Not All Big Moves Are Created Equal: Volatility and Probabilities

Watch the financial news long enough, and you’re bound to hear someone channeling Johnny Most at a Celtics game: “XYZ is up 5 points!” “ABCD is down 3%!” Surprise slam dunks—and stock price changes—often mean big stories. And market news could be pretty boring without the drama. But if you don’t put volatility (“vol”) in context, it’s hard to tell whether all the excitement over a stock is caused by a game-winning three-pointer, or a mere first-quarter free throw.

Create a Game Plan

This is where you come in, the savvy trader, who can interpret and frame certain market events by relying on a mix of vol and statistics. Nothing too complex, but enough to answer the question, how big is big?

This is important because it can help you incorporate market news into a trading strategy. Are you a momentum trader looking to buy into strength or short into weakness? A contrarian looking to buy stocks with big selloffs, or short stocks after big rallies? Whatever your strategy, it’s vital to have a metric that helps you determine whether a big price change warrants your attention. Let’s get started. 

First, a few statistics. In some financial models and theories (e.g., Black-Scholes), stock and index percentage price changes are assumed to be normally distributed. Think of a bell curve with a peak in the middle that theoretically represents 0% change. You’ll find big down moves on the left-hand side, and big up moves on the right-hand side. In reality, price changes in all cases may or may not be normally distributed, but the normal distribution lets us determine a couple useful things about how big is “big.” 

One standard deviation up and down from the mean theoretically covers about 68% of price changes. Two standard deviations up and down cover about 95%. And three standard deviations up and down cover about 99%. Further, a stock or index’s vol determines the size of a standard deviation in terms of price. The higher the vol, the bigger the dollar change in the stock price that standard deviation represents. Yes, one standard deviation covers 68% of a theoretical price change. But vol determines whether those price changes are $1 or $10. A $1 change in a $10 stock is a much bigger percentage (10%) than a $1 change in a $500 stock (0.2%). And whether the $10 stock might change that 10%, or that $500 stock might change that 0.2%, depends on each stock’s volatility.

Consider a Level Playing Field

A $10 stock with 15% volatility would have a theoretical range of $8.50 to $11.50 68% of the time in one year. To get that, multiply the stock price ($10) by the volatility (15%), then add or subtract that from the prevailing stock price. Multiplying stock price by its vol gives you a theoretical standard deviation for a year. 

Now, say you want to know the standard deviation for a day, week, or month. No problem. Just multiply that vol number (always a one-year number on the thinkorswim® platform from TD Ameritrade) by the square root of the time period to adjust it to your desired time frame. For example, for the standard deviation of one trading day, divide one by the number of trading days in a year (262 is used here), take the square root, multiply by the vol, then multiply that by the stock price.

For that $10 stock with a 15% vol, the one-day standard deviation would be the square root of 1/262 (or 0.0618) x 0.15 x $10 = $0.093. Theoretically, that stock could land in a range between $9.907 and $10.093, 68% of the time. If the stock moved down $0.19 in one day from $10 to $9.81, it would have theoretically dropped just over two standard deviations based on that 15% volatility. Two standard deviations is a pretty large move according to the normal distribution, even though the price changes only $0.19. 

Run Your Plays

Let’s put it all into practice. Say a stock has rallied from $80 on Monday to $85 on Tuesday. On Monday, the stock had an overall vol of 30%. So, 0.0618 x 0.30 x $80 = $1.48. And $1.48 is one standard deviation based on Monday’s price and volatility. Theoretically, 68% of the time, the stock might have closed in a range between $78.52 (down $1.48) and $81.48 (up $1.48) on Tuesday. But instead, it rose $5 on Tuesday. Divide the $5 change in the stock price by the $1.48 theoretical standard deviation to see how many standard deviations it rallied ($5/$1.48 = 3.38 standard deviations). Theoretically, with 99% of the potential stock prices being up or down three standard deviations, a 3.38 standard deviation price change is pretty unusual.

If the vol of that $80 stock was 60% on Monday, then 0.0618 x 0.60 x $80 = $2.97. That’s theoretically one standard deviation, and $5/$2.97 = 1.68. A 1.68 standard deviation price change is big, but not unusual, theoretically. 

The $5 price change in the $80 stock is that same p/l for 100 shares. But in statistical terms, it means different things. The $5 change when vol was 30% is worthy of some excitement. The $5 change when vol was 60%, not so much. In other words, when vol was 60%, the market was perhaps expecting a big price change, and the $5 move wasn’t as big as it might have been.

To get the vol and stock price numbers to do this analysis, hit the Charts page of thinkorswim (Figure 1). 

1—From “Studies,” add the “ImpVolatilty” study to the Charts, which shows the overall implied vol of a stock’s options. 

2—Set the cursor over a date that’s before the price change in question.

3—You'll now see the closing price of the underlying stock or index on the upper left of the chart, and the overall implied vol of the stock or index in the upper left-hand corner of the ImpVolatility study window. 

Then consider the stock or index’s price after a big change, and subtract the closing price of the previous date from that post-move price to get the price change. 

Adjust the vol for time, do some multiplication and division, and determine the price change’s standard deviations.

Fair Game

Why do you have to adjust vol by the square root of the time frame? If a stock moves up +1% one day, and down -0.999% the next, the stock price has had almost zero net change. But was it volatile? Yes. To make sure positive price changes don’t offset negative price changes (which would give the impression that there’s no vol), all the price changes are squared to make them positive. By averaging squared changes, you get a variance that’s directly related to time. Because it’s a square of the stock returns, that variance is harder to interpret. So, we take its square root to get back to the vol of stock returns. If you take the square root of the variance, you must take the square root of time, too. That’s why vol is related to the square root of time.

Now, past performance does not guarantee future performance, and vol’s not a perfect predictor of future potential returns. Sometimes it can underestimate a stock’s potential price changes, while other times it can overestimate. In other words, vol might predict a stock’s 3% move in a month, when it actually moved 5% (underestimating). Or vol might predict a stock’s 10% move in a month when it actually moved 8% (overestimating). Also keep in mind that the normal distribution at the base isn’t a perfect descriptor of returns. In practice, returns are rarely distributed along a “clean” normal distribution.

All in all, this analysis gives price movement a context. Going back to the $80 stock, if the $5 rise in price represented a statistically less likely 3.38 standard deviation change, a contrarian bearish trader might seize that potential opportunity to enter a trade, while a momentum bullish trader might wait for the stock to drop before entering. If the $5 price rise represented a statistically more likely 1.68 standard deviation change, the contrarian bear might wait for the stock to rally before shorting it, while a momentum bull might get long at that point and see more upside potential. 

No Free Throws

Use vol and statistics as one more metric in your trading toolbox. It’s not a strategy in and of itself. But it may help you determine entry and exit points for certain trades by quantifying the “bigness” of price changes.