Option Skew — Part 1: Put-Call Parity and Volatility Smiles
In this article, first I will explain how put-call parity indicates that the implied volatility used to price call options is the same used to price put options and then I will define volatility smile and volatility skew.
Put-Call Parity
The put-call parity is a no-arbitrage equilibrium relationship that relates European call and put option prices to the underlying asset’s price and the present value of the option’s strike price. In its simplest form, put-call parity can be presented by the following relationship:
PV(X) can be represented in continuous time by:
Since put-call parity is a no-arbitrage relationship, it will hold whether or not the underlying asset price distribution is lognormal, as required by the Black-Scholes-Merton option pricing model.
If we simply rearrange put-call parity and denote subscripts for the option prices to indicate whether they are market or Black-Scholes-Merton option prices, the following two equations are generated:
Subtracting the second equation from the first give us:
The relationship shows that, given the same strike price and time to expiration, option market prices that deviate from those dictated by the Black-Scholes-Merton model are going to deviate in the same amount whether they are for call or puts.
Since any deviation in prices will be the same, the implication is that the implied volatility of a call and put will be equal for the same strike and time to expiration.
Volatility Smile and Volatility Skew
Actual option prices, in conjunction with the Black-Scholes-Merton model, can be used to generate implied volatilities which may differ from historical volatility. When option traders allow implied volatility to depend on strike price, patterns of implied volatility are generated which resemble “volatility smiles”.
These curves display implied volatility as a function of the option’s strike (or exercise) price.
In this series of articles, I will examine volatility smiles for both currency and equity options. In the case of equity options, the volatility smile is sometimes referred to as a volatility skew since, as we will see in the next article, the volatility pattern for equity options is essentially an inverse relationship.
Summary
Put-call parity indicates that the deviation between market prices and Black-Scholes-Merton prices will be equivalent for calls and puts.
Hence, implied volatility will be the same for calls and puts.
About the Author
Actuary Roi Polanitzer is the owner and chief valuator of Intrinsic Value. He is one of the three leading experts in Israel in the areas of quantitative finance, option valuation, risk management and financial engineering. He has consulted for accounting firms, financial advisory firms, investigative auditing firms and publicly-traded and privately-held companies in Israel on risk analysis, valuation, and real options, and has written numerous papers and articles on those topics. He is the founder and Chairman of the Israel Association of Financial Valuators and Actuaries. Actuary Polanitzer is also the founder and CEO of the Professional Data Scientists’ Israel Association and the owner and chief data scientist of Prediction consultants. He has an MBA in business administration and BA in economics. He is a certified Financial Risk Manager (FRM), a certified Chartered Risk Manager (CRM), a certified Quantitative Finance Valuator (QFV), a certified Financial and Economic Modeler (FEM), a certified Market Risk Actuary (MRA), a certified Credit Risk Actuary (CRA), a certified Python Data Analyst (PDA), and a certified Professional Data Scientist (PDS).