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Using Put-Call Parity to Align OTM Implied Volatilities

Article Quant Q&A · Author: Anouer Bhy
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Summary

This note addresses apparent asymmetry between out-of-the-money put and call implied volatilities near at-the-money. It attributes the discontinuity in the displayed data to an implied-volatility calculation that uses the spot index in place of a matching-expiry forward price, with a zero discount rate. Under that setup, equal-strike puts and calls can produce different reported implied volatilities.

The proposed method infers the forward and discounting inputs from quoted call and put mid prices using put-call parity, then recalculates implied volatilities with a forward-based Black model. The reported example produces a smoother curve where both option types are quoted at the same strike. The discussion treats parity violations as potential arbitrage, but the inference depends on reliable paired quotes and suitable pricing assumptions; the described data source and example do not establish a universal calibration procedure.

Key ideas

  • Using spot instead of the appropriate forward can create apparent put-call implied-volatility differences at the same strike.
  • Put-call parity relates paired option prices to the forward price, strike, and discount factor.
  • A regression across strikes can estimate forward and discounting inputs from quoted mid prices.
  • Recomputing volatility with forward-based pricing can remove an artificial discontinuity, subject to data and model assumptions.

Tags

OptionsDerivatives pricingVolatilityArbitrage
Full text
# How to Enforce Symmetry in Implied Volatilities Around ATM for OTM Puts and Calls?


# How to Enforce Symmetry in Implied Volatilities Around ATM for OTM Puts and Calls?












I am analyzing implied volatilities (IVs) for options on an underlying asset, and I noticed discrepancies in IVs for out-of-the-money (OTM) puts and calls near at-the-money (ATM). The attached plot shows the market IVs, bid IVs, and ask IVs for a range of strikes. The issue is that the IVs are not symmetric around the ATM strike. What are the best practices to enforce symmetry in IVs for OTM puts and calls near the ATM region? Is it some form of arbitrage ?

## Answer by Chris Taylor (score 8, accepted)

https://quant.stackexchange.com/a/81592

It looks like you pulled the data from the Deribit web page.

Their implied volatility calculations are very crude. They use the Black76 model, but rather than a forward price, they use the spot index price, with a discount rate of zero. This results in implied volatilities which are not equal for puts and calls with the same strike, which appears as a discontinuity around zero, as you noticed.

Since the exchange does not appear to have traded futures or forwards on the underlying index with a matching expiry date, we need to infer the forward price from looking at options prices, using put-call parity. Each put/call pair with a price should satisfy

$$ C - P = D (F - K) $$

or else there is an arbitrage. We can rearrange this to

$$ F + 1/D (P - C) = K $$

and run a linear regression using the quoted mid prices for calls/puts at each strike, to get $\hat{F} = 193.55$, which implies a discount rate of r = 13% (not that implausible for a stablecoin like USDC).

If we then use the Black76 model with this forward rate and discount rate to back out the implied volatilities, we get a much nicer looking chart with no discontinuity where puts and calls are quoted for the same strike.

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.