- Noel Smith, Leo Capalleja & Sebastian Pachon

# Dispersion Trading Part 3

In part three of this series, we provide a toolkit for anyone who is interested in trading dispersion. We begin by highlighting an important property of correlation and volatility. We then provide a basic methodology to estimate the cheapness/richness of implied correlation (IC). Finally, we discuss basic approaches to portfolio construction and monetization.

### Mean reversion

Imagine a pufferfish in a fish tank. It is initially in its normal, uninflated form. After a couple of minutes, the pufferfish feels threatened and inflates itself as a self-protection mechanism. If we asked you to bet on the fish either returning to its normal state or remaining inflated, which outcome would you choose? Most people would bet on the former, because they know that the fish will eventually return to its uninflated form. We assume that the size of the fish might vary from time to time, but the fish will be in its normal form for the majority of the time. In other words, the size of the pufferfish follows a mean-reverting process, where deviations might occur, but over the long term, it will revert to its mean.

Financial markets offer examples of both mean-reverting and non-mean-reverting assets, and identifying which assets exhibit which behavior can be useful and profitable. Stocks, for example, do not exhibit mean reversion. A stock’s expected return should be above the risk-free rate, which is positive in the vast majority of cases. Interest rates (and interest rate volatility), on the other hand, do mean revert. While we may talk about high-interest-rate and low-interest-rate environments, risk-free rates cannot have a positive or negative drift forever. Other metrics that mean revert include correlation and volatility. This fact is actively used and traded in the market.

While price can go anywhere, volatility and correlation tend to mean revert.

Traders tend to sell (buy) extreme spikes (falls) in volatility because the behavior of the underlying asset is not expected to remain that unstable (stable). Same idea for correlation: if asset A has a long-term correlation of -0.6 with asset B, but recently has been realizing a correlation of +0.8, most would expect this metric to return to normal, or at least decrease from the extreme level.

The first plot shows that when spikes in implied volatility occur, they revert to the median rapidly because there’s no expectation for such turbulent behavior to last for long. The second graph, similarly, shows that overshoots/undershoots on correlation do not last long and, as we will see next, create opportunities for dispersion traders.

### Timing a dispersion trade

Dispersion traders are incentivized to increase risk when IC is high and decrease risk when IC is low. As such it is helpful to understand the bounds of IC. The maximum value IC can reach (under zero arbitrage conditions) is 1. Any IC value observed above 1 indicates an arbitrage opportunity (index volatility too high or constituent volatilities too low). The minimum value IC can reach is technically -1 (recall the calculation for IC in __Part 2__). However, the lowest values observed for the one-month IC are around 0.07, so we can think of zero as a *soft *lower bound. Therefore, a rational trader would maximize dispersion risk at an IC of 1 and minimize risk at an IC of 0. However, IC will rarely reach its extreme bounds. For example the S&P 500 3-month IC has mostly oscillated between 0.20 and 0.60 over the last decade (with an exceptional 0.90 high during COVID). So a dispersion trader must typically find more nuanced ways to spot opportunities than simply waiting for the minimum/maximum possible IC.

### IC rolling percentile

Determining whether the observable IC is high or low, like many financial metrics, depends on the context. As such, a dispersion trader must use their available tools to estimate if current levels are favorable for increasing or decreasing risk. Here, we present an example of a quantitative approach that a trader may use to perform this estimation.

Create a time series of IC using historical data from options on the index and options on the constituent stocks. This will show the market’s historical expectation of correlation between the constituents.

Set a rolling look-back window. There is no a priori correct answer as every trader might have different reasons for choosing a particular time horizon. For example, we may choose 252 trading days.

For every point of the IC time series obtained in the first step, compute its percentile if put in a set with the previous 251 trading days.

### building a dispersion portfolio

Theoretically, the most precise way to create a dispersion portfolio is to sell options on the desired index and buy options on every single one of its constituents according to the constituents’ corresponding index weights. This works in theory but is impractical for multiple reasons. First, this would require the trader to manage options on 500+ stocks and adjust the weights almost continuously. Second, this would require the trader to buy and sell very illiquid options as the majority of the constituents in the S&P 500 do not have very liquid options. Trading these illiquid assets back and forth would be prohibitively expensive and erode PnL drastically. Lastly, some of the constituents of the S&P 500 are not even optionable.

In reality, a more practical approach is to design a replicating portfolio for the index using fewer (e.g. 30 to 60) stocks. The list of eligible stocks can be filtered for liquidity to reduce transaction costs. Additionally, a trader may optionally employ an altered weighting scheme. For example, stocks may be weighted such that the index’s sector weights are proportionally represented even if it means using stock weights that deviate from the stocks’ index weights. This modified approach drastically reduces the complexity of creating a dispersion portfolio and makes it easier to re-balance the portfolio over time since the options in the portfolio are more liquid.

### monetize a dispersion trade

A __delta-neutral__ dispersion portfolio is profitable under two scenarios:

The future realized volatility of the single-name stocks in the portfolio is higher than the implied volatility that was bought.

The future realized volatility of the index is lower than the implied volatility that was sold.

If either (or both) of these scenarios take place, delta-hedging may be used to realize a positive PnL and “re-center” the portfolio to delta-neutral. In practice, dispersion portfolios are not always delta-neutral and some delta risk is typically present. Furthermore, a delta-neutral (according to standard definition) dispersion portfolio will typically have implicit long delta risk due to its short index __vega__ exposure. Note that optimal delta-hedging is an ongoing topic of discussion and research in options circles and is beyond the scope of this article.

### Short dispersion?

So far, we have discussed long dispersion (short index volatility) but not short dispersion (long index volatility). Although it is straightforward to define a short dispersion portfolio as short the single-name straddles and long the index straddle, this is in our opinion a much trickier trade. As we have mentioned before, there are advantages to long dispersion. For instance, the unpredictability of future events for single-name stocks and the liquidity premium present in index options. Moreover, if you recall the mean-reverting property of correlation, it is true that while a positive spike reverts quickly, a low correlation print does not tend to revert as quickly. In other words, it is easier to time a long dispersion trade than a short one.

We have our own answers to the above question about when/how/if to be short dispersion (long index volatility). Reach out to our team to discuss.