The Volatility Surface: Everything You Need To Know

Let’s take a dive into everything you need to know about the incredibly versatile volatility surface.

The volatility surface is definitely one of the cooler-looking graphs in finance (I mean just look at it), but beyond its visual appeal there lies an incredibly useful tool. Let’s take a look under the surface and really understand how the beast works.

Background

A volatility surface is a 3D graph of implied volatilities plotted against strike prices and time to maturity for options on a particular asset.

Here’s what each axis represents:

1. X-axis: Strike prices (Moneyness)— The horizontal axis of the volatility surface represents the strike prices of the options contracts. Strike price is the price at which the underlying asset can be bought or sold if the option is exercised.
2. Y-axis: Implied Volatility — The vertical axis of the volatility surface represents the implied volatility, which is the estimated level of volatility that the market is expecting for the underlying asset over the life of the option contract. Implied volatility is calculated based on the price of the option contract and other factors such as the time until expiration, interest rates, and dividend yields. Implied volatility values are typically displayed as a color gradient or contour lines on the surface of the graph, with higher levels of volatility shown in warmer colors like red and lower levels of volatility shown in cooler colors like blue.
3. Z-axis: Expiration (Time-to-Maturity)— The third dimension of the volatility surface represents expiration dates of the options contracts. Expiration date is the date on which the option contract expires and becomes worthless if not exercised.

By plotting implied volatility against strike prices and expiration dates, the volatility surface provides a visual representation of the market’s expectations for how volatile an underlying asset will be over time. It is a useful tool for option traders and analysts to analyze and compare the prices of different options contracts and make informed trading decisions based on their views on market volatility.

Trading Opportunities

Volatility surfaces can be used to identify various trading opportunities, such as arbitrage, relative value trades, and dispersion trading. Here‘s a few examples:

1. Implied Volatility Arbitrage: “Arbitrage” opportunities can arise when the implied volatility of an option is significantly different from its historical volatility or from the implied volatilities of related options. Traders can take advantage of these discrepancies by buying options with low implied volatility and selling options with high implied volatility. This strategy is based on the assumption that implied volatilities will eventually revert to their historical levels or to the levels of related options.
2. Calendar Spreads: A calendar spread involves buying and selling options on the same underlying asset with the same strike price but with different expiration dates. The term structure of volatility, represented by the volatility surface, can indicate whether a calendar spread is attractive or not. If the volatility surface shows that short-term options have higher implied volatilities than longer-term options, a trader can sell the short-term option and buy the long-term option, betting that the term structure will revert to its normal shape.
3. Volatility Skew Trades: A volatility skew occurs when out-of-the-money (OTM) puts have higher implied volatilities than OTM calls. Traders can take advantage of this skew by implementing strategies such as a risk reversal, which involves selling an OTM put option and buying an OTM call option with the same expiration date. If the skew reverts to a more neutral or smile shape, the value of the risk reversal position could increase.
4. Dispersion Trading: Dispersion trading involves taking advantage of the difference between the implied volatility of an index and the implied volatilities of its individual constituents. The strategy usually involves selling options on the index and buying options on its individual stocks. If the implied volatility of the index is higher than the average implied volatility of its components, a dispersion trader can profit from the difference if the discrepancies between the index and individual stock volatilities converge.

These examples illustrate how traders can use the information contained in the volatility surface to identify trading opportunities. However, it’s essential to consider transaction costs, liquidity, and risk management when implementing these strategies.

The Mechanics

Now, here’s the nitty-gritty of how a vol surface is built from scratch.

1. Implied volatility calculation:

• Implied volatility is the volatility implied by the market price of an option. It is the key input to the Black-Scholes-Merton (BSM) model, which is widely used for pricing European options. To calculate implied volatility, one needs to solve the BSM model for the volatility that equates the model price to the market price. This is typically done using numerical methods, such as the Newton-Raphson method or the bisection method.

2. Interpolation and extrapolation of the volatility surface:

Once implied volatilities are calculated for a range of options with different strike prices and maturities, the volatility surface can be constructed. However, not all options are actively traded, and there might be gaps in the data. Therefore, interpolation and extrapolation techniques are required to fill in the missing data points and create a smooth volatility surface.

• Interpolation:

Interpolation is the process of estimating the implied volatility for an option with a specific strike price and maturity, based on the known implied volatilities of other options. Common interpolation methods include:

• Linear interpolation: A simple and widely used method that assumes a linear relationship between two known data points. It is computationally efficient but may not capture the true curvature of the volatility surface.
• Cubic spline interpolation: This method fits a piecewise cubic polynomial to the known data points, providing a smoother estimate of the volatility surface. It requires more computation but can better capture the curvature of the surface.

Extrapolation:

Extrapolation involves estimating implied volatility beyond the range of known data points. This is typically done using the same interpolation methods as described above, but additional assumptions or constraints may be applied to ensure the extrapolated surface behaves reasonably.

For instance, one might assume that the implied volatility converges to a long-term average or follows a specific functional form, such as the constant elasticity of variance (CEV) model.

3. Model calibration:

The final step in constructing a volatility surface is to calibrate the model. This involves adjusting the parameters of the chosen interpolation and extrapolation methods to minimize the difference between the model-implied volatilities and the observed market-implied volatilities.

Various optimization techniques can be used for this purpose, such as least squares or maximum likelihood estimation. The calibrated model can then be used to price and hedge options, as well as to analyze the market’s expectations of future volatility.

All in all, the volatility surface is one neat tool.

If this article piqued your interest, you’d likely enjoy some of my other posts just like this one:

• Exploiting The Volatility of Volatility
• Arbitraging The Silicon Valley Banking Crisis
• It’s Showtime: Our Sports Betting Algorithm is Live and Profitable! [Code + Data Included]

Happy trading! :)