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How would I price a 1-year ATM European call option, given I know that for the first 6 months, the realized volatility will be 20% and the latter six months, the realized volatilty will be 90%?

One estimation is computing the vega pnl at the six month mark when we remark the option from 20% to 90%. But this relies on an assumption about the spot price. Is there an actual solution to this problem without making too many assumptions?

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  • $\begingroup$ Just use a time dependent instantaneous volatility? $\endgroup$
    – river_rat
    Commented Apr 17 at 15:21

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The implied vol should be the sqrt of realised variance over the option.

This is $sqrt(0.5*(20)^2+0.5*(90)^2)$.

This is just a result of the fact that for a European option, only the terminal distribution matters, so the vol of the terminal distribution determines everything.

Edit: You don't need to make any assumption, option is priced by the information you have provided.

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  • $\begingroup$ thanks - think general integral is: integral_(t0 to t) of sigma^2(s) ds / (t-t0) $\endgroup$
    – Sameer Lal
    Commented May 5 at 5:34

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