How Implied Volatility Calibrates Local and Stochastic Volatility Models
Summary
The discussion explains why local volatility and stochastic volatility models are used when Black–Scholes assumptions do not describe observed option markets well, especially for pricing exotic products. Under a true Black–Scholes model, implied volatility would be flat; market volatility smiles and skews indicate that geometric Brownian motion alone is misspecified. More advanced models can represent features such as changing volatility or jumps.
Calibration is to market option prices expressed as Black–Scholes implied volatilities, rather than to a theoretical Black–Scholes surface assumed to be correct. The Black–Scholes formula serves as a common translation: for each observed option price, solve for the volatility that reproduces that price. Models are then fitted to those implied volatilities, aiming to reproduce today’s vanilla option prices and provide a framework for valuing exotics. The exchange offers a conceptual explanation, not a comparison of model performance; calibration to vanilla prices alone does not establish that a model captures future dynamics or exotic-option risks accurately.
Key ideas
- A flat implied volatility surface would be consistent with the Black–Scholes assumptions, while market smiles indicate model misspecification.
- Local and stochastic volatility models add dynamics intended to represent market features and support exotic pricing.
- Observed option prices are converted into Black–Scholes implied volatilities for calibration.
- Black–Scholes implied volatility is a quoting and translation convention, not an assumption that Black–Scholes is true.
- Matching vanilla market prices does not by itself guarantee accurate exotic prices or future dynamics.
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# Different volatility surface ( Local vol, Stochastic vol etc.)
# 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?
## Answer by Kevin (score 11, accepted)
https://quant.stackexchange.com/a/49443
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.