Why Put and Call Implied Volatilities Can Differ Within Bid-Ask Spreads
Summary
The document considers whether a call and put with the same strike and maturity must have equal implied volatility, including for deep in-the-money or out-of-the-money options. It explains that put-call parity constrains prices when the options share an implied volatility under the model assumptions, but differing implied volatilities do not automatically imply an executable arbitrage.
The key practical issue is transaction costs. A conversion or reversal uses a combination of option and underlying bids and asks, so the spread affects whether the trade can lock in a profit. These execution prices create a range within which call and put implied volatilities may differ without an arbitrage opportunity. The answer refers to a trading grid as an example, but the document provides no details about its data, market, or calculation. Its conclusion is therefore about the effect of spreads, not a universal size or pattern of volatility differences.
Key ideas
- Put-call parity constrains option prices under shared assumptions, but does not require identical quoted implied volatilities in every market condition.
- Different implied volatilities alone do not establish an executable arbitrage.
- Conversion and reversal trades must be evaluated using the relevant bid and ask prices.
- Bid-ask spreads create a range in which call and put implied volatilities may differ without arbitrage.
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Full text
# Implied volatility equality for deep in/out-of-the-money put and call
# Implied volatility equality for deep in/out-of-the-money put and call
Someone posed the following question.
Given a strike $K$ and the stock price $S$ and the same maturity are the implied volatilities of the call and put with these same parameters equal for $|S-K|\gg0$ (deep in/out-of-the-money) for a bid (ask) price?
I think they are the same by virtue of the put-call parity. Is there any peculiar situation where the equality breaks down?
## Answer by Chenghao LU (score 2, accepted)
https://quant.stackexchange.com/a/47101
Put-Call Parity $C - P = S - K*e^{-rt}$ provided the implied volatility of $C$ and $P$ are the same. If the implied volatilities are different, there could be arbitrage taking opportunities exist. However, it doesn't mean there must be an arbitrage opportunity. If the implied volatilities being different doesn't result in arbitrage opportunity, then the different implied volatility can exist in the market.
E.g. the implied volatility of $C$ can be lower than $P$, because by arbitrage we will try to execute reversal $C - P - S$, which is Ask - bid - bid. Here the spread cost us more. Similarly, if the implied volatility of $C$ is higher than $P$, we can try to execute conversion $P - C + S$, which is Ask - bid + Ask.
As shown, when executing reversal or conversion, because of spread, the two boundaries are different. It gives an upper boundary and a lower boundary of the difference of implied volatility between the put and call to float within.
As shown in the trading grid, the implied volatilities are indeed different: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.