### Does implied vol vary for calls vs puts?

• Volatility skew tells us that options with the same maturity at different strikes can have different implied vol. However, can a corresponding call and put for the same strike and maturity have different implied vol?

8 years ago

Taking away all frictions and incomplentess of the market, the theory says that European Call and Puts do have the same implied volatility unless there is an arbitrage opportunity by put call parity $$C(t,K) - P(t,K) = DF_t(F_t - K)\ .$$ If you plug the Black-Scholes formula here for the prices of the call and the put, you will see that the equality only holds if and only if volatilities are equal. $$DF_t[F_t(\Phi(d_+^{Call}) + \Phi(-d_+^{Put})) - K(\Phi(d_-^{Call}) + \Phi(-d_-^{Put}))] = DF_t(F_t-K)$$ Since $\Phi(x)+\Phi(-x)=1$, put call parity holds if and only if $d_\pm^{Call} = d_\pm^{Put}$, so if and only if $\sigma_{Call}(t,K) = \sigma_{Put}(t,K)$.

In practice there are bid-ask spreads and liquidity issues which implies that observable prices of European options do no align necessarily to the theory.

For American options (the standard options traded on Equity stocks) we can still think in terms of implied volatility but there is no such thing as a put-call parity so implied volatilities are not necessarily equal anymore. There are some put-call parity style inequalities but those are not strong enough to guarantee the equality of volatilities.

exactly what I said, but +1 for writing down second formula