Volatility cone is a visualisation tool for the display of historical volatility term structure. It was introduced by Burghardt and Lane[1] in early 1990 and is popular in the option trading community. Using the same methodology, we can extend the use of such chart for periodic return data. I find these charts useful not only for options but also for the general market.

**Volatility cone construction**

First, let’s look into a sample volatility cone and explain how it is built. Figure 1 shows the volatility cone derived from daily S&P500 return data between 1990-01-03 and 2014-08-22. The horizontal axis represents the size of the sliding window with the vertical axis represents the annualised volatility. The variation in annualised volatility is large when the sliding window is small and gradually converges to a certain stable value when examining a longer time horizon. To construct the volatility cone, we start with the smallest sliding window size say 5 days. The volatility calculation for the first data point is calculated as the annualised standard deviation for the daily returns of the first 5 trading dates since 1990-01-03. Then we move the window one day forward (ie 5 trading dates since 1990-01-04) and repeat the process until the end of the sequence. We then sort the annualised 5-day volatility series in ascending order and identify the min, lower quartile, median, upper quartile and max. After completed the calculation for the smallest sliding window size, we repeat the procedure for next sliding window size and so on.

Figure 2 shows the S&P periodic return cone. Its construction follows the same methodology. But instead of calculating the volatility calculation for each sliding window, we calculate the aggregate daily return for each period. The return distribution diverges when we consider longer and longer time horizon.

**What does a volatility cone tell us?**

To help visualising what volatility cone actually tells us, two different sets of random return data are used to construct volatility cones. In the first set (see Figure 3), we generate a set of random normal numbers (mean=0, s.d.=0.01). In the second set (see Figure 4), we simulate serial correlation effect (which is very common in financial time series) using the following formula: where is a random normal variable. We use mean=0, s.d.=0.01 and correlation coefficient in this case.

While the volatility of both sets converges to the same stable value when using large sliding windows, the case with serial correlation not only has a narrower distribution but most of the percentile bands (min, 25/50/75-percentile) also points downward when the sliding window being used is relative small. It is because when a sequence of random numbers is serial correlated, a number tends to move in tandem with its immediate ancestors and is largely independent to those ancestors that are far away. When the sliding window is small, all the return samples are likely to move in the same way and thus have reduced volatility. This is a pattern to watch out for if serial correlation is suspected.

**Applications**

1) The return and volatility cones show the base rate, marginal/unconditional probability for different time spans. It helps us to answer questions and builds a mental map for questions such as the likelihood of a x% rally in a y-week period and whether the implied volatility term structure is too steep or too flat when compared to the history. For Bayesian estimation framework, we can start from the base rate and combine our domain knowledge (which can be quantitative or from more subjective understanding of the market) to come up with more reliable estimates.

2) While marginal probability estimation is too coarse for many applications, we can merge our market knowledge (anything from calendar effect to regime switch model) and turn the unconditional probability to conditional.

3) Option RV: it is the area covered initially by Burghardt and Lane. It is best to refer back to the excellent original article for this.

[1] G. Burghardt and M. Lane, “How to tell if options are cheap”, The Journal of Portfolio Management Winter 1990, Vol. 16, No. 2: pp. 72-78