# Modelling option price change in N days

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?

• what options are you looking at? I imagine you'd need to model the dynamics of implied vol surface if it's exotic. I don't think local vol contains such dynamics. – Will Gu Feb 21 '17 at 18:35
• I am looking at simple call option – Alexandr Proskurin Feb 22 '17 at 9:06
• @WillGu he could price the vanilla option in local vol, and then bucket all the paths across strikes at some future time (30 days), and average each of those. This would be better (i think) than doing something implying price changes from current greeks, and since it's a single option, we don't have to worry about the fact that local vol is poor for forward start options. And you could always just try it again with SLV and see if it makes a difference. – will Jun 22 '17 at 11:43

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).

• The problem is that Greeks also change in time. That is why I can't just use greeks values at time t in order to model the option price change if the underlying price changes at t + 50, for example – Alexandr Proskurin Feb 22 '17 at 9:08
• For sufficiently small variations of the underlying and implied vol, then you can assume that others parameters (time to maturity, interest rate, dividend yield) are negligible and that the underlying and implied vol are the only drivers of your option price. In that case, you can approximate your option price at t+50 by linearizing your option price at time t along the underlying and implied vol dimensions. – JejeBelfort Feb 22 '17 at 9:55
• The problem is that I assume that in t+50 underlying price will change dramatically so will IV and I am trying to model this. What do you mean by linearizing? – Alexandr Proskurin Feb 22 '17 at 15:50
• Your option price is not linear in the underlying price. Therefore, you can linearize it by a first order taylor expansion: $\Delta P = P_{t+50} - P_t$ can be approximated by $\Delta_{option} \times (S_{t+50} - S_t) + \Upsilon \times (\sigma_{t+50} - \sigma_t$) (with $\Upsilon$ being the vega of your option) for small variations of $\Delta S$ and $\Delta \sigma$. However, if you consider dramatic changes, this approach might not be accurate. – JejeBelfort Feb 23 '17 at 8:57
• Yes! That is the question. I am trying to model dramatic changes (i.e, +15% in one day). By the way I would also add theta to your equation in order to take into account time decay. – Alexandr Proskurin Feb 23 '17 at 10:08