Put–Call Parity and Matching Implied Volatility Skews

Summary

The note explains why calls and puts at the same strike share the same implied volatility in a standard pricing framework. Put–call parity links their prices through a forward-related value that does not depend on volatility. Given one option’s price, parity therefore determines the price of its counterpart at the same strike.

Because the same volatility input prices both options consistently under the Black–Scholes model, their implied volatilities match strike by strike, producing the same skew curve. The explanation is conceptual rather than a treatment of market data or trading practice. It assumes comparable options and pricing inputs; the note does not discuss practical differences such as bid–ask spreads, dividends, exercise features, or mismatched market quotes.

Key ideas

• Put–call parity relates call and put prices at the same strike through a volatility-independent forward value.
• A call’s price and the parity relation determine the corresponding put’s price.
• Under the same pricing assumptions, both options therefore have the same implied volatility at a given strike.
• Matching implied volatilities across strikes yield the same skew curve for calls and puts.

Tags

# Why vertical skew is same for puts and calls


# Why vertical skew is same for puts and calls












What is the reason that the vertical volatility skew graph(decreasing IV as the strikes increase) is the same for the puts and calls? The loose explanation is because of put call parity, but I am not able to find any more details.

I am looking for an explanation that makes sense intuitively.

## Answer by Mark Joshi (score 2)

https://quant.stackexchange.com/a/15593

put call parity guarantees that the implied volatility of a call and put with the same strike is the same. So the smile graph is the same as well and so are all quantities derived for it.

In more detail, $$ C(K) = P(K) + F(K) $$ The value of $F(K)$ is model independent and does not depend on volatility. So knowing the implied of $C(K)$ gives you the price of $P(K)$. The same implied vol will work for $P(K)$ since put-call parity works for all models so it must hold for the BS model with that vol.

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.