What does it mean to “calibrate vols”

As a beginner, it can sometimes be hard to discern what different terms and phrases mean in QF. I've heard multiple people such as academics and market-makers say things like "calibrate vols" or "calibrate to the market" but I'm not exactly sure what this means.

Focusing mainly on Vanilla American and European Options on equities, I know there are multiple models to price these Options such as Black-Scholes, Local Vol, SVI, etc...

Most of the time, the observed prices of Options you see in the market are quoted from the Black-Scholes model and therefore the "market" implied volatility is from BSM as well. If one were to transform the observed prices and volatilites into a 3D surface (vol surface), we would usually observe skew because the Black-Scholes assumption that all vols are constant along strikes is false.

So when it's said to "calibrate vols" what does that exactly mean? I assume the premise of that is to see whether or not the Options prices or implied vols from Firm XYZ's model line up with the market's quotes and from there underpriced vol would be bought while overpriced vol would be sold.

But when they "calibrate vols to the market", does that mean they input the Market price of an Option into their own model and see whether or not the BSM implied vol from that market price lines up with their (presumably better informed and more correct model) own model's implied vol? Or is it vice versa?

You are an investment bank. You trade a multitude of vanilla and exotic options. You want to make sure the option prices you quote as a client are arbitrage-free with respect to liquid option prices quoted in the market $$-$$ and also consistent between the different trading desks within your bank.

Basically you want to avoid other market participants taking advantage from you because you are quoting inconsistent prices $$-$$ or desks within your organisation trying to profit from each other.

Because you trade complex options, a simple model such as Black-Scholes or Bachelier is not enough. You need a more sophisticated model $$\mathcal{M}(\Theta)$$ which depends on a set of parameters $$\Theta=(\theta_1,\dots,\theta_n)$$.

Now, you need to set a value to those parameters. Because your constraint is that you want your model to be arbitrage-free, it makes sense to be able to back out the price of liquid options which somehow are related to the complex option you want to price: for example, if you want to price a Bermudan option, namely an option which you can exercise on a set of dates $$T_1,\dots,T_m$$, you might want your model prices for the $$m$$ European options expiring on $$T_1,\dots,T_m$$ (i.e. these are called the "co-terminal Europeans") to match the market prices.

So let's assume there are a set of $$m$$ options with market prices $$O_1,\dots,O_m$$ for which you want your model $$\mathcal{M}$$ to match the market price. Each option has a set of characteristics $$C_i=(c_{i,1},\dots,c_{i,k})$$, for example strike and expiry, that defines its payoff. So you want the following to hold as best as possible for each $$i$$: $$\mathcal{M}(\Theta;C_i)=O_i$$ where $$\mathcal{M}(\Theta;C_i)$$ is your model price.

Calibrating the vols, calibrating a model, consists on performing an operation along the following lines: $$\text{arg min}_{\Theta}\sum_{i=1}^m\left(\mathcal{M}(\Theta;C_i)-O_i\right)^2$$

It amounts to a procedure which allows you to recover the values for $$\theta_1,\dots,\theta_n$$ which generate the best fit between your target option prices, and the option prices generated by your model.

Specifically, "calibrating the vols" means we are trying to recover the implied option volatility from the liquid options $$-$$ given options are normally quoted in terms of implied volatility. Alternatively (but equivalently), you might have a volatility model, such as local volatility or stochastic volatility, and you want to fit its volatility function to the market data.

But the gist is as above: you want your model to generate prices which are consistent with prices of liquidly traded products.

• Let us suppose that a firm has fed their model the market prices of liquid options in stock ABC, and their model does indeed fit between the bid and ask vols of the market and have consistent prices to those of the market. But say along strike X and maturity Y, there are market vols that are greatly outside of the model's fit, some might be "overpriced" and some "underpriced". What would the bank or MM do in this scenario? Would they realize there's an error in their model, fix it and re-calibrate? Or would they buy/sell the vols and hedge accordingly on the basis that their model is "better"? – Jack Bueller May 21 at 23:40
• @JackBueller this really depends on the specific situation. A derivative pricing model is a relative pricing procedure: it gives you the price of C given the prices of A and B. Banks will rarely use a pricing model to identify a market mispricing, unless there is a blatant arbitrage. If the market quotes are liquid, then that is where the market stands and the bank will not go against that. Various pricing models cannot calibrate perfectly to all strikes/maturities/etc, if there is such a mismatch it’s most probably because the bank does not care about matching those areas. – Daneel Olivaw May 22 at 9:14
• Thank you. So as I understand now, Investment Banks are making sure what they quote is consistent with the market, etc. But what about in the case of an Options market-maker such as Citadel or a vol-arb fund? – Jack Bueller May 23 at 1:16
• @JackBueller I am not familiar with the business model of independent option market makers. However, my understanding is that these shops only trade highly liquid options, for which I am not sure you need sophisticated pricing models that require calibration. Volatility arbitrage funds by definition do probably take views on whether market-quoted implied volatilities are over-/under-priced with respect to their in-house models. However I suspect these models are probably econometric models, different from those used for pricing at investment banks. – Daneel Olivaw May 23 at 11:23

The answer in Implied Vol vs. Calibrated Vol as suggested by noob2 is more complete. But it may be slightly misleading in your last example. I've been a vanilla option market maker for ten years, so I'll chime in on what I would mean by that.

If a market maker says he's calibrating his vols to the market it means exactly what you're saying: getting prices from brokers or electronic market and compare the bid/offer, derive the implied vols and adjust your own vol to fit inside the bid/offer.

I'll add a caveat, depending on the underlying instrument they may be some correlation element to it, so that may also imply recomputing a correlation matrix; but that's outside of my knowledge.

The same might not hold true for a non-market marker as they may be using a non standard parametric model, or even a non-parametric model for their vols.

• Thank you. Having said that you were a vanilla option market maker, could you take a look at my reply to Daneel and elaborate on what you would do if/when you encountered the scenario I described? – Jack Bueller May 22 at 2:39