**Introduction**

One of my long-term research interests is how to use options prices to infer useful information from the market. I have written quite a few articles at Seeking Alpha on this topic over the years. The most common information that traders derive from options prices is the implied volatility, and the well-known VIX index is a measure of short-term implied volatility. In general, I am more interested in longer-dated options because they provide the options market’s consensus view for longer time scales. Back in 2007, for example, before the crash, I wrote articles noting that the implied volatility for long-dated options was much higher than the realized volatility at the time, and that this was an important warning to investors to carefully manage risk. After the crash, I revisited this topic and reviewed the outcomes.

More recently, I have been researching a new avenue of information that can be obtained from options prices. A body of research shows how to derive considerably deeper information from options prices than just the implied volatility. In fact, you can derive the options market’s consensus view on the most likely return of the underlying instrument or index, along with the various risk measures. I have built my own version of this type of model. Using this model, I can take options prices on a liquid ETF, SPY, for example, and calculate the implied range of returns and their relative probabilities. There is nothing especially complex about this type of calculation — there is considerable research showing the way.

In this article, I will show the implied probability distribution of returns for options on SPY expiring in June of 2018 and discuss what these results imply.

**Options Prices**

For this type of analysis, I look for options that are very liquid, as evidenced by a high degree of open interest and a tight bid-ask spread. The closing price of SPY on January 31 is $281.90. I select a series of put options that have strike prices below $281.90 and a series of call options with strike prices above $281.90 (below). The **$280 Put** option in the table below gives the buyer the right to sell a share of SPY for $280 between today and June 15^{th}, when the option expires. Even if SPY drops dramatically between now and June, the buyer of the put option can sell a share of SPY for $280. If the price of SPY drops to $270, you can buy a share of SPY at $270 and sell it immediately for $280 by exercising the put and you have made $10. In practice, the option buyer would simply exercise the option and receive $10. The **$285 Call** option gives the buyer the right to buy a share of SPY at $285, no matter what the price of SPY does. If SPY goes to $300 between now and June and the buyer of the $285 Call options exercises the option, he receives $15.

The prices of options reflect the market’s consensus estimate of the probability that SPY will move by a certain amount between now and June 15. By looking at a series of put and call prices, we can estimate the probability of all possible returns that are implied by the option prices.

*June 15, 2018 call and put options prices (the mid-point price here is the average of the bid and ask)*

**Statistical Model**

When I run the statistical model with the options prices in this table to back out the distribution of returns, I obtain the following:

*Option-implied probability of SPY return between January 31, 2018 and June 15, 2018*

When I calculate the fair prices for the options in the table above, using this distribution of returns, the resulting fair prices for the options match the actual options prices with an average error of 0.003%. In other words, the options prices imply the distribution of returns in the chart above.

The first thing to note is that the distribution is strongly skewed to the negative — the downside tail is more extreme than the upside. This asymmetry partly reflects the natural risk aversion of investors and is not an unbiased estimate of the relative probability of losses versus gains.

There are three features of these implied return distributions that are most important. The first is the mode of the distribution — the return that corresponds to the peak. This is the return that the options prices suggest is most likely between today and June 15. The value of the mode is 3.53%. This means that the options market consensus is for 3.53% price return (e.g., not including dividends) over the next four and half months. The estimated standard deviation of these returns (which is the implied volatility) is 8.68%. This is the 4.5-month volatility - not annualized volatility. If we annualize these values, the results suggest a 9.4% annual return and 14.2% volatility. The options only go out to June. The annualized values are to give a sense of how attractive the situation looks. Over the ten years through the end of 2017, SPY has returned an annualized 8.4% with a volatility of 15.04%. In other words, the current options prices on SPY for June 2018 suggest that SPY will return more than average, relative to the last decade, and with slightly less risk (lower volatility) than we have experienced over the last decade.

If we convert the probability distribution to the cumulative probability distribution, we can see the percentiles of return — a useful view as well (below).

*Cumulative implied probability of returns over the next 4.5 months*

The 70^{th} percentile implied return over the next 4.5 months (between now and Jun 15^{th}) is 5%, which means that there is a 30% chance that SPY will return more than this amount and a 70% chance that SPY will return less than this amount. Of particular interest is the 0% return percentile, which suggests (in this case) that there is about a 36% probability that SPY will return less than 0% (and a 64% chance that SPY will return more than 0%). As noted previously, the skewness in the returns (with the low percentiles being more extreme than the high percentiles) is a standard feature that reflects risk aversion. This shows that you pay more for a put than for an equivalent call because investors are more afraid of downside (loss) than they are optimistic about gains.

When I performed this same calculation on November 26^{th}, 2017, almost exactly two months ago, for the June 2018 options, the annualized implied mode return was 9.0% and the annualized implied volatility was 14.0%. The current option outlook is for slightly higher returns for the next several months.

**Conclusions**

I have explained the basic mechanics of using options prices to back out the implied distribution of returns. This can be done for different time horizons. I have shown results for June 2018. It is also of interest to compare the mode return (the most probable outcome implied from the options prices) and the implied volatility between different asset classes, and this will be a topic for future articles. The general takeaway from this analysis is that the consensus view of the options markets is that SPY has a good chance of continuing to perform well over the next several months.

**Disclosure:** I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it. I have no business relationship with any company whose stock is mentioned in this article.

**Additional disclosure: **I am long U.S. equities, although I do not own SPY