Implied Correlation
Implied Correlation Options prices are derived from a series of non-constant dependent variables.
- The price or value of the underlying asset the contract is covering
- The time to expiration of the contract
- Risk free rate of return
- Implied volatility
The relationship between the price of the underlying asset (1), and the price of the contract is direct in a call option, and inverse in a put option. When considering an option contract on an equity security, the closer the stock price gets to the strike price of the contract the higher the likelihood the price or the underlying will reach or exceed this strike. For this reason, the premium on a call option is directly correlated. When the price of the underlying increases, the opposite is true for a put option. As the price of the underlying approaches, or exceeds the strike price, the purchaser of a put option will find less value in their contract.
When considering the time to expiration (2) of either contract, while assuming a constant underlying price, the passage of time will deplete the value of the contract.
Risk free rate of return (3) is based on the current interest rate levels. The impacts of which are both macroeconomic and demonstrate the minimum required rate of return of an investment. Call options increase in value as interest rates increase and the opposite is true of put options.
The primary focus in this case will be implied volatility (4). Implied volatility is a measure of the potential risk of the underlying asset as it depicts market sentiment on the future fluctuations of the underlying. In an options contract in which the underlying is a single security, the implied volatility can be calculated by finding the other input variables (1-3) in conjunction with the current contract price and solving Black-Scholes formula for the only unknown variable.
In indices this and their derivative options contracts implied volatility is based on multiple assets. This implied volatility will instead be a factor of two variables. The implied volatility of its constituent underlying assets, and the correlation amongst these assets. This was recognized as in numerous instances the implied volatility of the constituents of an index would increase or decrease without a succeeding change in the implied volatility of the index itself [1].
Calculating Implied Correlation
The implied correlation of an index options contract is calculated by using the relative weights of the underlying companies and their individual implied correlations.
Where Wi , is the weight of the ith company, Pi , is the price per share of the ith company, and Si, represents the shares outstanding of said company.
By using weights of individual companies and their individual volatilities the overall variance of the index can be calculated as:
Where σ represents implied volatility, w represents the weight of the respective companies, and p represents the pair-wise correlation between the components.
The implied volatility of each company is found by calculating the weighted average of both at-the-money call and put options. The implied volatility of each contract is found using the Barone-Adesi, Whaley method [2].
Where Wc represents the weight of the nearest call option by price, and Wp represents the weight of the nearest put by price. The weights of these individual options are calculated as such;
Xc = Strike price of the nearest call option above current price
Xp = Strike price of the nearest put option below current price
S = Spot price
The individual variances, in conjunction with the weights of each component are plugged into the formula below to find the average implied correlation amongst index components.
If breaking down the formula for average correlation one will notice that the relationship between increases in the pair-wise correlation between the components of the index and the average correlation is direct.
This is because the index volatility is a function of the volatility of individual components as well as their correlation, denoted by p. The larger the difference between index volatility and the sum of weighted individual volatilities, the larger the overall correlation. This is supportive of the initial claim that the implied volatility of indices is individual volatility and correlation. If index volatility is high in comparison to its individual constituents, high correlation will be the driving factor.
This method of calculating is applicable to major indices as well as sector centric indices, such as the PHLX Semiconductor index, or indices based on market cap, such as the S&P 600, which exclusively includes small cap stocks.
Time Series Analysis
Inverse Relationships
An inverse relationship exists when considering the effects of price change on correlation and volatility. In periods of positive index returns, there are negative implications on correlation and volatility. COR3M is a trackable implied correlation derivative security which focuses on the 50 largest components of the S&P 500. As a result of the inverse relationship between index performance and correlation, implied correlation, similar to volatility, can be used as a speculative tool or a method of hedging for risk in volatile markets.
Empirical data has signaled that volatility is relatively mean reverting in comparison to other metrics and assets[3]. This may establish some framework for considering the behavior of implied correlation as they are both a depiction of market sentiment which usually spikes or rapidly declines when there are some extreme events.
If one considers that the implied volatility of an index, and any standalone asset is mean reverting, this may impact the understanding of implied correlation. If considering that the implied volatility of an index is a factor of individual implied volatilities, which are mean reverting, then implied correlation in equity indices should be mean reverting as well to fit the definition of mean reverting implied volatility. This method of thinking may establish some framework when considering the movement of implied correlation.
Because high correlation amongst companies with clear fundamental and operational differences is not regular the likelihood of consistently above average implied correlation in a diverse index is low. The same may be true for consistently sub average implied correlation.
After conducting an Augmented Dickey-Fuller (ADF) test to test for potential mean reverting time series processes, it was found that in short time spans (3 months), volatility indices show greater signs of mean reverting behavior than implied correlation. However, the implied correlation index, COR3M, showed greater mean reversion than equities assets such as the SPY. Results showed more rapid depletions in ADF statistics for equities assets as lag values were increased, than for the implied correlation and volatility indices.
Implications
The implications of this lie in potential dispersion trading schemes, where volatility trades are made on the individual components of an index, and an opposing trade is made on the volatility of the index itself, to make bets on assumptions regarding the implied correlation of the index. When predicting that an index will demonstrate high implied correlation, one would take a long position by buying call options on the index, while simultaneously selling call options on a portfolio of the individual components. If the opposite belief is assumed, the actor would buy call options on the individual stocks and sell call options on the index or ETF.
Written by HAKIL HAXHIU
- CBOE S&P 500® Implied Correlation Index
https://d1-invdn-com.akamaized.net/pdf1de153091a4e12d7d6ac9bcae0f8b553.pdf
- Barone‐Adesi, Giovanni, and Robert E. Whaley. “Efficient analytic approximation of American option values.” the Journal of Finance 42.2 (1987): 301-320.
- He, Ling T. “Mean Reversion of Volatility Around Extreme Stock Returns: Evidence from US Stock Indexes.” The International Journal of Business and Finance Research 7.4 (2013): 91-101.
