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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 :).

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  • $\begingroup$ I think your expression for $h_{t+1}$ is wrong, it should rather be: $$ h_{t+1} = \underbrace{\exp\left( \beta_0 - \beta_2\gamma\lambda \right)}_{\alpha} h_t^{\beta_1} \exp\left( \beta_2 \left( \vert \epsilon_t-\lambda \vert + \gamma \epsilon_t \right) \right) $$ Anyway, the conditional variance process and return process do seem to have finite unconditional moments under E-GARCH + Gaussian innovations. See original paper by Nelson 1991 paragraph between equations (2.3) and (2.4) and the developments in APPENDIX 1 that may help you: math.nus.edu.sg/~ma2222/E-vol59n2p347.pdf. $\endgroup$ – Quantuple Aug 9 '17 at 8:38

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