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_1this can be used to price options after a crash when it's assumed that options are more expensive in the short term and over the long term reverting to the old volatility. There is also a time perameter 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 Pricing options with two volatilitiesWhen the volatilities are equal the equation reduces to black-scholesan open problem is to prove that an option priced with this formula is always less than with black scholes , assuming the long run volatility is less than the short run (it is supposed to be). 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 longrun 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.

can't you simply use two barrier options, like a down and out and an up and in, then add the value together?