The volatility surface is a three-dimensional chart that shows how implied volatility varies across different strike prices and expiration dates for options on the same stock. These volatilities differ because of market factors and pricing discrepancies, which means not all options are valued equally. It helps to know the basics of stock options and the Black-Scholes model, which is used to estimate option prices. However, the model’s assumptions, like constant volatility and perfectly efficient markets, often don’t match real-world conditions, leading to the variations seen on the volatility surface.
Key Takeaways
- The volatility surface is a 3D plot showing implied volatilities of options based on strike prices and expirations.
- Implied volatility reflects the expected volatility of an option’s underlying stock over its lifespan.
- The Black-Scholes model uses volatility to price options but assumes a flat volatility surface, which isn’t often the case.
- Market discrepancies result in a non-flat volatility surface, often displaying volatility smiles or skews.
- The volatility term structure describes how an option’s implied volatility changes across different expiration dates.
Understanding Stock Options
Equity stock options are a certain type of derivative security that gives the owner the right, but not the obligation, to execute a trade. Here we discuss some basic types of stock options.
Call Option
A call option gives the owner the right to purchase the option’s underlying stock at a specific predetermined price, known as the strike price (or exercise price), on or before a specific date, known as the expiration date. The owner of a call option makes a profit when the underlying stock increases in price.
Put Option
A put option gives the owner the right to sell the option’s underlying stock at a specific price on or before a specific date. The owner of a put option makes a profit when the underlying stock decreases in price.
Other Option Types
Also, while these names have nothing to do with geography, a European option may be executed only on the expiration date. In contrast, an American option may be executed on or before the expiration date. Other types of option structures also exist, such as Bermuda options.
Fundamentals of Option Pricing
The Black-Scholes model is an option pricing model developed by Fisher Black, Robert Merton, and Myron Scholes in 1973 to price options. The model requires six assumptions to work:
- The underlying stock does not pay a dividend and never will.
- The option must be European-style.
- Financial markets are efficient.
- No commissions are charged on the trade.
- Interest rates remain constant.
- The underlying stock returns are log-normally distributed.
The formula to price an option is slightly complicated. It uses the following variables: current stock price, time until option expiration, strike price of the option, risk-free interest rate, and standard deviation of stock returns, or volatility. On top of these variables, the formula uses the cumulative standard normal distribution and the mathematical constant “e,” which is approximately 2.7183.
Exploring the Volatility Surface
Of all the variables used in the Black-Scholes model, the only one that is not known with certainty is volatility. At the time of pricing, all of the other variables are clear and known, but volatility must be an estimate. The volatility surface is a three-dimensional plot where the x-axis is the time to maturity, the z-axis is the strike price, and the y-axis is the implied volatility. If the Black-Scholes model were completely correct, then the implied volatility surface across strike prices and time to maturity should be flat. In practice, this is not the case.
The volatility surface is far from flat and often varies over time because the assumptions of the Black-Scholes model are not always true. For instance, options with lower strike prices tend to have higher implied volatilities than those with higher strike prices.
As the time to maturity approaches infinity, volatilities across strike prices tend to converge to a constant level. However, the volatility surface is often observed to have an inverted volatility smile. Options with a shorter time to maturity have multiple times the volatility compared to options with longer maturities. This observation is seen to be even more pronounced in periods of high market stress. It should be noted that every option chain is different, and the shape of the volatility surface can be wavy across strike price and time. Also, put and call options usually have different volatility surfaces.
Important
As you move up or down the strike price from the at-the-money strike, implied volatility can be either increasing or decreasing with time to maturity, giving rise to a shape known as a volatility smile because it looks like a person smiling.
Why Does the Volatility Skew Exist?
Since the late 1980s, options traders have recognized that downside put options have higher implied volatilities in the market than their models would otherwise predict. This is because investors and traders who are naturally long will buy protective puts for insurance purposes. This bids up the prices of the puts relative to upside options. As a result, there tends to exist volatility skew. If upside options are also bid, sometimes due to expectations of a potential takeover, then a volatility smile occurs as both extremes have increased implied volatilities.
What Is Local Volatility?
Local volatility considers the implied volatility of just a small area of the overall volatility surface. It may hone in on just a single option, either a call or a put of a specific strike price and expiration. The volatility surface may be thought of as an aggregation of all the local volatilities in an options chain.
What Is Volatility Term Structure?
Volatility term structure is part of the volatility surface that describes how options on the same stock will exhibit different implied volatilities across different expiration months, even for the same strike. Similar in concept to the term structure of bonds (where interest rates differ based on maturity), the volatility term structure may be either upward or downward sloping depending on market conditions and expectations. An upward-sloping term structure indicates that traders expect the underlying stock to become more volatile over time; and a downward slope that it will become less volatile.
The Bottom Line
The volatility surface reveals that implied volatility isn’t constant, which shows how market forces and investor expectations influence option pricing across strikes and expirations. This variation challenges traditional models like Black-Scholes, which assume stable volatility. Implied volatility can help you determine an option’s price and assess market sentiment, making it a critical concept for traders and analysts alike.
