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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?

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I'll answer both of your questions in one go:

Your ideas are correct. If the Black-Scholes model was true, the implied volatility surface would be flat but it is not in real life. Thus, the geometric Brownian motion as stock price model is misspecified and we need more sophisticated models (sto vol, jumps etc), in particular if we want to price more advanced (exotic) products.

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.

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  • $\begingroup$ Thank you such a thorough response. Few follow ups 1. For calibration, when you use BSM to translate prices into vols , you assume that BSM iimplied vol is correct. It does further imply that BSM model is correct. Please correct me if I am interpreting wrongly. $\endgroup$ – Ussu Oct 30 at 10:27
  • $\begingroup$ @Ussu that’s not quite correct. You do not assume that BS is right. You merely use this formula to convert prices into volatilities. By definition, if you then use the BS formula with this implied vol, you get the observed market price. You only use the BS formula as a one-to-one map (provided 'correct' data) between market prices and market implied volatilities. These market implied volatilities are then used for calibration purposes. $\endgroup$ – KeSchn Oct 30 at 11:18
  • $\begingroup$ Makes sense. Thanks !! $\endgroup$ – Ussu Oct 30 at 11:20
  • $\begingroup$ @Ussu If you’re happy with answers, not just this question but in general, you could accept them (the check mark). Then, everyone knows that this particular question is dealt with and you don’t require further information $\endgroup$ – KeSchn Oct 30 at 11:35
  • $\begingroup$ Yeah sure. Will do that. Thanks. $\endgroup$ – Ussu Oct 30 at 11:36

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