The Volatility Surface Explained

The volatility surface is a three-dimensional plot of stock option implied volatility seen to exist due to discrepancies with how the market prices stock options and what stock option pricing models say that the correct prices should be. To gain a full understanding of this phenomenon, it is important to know the basics about stock options, stock option pricing, and the volatility surface.

Equity stock options are a certain type of derivative security that gives the owner the right, but not the obligation, to execute a trade. 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, on or before a specific date, known as the expiration date. 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. Also, while these names have nothing to do with geography, a European option may be executed only on the expiration date, while an American option may be executed on or before the expiration date. Other types of option structures also exist, such as Bermudan options.

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:

1. The underlying stock does not pay a dividend and never will.
2. The option must be European-style.
3. Financial markets are efficient.
4. No commissions are charged on the trade.
5. Interest rates remain constant.
6. The underlying stock returns are log-normally distributed.

The formula is slightly complicated, but to price an option, 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.

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. And for a given strike price, implied volatility can be increasing or decreasing with time to maturity, giving rise to a shape known as a volatility smile, because it looks like a person smiling.

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 shorter time to maturity have multiple times the volatility than 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.

The fact that the volatility surface exists shows that the Black-Scholes model is far from accurate; however, market participants are aware of this issue. With that said, most investment and trading firms still use the Black-Scholes model or some variant of it. (Read more about contending with the limitations of Black-Scholes.)