# Implied Vol Smile: from Calls, Puts or Both?

This might be a simple question, but I couldn't find the answer anywhere: is there a separate Volatility smile (and surface) based on Calls and a separate Volatility smile (surface) based on Puts? Or is there simply one Volatility smile (and one surface) based on the most liquid options, mixing calls and puts?

I would have thought that if you plot strike on the x-axis and IV on the y-axis, then:

(i) To the left of ATM strike, you'd use OTM puts

(ii) To the right of ATM strike, you'd use OTM calls

But various pictures I have found online simply show the smile as if it can be constructed just from calls or just from puts, i.e.:

The problem I see with constructing the surface just based on calls, or just based on puts, is that ITM options may not be liquid enough or traded at all.

Last but not least: say that you'd use OTM puts to the left of ATM, and OTM calls to the right of ATM: what about the ATM point? What if the ATM call IV is different to the ATM put IV?

• In academic theory the IV of a call and the IV of a put at same strike are the same due to put_call_parity. In practice it may matter and you use puts on one side and calls on the other (i.e. always the OTM option) for maximum numerical precision and to ensure a unique result which can be reproduced by other researchers. Jul 9 '20 at 16:48
• From what I've seen the surface is usually given with moneyness rather than strike on one axis. But that would mean the IV isn't the same for the same strike, right, since the call and put will have difference moneyness, how does that add up? Jul 9 '20 at 17:17
• By Moneyness you mean $\ln(S/K)$ ? Jul 9 '20 at 17:45
• Hmm, no I believe I've seen it quoted as simply S/K, and in some cases F/K (F as in forward price). I think you would run into the same situation I mentioned in my comment with either definition though Jul 9 '20 at 17:53

In practice, things are actually quite different and a bit more subtle. You really need to differentiate between the underlying being an index or e.g. a single stock. I will try to provide some insight:

• Index options are, in general, of European type. The market quotes prices for calls and puts and you can back out the implied vols via the usual BS formula. OTM options are clearly more liquid in the interbank market. As an example, for an index like the EuroStoxx, bid-offer vol spreads for OTM options are in a range 0.3 - 0.5% for short term options (sometimes even tighter). ITM quotes are usually wider. Hence, when you calibrate a mid-market vol curve, which is in line with bid-offer vols for OTM options, it will, in general, also be in line with bid-offer vols implied from ITM quotes. But as other authors above mentioned, it is important to view this also from your operational set-up. For example, as a market maker, you will need to price both OTM and ITM options. Hedging costs may be more significant for ITM options based on your calibrated mid-market curve (on OTM options)

• For single stocks, this is completely different due to several aspects: 1) They are of American type, 2) market quotes are much wider than for an index, even for ATM options. What really is an issue for single stocks vol surfaces is the early exercise feature. One can show that implied vols for calls and puts with the same strike may differ significantly due to the potential early exercise of American options (even though they may both be very liquidly traded in the market). This is very prominent for underlyings with high dividend yields and in a negative interest rates environment.

To answer your original question, I think that market participants do not maintain two different vol surfaces (calls and puts) for one and the same index.

But for me, the more interesting question (based on the comments on differing implied vols due to early exercise above) is:

Do market makers maintain two different vol surfaces (calls and puts respectively) for one and the same underlying single stock?

Further notes:

1. One shouldn't build an implied volatility surface just from call prices or just from put prices. One should build it from liquid instrument quotes and, if necessary, some less liquid ones. Some markets, like FX option one, quote package prices (butterfly, risk reversal, ATM straddles).

2. Deciding how to parameterize the implied volatility surface, (tte, moneyness, volatility), is important as it impacts the surface construction (interpolation space) and it's supposed to reflect what the respective market empirically suggest about its 'dynamics' (strike-stickiness, delta-stickiness). Moneyness (forward-moneyness, log-moneyness, delta-moneyness etc.) definition itself is also relevant (definitions for what ATM means are also non-trivial choices, e.g. delta-neutral, forward, spot). Back to FX option market example, see such complexity explained well here.

3. Parameterization can be sofisticated as in interest rate derivative market (SABR volatility model), with those parameters ('stochastic' $$\alpha$$, $$\beta$$, $$\rho$$ and $$\nu$$) themselves becoming a 'real thing' (that is vega risk with respect to them is the measure that's intensely monitored/limited).

1. Call and a put of the same strike have the same I.V, in theory.

2. The ONLY reason for this to differ is the limits to arbitrage on call put parity. Now this is a static strategy that has no rebalancing - so the only problem here is transaction costs in buying/shorting the stock. So if you have reason to believe that this strategy is difficult to implement, call and put IV's can differ.

3. If you answered yes to the above, then the choice of instruments depends on what you want to achieve. If you want to use these as calibration instruments, then the choice depends on the product you're pricing.

4. If you answered no (i.e. you believe it is easy to implement a strategy on call put parity), then you conclude that the difference in implied vols is due to noise in the price data. In this case you may use the more liquid of the calls and puts.

In general what happens is more of a combination of the two.

just to add to the other answers, the smile is essentially theoretical, in practice since the 87 crash, investors value more downside protection and the demand is higher for out of the money puts, leading to a volatility skew/smirk.

also bear in mind that these effects are more pronounced closer to the maturity, at inception the IV curve is essentially flat.

• While definitely interesting to note, this doesn't relate to the answers to the question, right? The actual shape of the smile or skew or smirk has no impact on whether the implied volatility is the same for puts and calls of the same strike, correct? Jul 9 '20 at 19:34
• are deep ITM calls as much in demand, as deep OTM puts?
– John
Jul 9 '20 at 19:42
• Hmm, well because of put-call parity I would imagine that least theoretically the demand for buying ITM calls since by put-call parity you can replicate the put with the call and get the same downside protection by also shorting the stock. So if I'm offering to take on someones risk and can do it by selling an OTM put or selling a an call and buying the stock, and cash (put-call parity) I would choose the cheaper option, driving the IV of the two to be the same Jul 9 '20 at 19:54
• Perhaps this was just a convoluted way of saying IV is the same for puts/calls because of put-call parity in theory, and practical considerations make this not so. Still that's getting to the actual question though where as my comment asked whether or not the shape of the smile/skew has any bearing on if it holds or not. Jul 9 '20 at 19:59