Back-to-Basics: The Volatility Surface

Teri Geske
Senior Vice President, Product Development

First, we’ll briefly review the concept of implied volatility. Given a market price for any option (whether on an individual stock, an equity index, an interest rate, a commodity or an FX rate), we can derive the volatility parameter that would cause our option pricing model to produce that price. This is the implied volatility, i.e., the volatility that is implicit in the quoted price. In an over-simplified example, assume an option pricing formula that takes the form, Price = $5 + Vol/2. If we are quoted a price of $7, the Implied Volatility is 4.0. In practice, market participants quote implied volatility using the Black-Scholes option model (the Black ’76 model for fixed income options such as caps, floors and swaptions). Because all market participants agree to use the Black-Scholes model for the purpose of calculating implied volatilities, these volatilities can be, and frequently are, quoted in lieu of dollar prices. For example, if a dealer is asked to quote a price on a 3 year LIBOR interest rate cap (with an at-the-money strike rate), the “price” quoted would be something like “26%”, which is the volatility parameter that would produce the dealer’s dollar price from the Black model. Quoting prices in terms of volatility (or “vols”) instead of dollar prices is consistent with the notion that options are a way of buying or selling volatility.

When observing the prices for actively traded options, we can see that implied volatilities are not constant across strike prices. Note that this is counter to a key assumption in the Black-Scholes model, namely that changes in stock prices or index levels are normally distributed with a volatility that remains constant regardless of the level of the underlying random variable. So, it is somewhat ironic that when the market uses the Black-Scholes pricing framework as a means of quoting market implied volatilities, the results reveal information that is inconsistent with the Black-Scholes model.

For a given time to expiration, implied volatility tends to be lowest for at-the-money options, and higher for deep in-the-money or out-of-the-money options. When we graph the implied volatilities against “in-the-moneyness” or strike levels (where P – K = Price – Strike), the result is often a curve in the shape of a smile (a), or a skew (b). It is also possible for the volatilities implied by market prices to produce a “frown”, where the at-the-money volatility is higher than the volatilities for out-of-the money strike levels, or some other non-linear shape.

The presence of the volatility smile (or skew, or frown) discloses the market’s view that volatility itself is likely to change if interest rates (or a stock price or index level) rise or fall from current levels. This is somewhat analogous to the notion of an implied forward term structure of interest rates, which in theory, contains information about the market’s expectations for future interest rate levels. One interpretation of a volatility “smile” is that volatility is expected to increase if the underlying asset price moves dramatically in either direction, which has some intuitive appeal. However, just as implied forward rates are not necessarily a good predictor of where interest rates will be in the future, the volatility smile does not necessarily do a good job of predicting future realized volatility at different market levels.

When we graph implied volatilities against time to expiration, the result is a “term structure of volatility”. For example, the quoted price a 2 year LIBOR interest rate cap might equate to an implied volatility of 30%, while the implied volatility of a 3 year cap might be 26%. Typically, the implied term structure of volatility for interest rate derivatives is downward sloping, meaning volatilities are higher for short expirations than for long expirations, although there can be a “hump” at the very short end of the volatility term structure. By combining the graph of volatility and strike levels with the graph of volatility and time-to-maturity, we produce a three-dimensional diagram, known as the volatility “surface”. Since implied volatilities are obtained from dealer quotes for a distinct set of strike levels and expirations, constructing a continuous volatility “surface” requires an algorithm to interpolate between the values obtained from dealer quotes. Thus, one volatility surface derived from a set of prices may not be identical to another volatility surface based on the same set of quoted prices.

In summary, the subject of volatility and asset pricing has many dimensions. Implied volatilities, which, for fixed income markets, are derived from interest rate cap, floor and/or swaption prices, can be used to construct a volatility surface, reflecting the market’s consensus estimate of future volatility across rate (price) levels and across time.

We hope this review of the basics of the volatility surface has been useful. If you have any suggestions for future Back-to-Basics topics, please let us know – email your ideas to fia.marketing@interactivedata.com.