measure $\mathbb{P}$. Although you can obviously use them for forecasting, the core information which is used to build the model is backward-looking in nature (historical behaviour of the stock)....


4

Assume that your stationary time series (here a daily close-to-close log-returns' series) is modelled as follows $\forall t \in \mathcal{T}=\{1,...,N\}$ \begin{align} r_t &= E_{t-1}[r_t] + \epsilon_t \\ &= E_{t-1}[r_t] + \sigma_t z_t \end{align} with $z_t \sim N(0,1)$ and $\{z_t\}_{t \in \mathcal{T}}$ are IID. The above equations suggest that, ...


3

If your question is: "Given all the information available up to time $t$, if I compute the 1 period ahead forecast $r_{t+1}$, is the conditional volatility over $[t,t+1[$ given by $\sqrt{r_{t+1}}$?", the answer is NO. To compute the 1 period ahead conditional variance, you should use your model equations (see this post which might help you better understand ...


2

You're right. Hansen and Lunde ran 330 specifications, and found GARCH (1,1) the best fitting volatility model. However, in some cases other specifications can beat the results of GARCH (1,1). Checking the ACF/PACF of the squared error term is necessary, although, not sufficient condition. Let's assume the following GARCH (m,s) model $$y_t=a_0+a(L)\...


1

The answer to your question is $$\text{Var}(w'R_1+w'R_2)=w'\Sigma_1w + w'\Sigma_2w + 2 w' \Sigma_{12} w $$ where: \begin{align} \Sigma_1(i,j) &= \text{Cov}(R_1 (i), R_1 (j)) \\ \Sigma_2(i,j) &= \text{Cov}(R_2 (i), R_2 (j)) \\ \Sigma_{12}(i,j) &= \text{Cov}(R_1 (i), R_2 (j)) \end{align} are positive definite $N \times N $ covariance matrices.


1

If you have options data with long enough history you could always construct a comparable index by computing the implied volatilities and using a similar weighting methodology to VIX or looking at the implied volatility of the 1 month call/put with strike closest to the price at the observation date (i.e. one closest to 100% moneyness). If you want an ...


1

let' s define a ARMA-GARCH model: $y_{t} = \mu_{t} + \epsilon_{t}$ where $\mu_{t} $ is the conditional mean process (ARMA(p,q) part, $\mu_{t} = E(y_{t}|\mathcal{F}_{t-1})$) . The errors (or mean residuals) re defined by: $\epsilon_{t} = \sigma_{t} \eta_{t}$ where $\eta_{t}$ is a white noise (0,1) Then : $Var[\epsilon_{t}]= \sigma_{t}^{2}$. next see ...


1

When calculating the simple arithmetic mean, each observation has an equal weight: $$ \hat \mu^{simple} = \frac{1}{T}\sum_{t=1}^T x_t.$$ If the observations are $i.i.d.$, $\hat \mu^{simple}$ is an efficient estimator of the population mean. When estimating the mean of a GARCH process, $\hat \mu^{simple}$ is no longer efficient. It makes sense to ...


1

Try the mgarch package, it's available at CRAN. In this link you will find an example from Prof. Zivot.



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