The instantaneous volatility today is probably right around 20, because if we stay around here, that's what the market expects us to realize. But the instantaneous volatility if we end up near the further option's strike is some number X such that the average of 20 and X is 15, because 15 is what the volatility of the entire path ought to be. So obviously X is 10. The instantaneous local volatility when the underlying is at the higher strike price should be 10.
And by the same logic, the instantaneous local volatility when the underlying is at the lower strike price should be 30
Â