Conceptual Questions about Vol-surfaces

[karstensteffensen]

Hi I am new to the forum, so forgive me any misappropriations with this post. I have searched existing posts and none come quite close to my question.

I am writing a thesis about volatility modelling, and have calibrated the relevant models to market data, such that I have the model parameters.

Now I want to fit the volatility surfaces of the model and compare to the empirically observed and want to do it with the Gatheral SVI model. I have some code from mathworks that seem to do the trick, however I just want to ensure that I input the right data.

My questions:

1. The Empirically observed surface: Am I correct to assume that this is the Black-Scholes implied volatility from actual option prices?
2. Implied volatility surface of any other option pricing model: This one I have some trouble with, but am I correct to assume that I re-price the options with the relevant option pricing formula and re-fit the surface?

I have read both Gatheral and Wilmott on the subject, and both seem to brush over this pretty rudimentary distinction. Hopefully my difficulty with this important issue, haven't impuned me too much.

Kind Regards,
Karsten

 

[diegosanaz]

Well, it would be like this. The Black Scholes model asumes constant volatility. If you calibrate you model then you have some theorical option prices. The you use the black scholes formula (in this scenario this is redundant) to get the volatility you would need in each option to get the obtained price. This is the implied volatility. In this scenario, it is just a straight horizontal line.

Under other models, for instance Heston, once calibrated you would get some option prices. Using the Black Scholes formula with those prices, again, to obtain the volatilities you would need to get the induced option prices from the model. In this scenario, this implied volatility would be curved (a smile).

In other word, think of the Implied Volatilities as the volatilities you would need to input to the Black Scholes formula in order to get the same option price induced by the model.

Hope this helps,