*Guest Post by Luis Cruz*

## A Quantitative Analysis of Volatility and Market Fundamentals for Trading in the Options Market Using Statistical Arbitrage

### Introduction

In the last few decades, there has been an astonishing demand in the market for derivatives. Options are a very powerful tool for investors and traders. They are complex financial derivatives that make it very compelling for institutional or individual investors because of their versatility. Options are a great instrument for speculation and hedging, however, their ability to leverage returns and their nature of being wasting assets (they lose their value as time progresses) make these financial instruments somewhat complex and a risky investment. This paper analyzes mean-reverting processes and the use of scraping tools to analyze how the popularity of a stock is changing across time with services like Google Trends.

### Trading Volatility

Different factors affect the price of an option and several approaches exist to determine the fair value of an option. One of these methods is the popular Black-Scholes method to arrive to the theoretical price of an option using the price of the underlying stock, the strike price, time until expiration, risk-free rate, dividends and one crucial factor that sometimes is left out by many novice traders, volatility. There are two types of volatility when dealing with options. One is the historical volatility, which is based on the past. Usually it is calculated by using an annualized standard deviation. Then, there is the expected volatility that the underlying stock will have in the future, that is, implied volatility. Implied volatility for European-style options is usually determined using Black-Scholes. For American-style options the price usually doesn’t differ that much from European options, except for the fact that they are usually more expensive because the buyer can exercise at any given time. One can neglect the theoretical price difference between the American and European options and therefore use Black-Scholes to calculate the implied volatility.

The Black-Scholes pricing formula for call options:

The given formula for Black-Scholes can be used to determine the expectations of the market for volatility in the future.

Volatility is basically what determines the options’ pricing. If there is a high probability point where a stock’s volatility will change in the future, the trader can determine the best approach to enter a trade. Volatility is usually considered mean-reverting, that is, there are extreme lows and extreme highs, historically speaking. By determining where implied volatility is low, the trader can buy undervalued options and then sell high when the volatility increases.

Another way to find undervalued or overvalued options is comparing the implied volatility to the historical volatility. If there is a significant divergence, an investor can be confident to enter a trade as it is a high probability entry point.

One example of such a case would be the following:

As you can see in the picture, Apple has had an implied volatility of around 30% on average. However, by looking at the last point, there is a huge divergence between implied and historical volatility. So at the moment, it’s better to enter a trade that’s based on selling options as they are considered overvalued.

Let’s consider another example. Teucrium Corn ETF (CORN traded at NYSEARCA) tracks the prices of corn. As we can see historically, implied volatility has been statistically above historical volatility.

However, if we calculate the cumulative distribute function, one can notice that implied volatility is low historically. The latest sample’s IV’s CDF is at 0.08, which is a good indication that this could be a good entry point for buying options as they are really cheap historically.

Remember that the cumulative distribute function for a process that is normally distributed is calculated with:

where ** p** is the probability that a single observation

**with mean**

*x***and standard deviation**

*µ***will fall in the interval (-∞,**

*σ***].**

*x*### Scraping Google Trends For Search Queries Statistics

An approach for trading options by volatility alone is usually a good trade by itself. However, there is no guarantee that volatility will increase in the near future (unless there is an upcoming earnings report or dividends will be paid out to shareholders). To compensate the uncertainty, Google Trends is used to determine the popularity of the stock. This does not give us the sentiment on the stock, just a prediction on whether there will be a move on the stock price based on its recent popularity compared to its past popularity.

However, if there is an indication that there will be a huge move in a stock, an investor can use a delta-neutral with a positive-vega options strategy, such as a straddle.

An example of how Google Trends work for stock prediction:

The huge peak is when apple launched the iphone 7. Since investors will usually educate themselves on a company before entering into a trade, these huge peaks are often an indication that a move on the stock is coming soon.

As a reminder, Google Trends gives out the samples based on a range defined by the largest peak. All other samples show how popularity is changing across time relative to the largest peak.

This shows how Apple has been gaining popularity in the past month. The following image is basically integrating the daily samples to have a better view of what’s going on in the long-term.

### Conclusion

When trading on the derivatives market, one should not base their strategy on only one single factor. There are many variables that affect how the price of the options change. However, by entering a trade with high probability of winning using statistics and combining it with a fundamental analysis indicator such as the one based on Google Trends, a solid trading strategy can be constructed. Other things to analyze before getting into a trade would be things like choosing the options with the least theta influence (time decay) and highest vega influence (volatility).