# 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?

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.

• Thanks the amazing answer! Commented Sep 20, 2023 at 8:09
• @Kai you may consider accepting the answer if it was helpful. Commented Sep 20, 2023 at 10:07
• @Hasek If you’re an in illiquid market and cannot accurately back-out the forward rate, how is calibrating with prices or vol still not equivalent? Because at each iteration, you still need to calculate option V which requires the dividend yield and interest rate. So I’m not sure how they’re still not practically equivalent as you will still need some estimate of the term-structure. In other words, if you had the vols from the market without knowing the forward, you still need to know it to get back to vol for the cost-minimisation during the calibration procedure Commented Jan 16 at 8:50