Finding the Unconditional Variance of an E-GARCH Model
Summary
The document explores how to derive unconditional variance for an exponential GARCH model, motivated by using the model in option pricing. It presents a log variance recursion with a Gaussian innovation under a pricing measure, then describes an attempted transformation into a variance recursion. The author flags a possible algebraic error and later cites a series expression for expected variance based on prior work.
The central unresolved issue is the role of the initial conditional variance in that expression. The author reasons that its contribution may vanish as the recursion index grows when the persistence coefficient is below one, but explicitly recognizes this is intuition rather than a proof. No final derivation or numerical example is provided, so the document is useful as a guide to the modeling question and its assumptions, not as a completed formula. It also does not establish the conditions under which the cited limiting argument is valid.
Key ideas
- The target quantity is expected conditional variance in an E-GARCH process.
- The model is expressed as a recursion for log variance with Gaussian innovations.
- The proposed transformation into variance form may contain an algebraic mistake.
- The initial conditional variance may lose influence asymptotically when persistence is below one, but the note does not prove this.
- The derivation remains unresolved and requires careful treatment of model assumptions.
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# Unconditional variance of an E-GARCH model
# Unconditional variance of an E-GARCH model
I am attempting to calculate the unconditional variance of an E-GARCH model: $$\log(h_{t+1}) = \beta_{0} + \beta_{1}\log(h_{t}) + \beta_{2}\left[|\varepsilon_{t} - \lambda| + \gamma(\varepsilon_{t} - \lambda) \right]$$ where $\varepsilon_{t} \sim \mathcal{N}(0,1)$ under the LRNVR measure, $\mathcal{Q}$.
I can calculate the unconditional variance of the GARCH and the GJR-GARCH models quite easily though I am completely stumped on this one.
I manipulated it down to: $$h_{t+1} = h_{t}^{\beta_{1}}\beta_{1}e^{\beta_{0} - \gamma\beta_{2}\lambda}e^{\beta_{2}(|\varepsilon - \lambda| + \gamma\varepsilon)}$$ though I may be wrong in this manipulation (Thankyou for pointing out my error Quantuple). If I am correct, I do not know how to continue from here.
I eventually want to use this results for option pricing under an E-GARCH model, hence the need to calculate this parameter.
EDIT: After reading Daniel Nelson's paper, and Option Pricing Using GARCH Models: An Empirical Examination by Caroline Sasseville, I have an expression for $\mathbb{E}[h_{t}]$. It is quite large and it has to be estimated, but the start of it is as follows: $$\mathbb{E}[h_{t}] = (h_{1})^{\beta_{1}^{i-1}}\left(e^{\beta_{0}\frac{\beta_{1}^{i-1}-1}{\beta_{1}-1}}\right)2^{1-i}\prod_{k=1}^{i-1} f(\lambda, \gamma, k)$$
I understand how to compute this but I am perplexed as to what $h_{1}$ is. I interpret $h_{1}$ as the initial conditional variance which is usually set to the unconditional variance. However, my thoughts are that as k tends towards infinity, $(h_{1})^{\beta_{1}^{i-1}}$ would tend towards one, as based off Sasseville's paper, $\beta_{1}$ is less than one and hence $\beta_{1}^{i-1}$ would tend towards zero. This is not a rigorous answer, just intuitive thinking but would love if someone had some clarification on this. Possibly I am missing something incredibly simple as to what $h_{1}$ is :).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.