# Different volatility surface ( Local vol, Stochastic vol etc.)

Despite many questions about local and stochastic volatility available on this forum, i still have a few doubts left. Essentially I am seeking validation whether I am interpreting things correctly.

Q1. A very basic question: Why are different volatility surfaces used?

My take (please correct if i am wrong):

Implied volatility surface generated using Black-Scholes model is not able to price exotic options (with barriers) correctly. So we need local volatility and stochastic volatility surfaces. The parameters are calibrated so that these volatility surfaces match the implied volatility surface generated from the Black-Scholes model. For calibration we use vanilla options. Once these local volatility and stochastic volatility surfaces match the implied volatility surface generated using the Black-Scholes model and are able to price vanilla options correctly, it is used for pricing exotic products.

Few follow-up questions:

Q2. Why local volatility and stochastic volatility models try to match the volatility surface generated by the Black-Scholes model? Once these local and stochastic volatility surfaces have been generated, we don't need the volatility surface generated by the Black-Scholes model, the new volatility surface can be used for pricing both vanilla as well as exotic options. Is it the correct inference?

However, we do not calibrate stochastic and local volatily models to the Black-Scholes model but to Black-Scholes implied volatilities. If the set of option prices you observe is nice (say arbitrage free), you can translate the observed option price into implied Black-Scholes volatilities (just numerically solve the equation $$\mathrm{MarketPrice}-\mathrm{BSPrice}(\sigma)=0$$).
Implied volatilities behave $$$$better'' than prices, so people prefer to calibrate models to implied volatilities rather than naked prices. But you still calibrate your stochastic and local volatily models to data observed in the market and you do not assume that the Black-Scholes model is correct in any way. You simply use it to translate market prices into market volatilities. These volatilities are then used to find the parameter of your more advanced models and thus guaranteeing that it matches today's market prices.