Defining Option Moneyness So Volatility Surfaces Preserve Put-Call Parity
Summary
The document clarifies how a volatility surface indexed by moneyness can remain consistent with put-call parity. Although the question assumes calls and puts use opposite moneyness ratios, the answer explains that moneyness is generally defined the same way for both option types. Each moneyness level then maps to one strike for a given spot price, and calls and puts at that strike and maturity share implied volatility under parity.
Examples relate strike levels to spot and distinguish out-of-the-money puts from out-of-the-money calls. The answer also notes that display conventions can express moneyness as a ratio to spot or as an in-the-money/out-of-the-money measure. These conventions and the parity explanation concern the stated surface setup; the document does not derive pricing formulas or address adjustments that may matter for American options or other market conventions.
Key ideas
- A moneyness surface usually applies the same moneyness definition to calls and puts.
- For fixed spot and maturity, each moneyness level corresponds to a single strike.
- Put-call parity implies equal implied volatility for a put and call with the same strike and expiry in the described setting.
- Moneyness can be displayed as a strike-to-spot ratio or through in-the-money and out-of-the-money measures.
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Full text
# How does a volatility surface based on moneyness instead of strike stay consistent with put-call parity?
# 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?
## Answer by AKdemy (score 4, accepted)
https://quant.stackexchange.com/a/74200
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?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.