# 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?

• If the option prices satisfy the put-call parity, then their implied volatility must be the same. So the question is: can the PCP be violated for deep OTM/ITM options e.g. due to market inefficiencies. This may occasionally occur. Either you can average the implied volatility or one typically prefers the implied volatility of the OTM option. – Kevin Jul 30 '19 at 12:13
• @KeSchn: Could you please be more specific about what exactly market inefficiency is? Is it because it is just quoted or surveyed price/implied volatility rather than bid or ask that are obliged to be executed? Why does one prefer the implied volatility of the OTM over that of the ITM option? – Hans Jul 30 '19 at 20:44
• Sometgijg else to consoder is thst deep itm options have delts while otm do not. Of you habe sold the option your margon rewuirements sill be grester for an itm option, this imoacts the price. – will Jul 31 '19 at 5:58
• @Cornholio: How does the discrepancy between the implied volatilities of the call and the put come in while the put-call parity holds? – Hans Aug 3 '19 at 19:22
• @will: How exactly do the different delta's impact the prices to break the put-call parity? – Hans Aug 3 '19 at 19:25

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.