Empirical Implied Volatility Surfaces and Model Calibration

# 'Empirical Volatility' Surface vs 'Heston Fit'


# 'Empirical Volatility' Surface vs 'Heston Fit'












In the "The Volatility Surface: A Practitioner's Guide" book by Jim Gatheral, there's chart, comparing 'Empirical Volatility Surface' (upper) vs 'Heston Fit' (lower).

> Comparison of the empirical SPX implied volatility surface with the Heston fit as of September 15, 2005. From the two views presented here, we can see that the Heston fit is pretty good for longer expirations but really not close for short expirations. The paler upper surface is the empirical SPX volatility surface and the darker lower one the Heston fit. The Heston fit surface has been shifted down by five volatility points for ease of visual comparison.

I understand how they get the 'Heston Fit' surface - by applying the Heston Formula backwards - feeding it the real option prices and getting back the volatility (the Z-axes not marked, but it should be implied volatility right?).

But what is the 'Empirical Volatility' surface? How do they got it? Did they got it by applying the Black–Scholes formula backward or is it something else?

P.S.

I also don't understand why the goal of Option Pricing Model is to maximise the fit of the Implied Volatility Surface (to empirical surface, or flat surface). Why should we care about the real market prices of options, when (with historical data) we know option price exactly (let's consider European Options only)?

And we can maximise the profit. Backtesting on historical data, and forcing the model to price options in such a way that after millions of trades of various options - we end up with the 0 balance (i.e. every option will be priced as close as possible to its real price at the time of expiration). In a sense like Maximum Likelihood Estimation.

## Answer by Frido (score 5, accepted)

https://quant.stackexchange.com/a/81757

Be it Heston model or some other model, I believe what Gatheral means by "empirical" volatility surface is simply the market price of vanilla options, or equivalently, by finding the (strike dependent) BS volatility parameter values to match the market prices at each strike, the market implied volatility surface as observed at calibration time.

Turning to your other question, which is why should you care about market prices of options. There are several reasons. For example:

a.) From a market maker point of view you want to sell/buy options at the right price (within the bid-ask) in order to be competitive and not to be arbitraged. Furthermore you need to mark-to-market your book from a risk management perspective.

b.) From an investor / "arbitrageur" point of view the market price matters even if you think it's incorrect, because especially when you think the market is incorrect that is the point an arbitrageur might enter a trade to monetize the perceived dislocation.

c.) For the pricing of exotic options with a thin, non-existent, or only OTC market, you usually need a model to price and hedge it, especially if it is a path-dependent derivative. So which model to use? Logic dictates that whatever model you choose the model should also price vanilla options correctly. And for that you need to calibrate the model you choose to the vanilla options market, and if possible a joint calibration to other instruments (such as VIX futures/options), because in an ideal world the model should correctly price all instruments (which is rarely the case).

You also wrote something about historical backtesting / option prices. I suspect here you're thinking about something along the lines of "break even volatilities", which is one of the analyses a buy-side vol trader could do to monetize (perceived) dislocations - see point b.) above.

HTH.