We have had much debate here at Risk Reversal over the best way to present this topic. Implied Moves of events, usually earnings, are something we frequently mention and use, but they are at best an estimate and we will go over some of the ways to calculate them.

Before venturing into this quite interesting but involved discussion about implied moves, let us introduce our decision tree for how to calculate implied earnings moves:

**The stock has weekly options**

What we all agree on is that the implied move of an event that is happening at or near expiration is the easiest and most accurate estimate. Options with weeklies are especially convenient for calculating implied move as the expiration is always within a few days of the event. In this case, the at-the-money straddle best estimates the implied value of the move and the straddle divided by the underlying price gives the percentage implied move. So an at-the-money straddle expiring tomorrow that is trading at $5 on a $100 stock implies a 5% move.

**The stock does not have weekly options**

What happens if there are no weekly options? The estimate becomes less precise the farther the nearest expiry is from the event. We will consider the extreme example where the event is the Monday after expiration. Say we have an April 21st expiration and our event is Monday, April 23rd. The April straddle expires before the event so we must consider the May options for our estimate.

The problem with just taking the at-the-money straddle with a long time to expiration is that the straddle still has time value. A way to take this into consideration would be to calculate the at-the-money straddle minus the theta (time decay) of the straddle times the number of days to expiration. However, theta is affected by volatility so if we assume the implied volatility will come in after the event then our calculation would be off. One could consider the historical post-earnings implied volatility and figure out theta from that, but that may or may not be something a person trading from home could easily do with most trading models.

**The stock also does not have liquid monthly options**

And all bets are off in very ill-liquid options. In this case, probably the best estimate would come from historical moves.

**Historical Earnings Moves**

Historical earnings moves can be calculated by determining if the earnings were before the market opened (BMO) or after the market closed (AMC). If before the open, then one takes the difference between the closing price the day before earnings and opening prices on the day of earnings of the underlying and divide by the closing price to get the percentage move. If after the close, one takes the difference between the closing price on expiration and the opening price the day after expiration and again divide by the closing price to get the percentage.

On the site we usually look back chronologically at the past 3-8 earnings moves. We also like to consider correlations between earnings moves and other underlyings or similar market situations. Some other considerations might be same quarter earnings year to year. Some people take historical earnings and average over a much longer period of time, sometimes weighting more recent earnings higher.

**The stock has liquid monthly options. Use forward volatility**

A popular method on many trading desks is to use the concept of **forward volatility**. Rather than bog you down with the math right here, let us try and explain the intuition behind this method. For those that are interested, the details of the calculations and further reading materials are included in the appendix of this post. The idea here is to first ascertain the market’s price of future volatility without the expected earnings event. Since the earnings event is before May expiry, our assumption is that the time period between May expiration and June expiration contains no such event. If we can use the market’s price of volatility for the period between May and June (the May/June forward volatility) as our “benchmark” volatility level, we can compare that benchmark to prevailing May and June options prices and determine what extra move is implied in those options prices.

There are a number of considerations with the assumptions in this model which we won’t get into, but it is widely used and therefore probably becomes a better estimate in very liquid options as it could be self-fulfilling. In the case of our above example, where May is not the front month, there may not be June options yet. Using a later month may introduce liquidity issues in that the far out month’s options may not trade frequently or may be affected by other events or factors so implied volatilities may be less relevant.

**Appendix:**

This is where things start getting nerdy. We will introduce some variables to make our equations simpler using our example of an April 23rd event coming after April expiration.

t_{1} = the days between today and May expiration

t_{2} = the days between today and June expiration

v_{1} = May implied volatility for the at-the-money straddle

v_{2} = June implied volatility for the at-the-money straddle

Let’s take 2012 for our example, today is April 11, and implied volatilities of 42 in May and 35 in June. So there are 26 weekdays between April 11th and May 19th (expiration) and 46 work days between April 11th and June 16th. Thus:

t_{1} = 26

t_{2} = 46

v_{1} = 42

v_{2} = 35

The formula for forward volatility is:

sqrt ((t_{2}(v_{2})^{2} – t_{1}(v_{1})^{2}) / (t_{2}-t_{1}))

*For a rich description of the forward volatility formula please see Nassim Taleb’s Dynamic Hedging: Managing Vanilla and Exotic Options, p. 154. *

Using our example we get:

sq rt (46x(35)^{2} – 26x(42)^{2}) / (46-26) = 22.9, let’s call it 23. We will refer to forward volatility as FV.

Going back to a daily implied vol estimate (see volatility education section) this gives us a daily implied move of 23 / sq rt 252 = 1.45% (People seem to use 252 for the number of trading days in a year, so we will go with that for these calculations instead of the 256 we used previously which has a much cleaner square root.)

If we take that to be a normal “non-event” period, and say, ok, we also expect the stock to move 1.4% every day except for Apr 23rd, what is the move on Apr 23rd needed to justify the 42 implied volatility in May options and the 35 implied volatility in June options. The forward volatility can be considered our “non-event” volatility, next we need the **variance of the earnings** announcement. The **variance** of a random variable or distribution is the mean of the squared deviation of that variable from its expected value or mean. Our formula for the variance of the earnings is:

t_{1}((v_{1})^{2} – (FV)^{2})/252

*Please see Stanford Business School Paper “Risk Premiums and Non-Diversifiable Earnings Announcement Risk,” section 3.1, by Mary Barth and Eric which details this calculation. *

Using the inputs from our example we get:

26((42)^{2} – (23)^{2}) = 127.42

And, grand finale, for the implied event move percentage we take the square root of the variance of the earnings plus the forward volatility/252.

In our example: sq rt (127.42 +23/252) = 11.29%

This extreme an implied move is an unlikely situation, but it’s a good example to illustrate the methodology of the calculation (normally, the difference between May and June expiry implied volatility would not be that great with more than a month from May expiry). Once again, this is not an exact science, and there are often other factors affecting the relationship between May and June implied volatility outside of purely the earnings event, but it is one estimate that we can use when weekly options are not available.

We want to make one last point. The Stanford paper is a discussion of earnings event moves more broadly, and finds that the options market routinely overprices the actual move, particularly for large cap names. Our experience as traders tends to confirm that thesis, and when using the forward volatility calculation above, it probably makes sense to “damp” the result by 10-20% based on that experience.

Here is a link to our Implied Move Calculator:

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