Calibrating Option Models to Prices or Implied Volatilities

# Calibration of Local or Stochastic Volatility Models to Prices vs Implied Volatilities


# Calibration of Local or Stochastic Volatility Models to Prices vs Implied Volatilities












As the title suggests, what is the difference between calibrating an option pricing model (say the Heston model) to market option prices instead of computing their implied volatilities using Black-Scholes and subsequently calibrating the Heston parameters to them?

I assume since "implied volatilities behave 'better' than prices", that would mean that the calibrated model parameters using option prices would be more inaccurate?

## Answer by Hasek (score 2, accepted)

https://quant.stackexchange.com/a/74744

> I assume since "implied volatilities behave 'better' than prices", that would mean that the calibrated model parameters using option prices would be more inaccurate?

I'm not quite sure what did you mean by that.

Speaking from a strictly theoretical Black-Scholes framework there is no difference between calibrating a model to option prices and calibrating a model to implied volatilities since there is a one-to-one correspondence between prices and volatilities.

In practice however everything happens. You may not have available price quotes (very rarely) and you are left with calibrating to vols. The opposite can also happen - you may not have volatility quoted but do have prices, then you're implying vols by yourself and calibrating your model to them.

It may also happen that you cannot match quoted prices with quoted vols! This is mostly the case for a very illiquid options or emerging markets. What do I mean by that is that in a Black-76 formula (Black-Scholes for interest rate options and options on futures) $$call = DF(T)\cdot\left(F\cdot N(d_1)-K\cdot N(d_2)\right)$$ with $d_1 = \frac{\ln(F/K)+(\sigma^2/2)T}{\sigma\sqrt{T}}$ and $d_2 = d_1 - \sigma\sqrt{T}$ there is also a forward $F$ and a discounting factor $DF(T)$ whose exact values are not quoted with options and are calculated from discounting and projecting curves. It may happen that your curves are slightly different from curves that a market maker used for pricing and so plugging in quoted volatility $\sigma$ and your $F$ and $DF$ will not match the quoted price exactly. The error may not be that big for a single option but can add up significantly for interest rate caps and floors which are series of caplets and floorlets priced with Black-76 formula. I think that sometimes such a mismatch between vols and prices can be intentional in order to make it harder to reverse engineer a counterparty model and calibrate to given quotes.

Nevertheless a model should reprice a given set of vanillas as closely as possible in order to provide reliable hedges so I would say that one should calibrate a model to prices when in doubt.

## Answer by KaiSqDist (score 0)

https://quant.stackexchange.com/a/81234

I would like to add an additional note I recently discovered that disagrees with the accepted answer. This alternative opinion is from a well-known paper, The Model-Free Implied Volatility and Its Information Content by Jiang & Tian (2005, RFS). The authors state:

"Among the approaches used in previous research, the curve-fitting method is the most practical and effective. Although some studies apply the curve-fitting method directly to option prices [e.g., Bates (1991)], the severely nonlinear relationship between option price and strike price often leads to numerical difficulties. Following Shimko (1993) and Ait-Sahalia and Lo (1998), we apply the curve-fitting method to implied volatilities instead of option price."

Thus, it might be a good idea to apply curve-fitting functions in the IV space.