I need to understand how will my option price change if the price of underlying asset changes by, for example, 15% in 30 days. I would like to use BS formula, but in this case I know all parameters except implied volatility. The problem is that IV today will differ from IV in 30 days. How can I model that? Should I use local volatility model?
First you have to assume that the main drivers of your option price are its underlying value and implied volatility, meaning that greeks like rho and theta are negligible with reference to delta and vega.
Then, could you enlighten us on how you determine the underlying change? If your approach is sound for the underlying, and the one used to find the implied vol changes is sound as well, you can find the change in the option price by plugging these "stress" parameters into BS formula.
Otherwise just compute the delta and vega of your option today and deduce the new option price my multiplying the greeks with the variations of the underlying and implied vol of your choice (this assumes a linearization of the option price around the spot and the current implied vol).