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Imagine you have two volatilities, the second which is "activated" when the stock crosses a barrier called $p_b$. The present price is $p_1$. ($p_b>p_1$).

This can be used to price options after a crash when it's assumed that options are more expensive in the short term and are reverting over the long term.

There is also a time parameter for the ratio of time for the duration first volatility ($t_1$) with the time duration until expiration ($t_2$).

I've tried pricing such an option and it's quite messy and involves some improvisation for tricky situations such as when the barrier is the same as the present price or when the time duration $t_1$ is very small relative to $t_2$.

This is the best I have done so far. It still has inconsistencies where the call exceeds the present price for very specific instances.

When the volatilities are equal, the equation reduces to Black-Scholes.

If the first volatility is very flat and the second is greater, the stock --- as $t$ progresses --- will eventually assume the longer-run volatility, but the option is slightly cheaper than simply pricing it with the long-run because the lesser short-run volatility is like friction. Likewise, if the finite frame volatility is greater than the longer run, then we should get a small "boost" for the call price from the short-run even as the option assumes the long-run lesser volatility.

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I do not clearly understand the payoff function of this derivative. How is it different from a regular barrier option? Do you mind explaining it a bit clearer or potentially write out the payoff function? –  Matt Wolf Aug 7 '13 at 4:27
    
The call doesn't become worthless upon crossing the barrier. The volatility over the longer term reverts to the original volatility in the infinite frame based on a probability integral N(d_5) If the probability of hitting the barrier in t_2 is very low then reversion to the long-run volatility takes a longer time than if it's high. –  myoptionexpr Aug 7 '13 at 6:09
    
Could you please write down the exact payoff function? I am still massively confused: Are we talking about an improvement of the BS formula (in which case I think most market participants do not care much about because the current model is already known to be inaccurate and merely used as a translation tool, so unless a perfect model is proposed that removes all faulty assumptions it will be hard to propose a "little less inaccurace") or are we talking about a different derivative than a standard put or call? –  Matt Wolf Aug 7 '13 at 8:08
    
...And is there any sort of relationship between you and the user of the following post because both ideas sound very similar. (quant.stackexchange.com/questions/8672/…) –  Matt Wolf Aug 7 '13 at 8:17
    
why are there two volatilities? –  Matt Wolf Aug 7 '13 at 12:45
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