# How does a volatility surface based on moneyness instead of strike stay consistent with put-call parity?

By definition due to put-call parity the implied volatility will be the same for puts and calls with the same strike price and time to maturity. Meanwhile, a volatility surface is often quoted in terms of moneyness and maturity rather than strike and maturity. So one node will correspond to e.g 1 year maturity and 1.25 moneyness. Now I take moneyness to mean S/K for a call, and K/S for a put, but for a given spot price S this would give different values of K for puts and calls. Say S=1.25 we have that K=1 for the call option, and K = 1.25^2=1.5625 for the put. Now this could hold of course with a slightly asymmetrical smile centered around K=1.25, but does this really always hold? I would think that the shape of the smile and put-call parity are completely unrelated. Perhaps I'm interpreting moneyness the wrong way?

• I think the issue may be resolved when using logarithmic moneyness, i.e. $\log(F/X)$. Commented Oct 22, 2020 at 8:59
• I don't see how. Could you provide a calculation example? Commented Oct 22, 2020 at 9:04
• If your call has a moneyness $m=\log(105/100)\approx 0.05$ then the corresponding put has a moneyness of approx. $-0.05$ and the two options would be tied by Put-Call-Parity, no? Commented Oct 22, 2020 at 10:02
• @Kermittfrog Hmm, not sure. How about we frame it another way: Say we are creating the surface and want to create the node with 0.05 log moneyness, we would look for the call with a strike that satisfies 0.05=log(S/K), which is S=105 would give K = exp(0.05)*105=110.38. We could also look for a put with a strike that satisfies 0.05=log(K/S) which leads to K=99.87. So then we would have that sigma(K=99.87)=sigma(K=110.38) which makes a perfectly pretty smile so it's not unreasonable, but there's nothing that says we need to have a perfect smile like this which this method seems to assume. Commented Oct 22, 2020 at 10:31
• I think you are using call and put convention at the same time, i.e. forward call moneyness = $F/K$ whereas forward put moneyness = $K/F$. I would stick with one notation, i.e. a forward call moneyness of $m=ln(F/K)$ equals a forward put moneyness of $-m$. And $C(m)+Ke^{-rt}=P(m)+S$. Does that work? Commented Oct 22, 2020 at 11:21

If you have a moneyness surface, moneyness is usually defined the same way for calls and puts. I have seen in another question from you that you use Bloomberg. On OVDV, moneyness is defined as K/S. Your screenshot from the linked question shows spot is 1814.79; since you ticked Strikes, you can also see the corresponding strikes (highlighted in Red right underneath the moneyness levels):

Therefore,

• $$\color{grey}{100 \% \ moneyness}$$: (also called At-the-money-Spot ATMS) is displayed as 1814.8.
• $$\color{blue}{<100 \% \ moneyness}$$: This corresponds to OTM Puts (ITM Calls) - for example 90% moneyness corresponds to a strike of 1633.3 (90% of 1814.8),
• $$\color{YellowGreen}{>100 \% \ moneyness}$$: This corresponds to OTM Calls (ITM Puts) - 105% is 1905.5 and so forth.

OVME allows you to display moneyness in a different way relative to ATM, in terms of ITM (OTM), but it is essentially the same. The screenshot below corresponds to a Spot of 3166.48.

• 1373.28 is 56.63% ITM for a call ($$1- 1373.28/3166.48$$, where $$1373.28/3166.48 \approx 42.37 \%$$ moneyness on OVDV)
• 2746.56 is 13.26% OTM for a put ($$1- 2746.56/3166.48$$, where $$2746.56/3166.48 \approx 86.76 \%$$ moneyness on OVDV)

So, each moneyness level corresponds to one strike - and put and call for the same strike have the same vol. You can look at Is it possible to have only one volatility surface for American options (that fits both calls and puts) to see why OVDV shows only one surface for calls and puts?