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The first thing you have to understand that volatility is an abstraction, and there are different possible implementations of this abstraction in terms of trading. When someone writes "short spot index volatility, long on implied volatility" they mean something like take a position in options (implied vol) and delta hedge in the underlying instrument, ...


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In this context, unconditional variance refers to the stationary variance level predicted by your GARCH model. This quantity need not coincide with the sample variance of the data on which the latter model has been calibrated. That being said, in an effort to reduce the complexity of the GARCH parameters' estimation process (nasty non-linear optimisation ...


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Let $X^h$ be your hourly process Let $X^d$ be your daily process Let $\delta$ be one day you have $$X^d_t=\frac{1}{\delta}\int_{t-\delta}^{t}X^h_s ds$$ $$dX^h_t = a(b-X^h_t)dt + \sigma dB_t$$ $$\Delta X^d_t := X^d_{t+\delta}-X^d_t =\frac{1}{\delta}\int_{t-\delta}^t\left(X^h_{u+\delta}-X^h_{u}\right)du$$ so it is a gaussian random variable by knowns ...


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You can aggregate your starting hourly data to obtain daily data and re-estimate the parameters, then simulate. Alternatvely, with your parameters already obtained, you can simulate hourly data and make a post-simulation aggregation to have daily data.


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Thank you for your answer @MarkJoshi. I followed you advice and achieved in deriving the approximation formula. However, I can not fully understand why the fact that Black's formula is linear in $\sigma$ for ATM strikes causes the Rebonato approximation only to be accurate for ATM strikes and not OTM and ITM strikes. I would be grateful if somebody can ...


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You are on the right track IMO, except that there are no exceptions to "the greater the expected value, the higher the option value" rule, since an option premium's is precisely the discounted expectation of its payout by absence of arbitrage - at least under the risk-neutral measure. As far as the influence of volatility on the option value is concerned: ...


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it certainly works best at the money. Why? I think it comes from the fact that Black's formula is approximately linear at the money. The approximation $$ \frac{1}{\sqrt{2\pi}} \operatorname{SR} \sigma \sqrt{T} A, $$ with $A$ the annuity is remarkably good. One way of deducing these formulas is to do an asymptotic/Taylor expansion about $\sigma=0.$


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For a mathematical model you can have a look at this paper: The Valuation of Compound Options by Robert Geske where after equation (17) it is shown that $\partial \sigma_s/\partial S<0$.



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