Wednesday, June 18, 2008

Call Options are Lottery Tickets

In theory, far out-of-the-money call options should offer extremely high expected returns as a percent of their price. As underlying stocks always, in practice, have positive betas against ‘the market’, all calls will have positive betas that exceed the beta of the underlying stock, and call betas will increase in the strike price as the calls get further out-of-the-money. Hence, all calls will have positive expected returns and the expected returns will be larger for greater strike prices, because the betas increase as you go out of the money. Theoretically, beta—or any covariance with the elusive SDF—measures the ‘how much’ of risk, and so if risk is priced, higher options with higher strike prices (ie, more out-of-the-money), have higher beta(s), which implies higher average return.

For example, say you have a stock with a price of 100, and buy a call with a strike price of 120, expiring in 3 months. If the stock price rises to 110 over the next month, the call option will rise 120%, while a long stock position rises only 10%. This is the implicit leverage in an option, that is, it is like being able to borrow 10 times one’s capital and invest in the market. It is exactly the same bet, just higher powered.

Sophie Ni (2007) looked at data from 1996 through 2005, and found that the highest out-of-the-money calls have average returns of −37% over a month! The chart above shows that if you bucket call options into groups based on their ‘deltas’, which is a measure of how sensitive they are to the market, you find that call options, indeed, are indeed highly levered stock positions. Remembering that the average stock has a beta of 1.0, these betas range from 4 to 15—giving one 4 to 15 times the juice of the daily return. An option's beta is the beta of the stock, times the 'omega', which is a measure of the percent return in the option price given a 1% change in the stock price. Not only is the average return negative for call options, these returns get worse, the more implicitly levered, the more ‘risky’, the options become. Returns are negatively correlated with the betas on those options. This has been corroborated by several other studies, including Coval and Shumway (2001). Investors basically are overpaying for lottery tickets when they buy options, and just like the lottery, the average payout is worse, they more ‘risk’ one takes. If there’s a risk premium in equities, it certainly is not amplified in options in any way, because you lose money, on average, buying leverage market positions via call options.


Anonymous said...

You should do a post about that old paper on the returns from buying puts, heres a more recent similar one:

Can't find the original

Btw, your X-axis label should be Call Option Strike. If it was delta, the, low should be on the right

Eric Falkenstein said...

Thanks, fixed that. Good article. I think from a retail perspective, buying options is really stupid, selling them just stupid. Be a market maker.

Anonymous said...

Indeed, I am.

Off topic, have you seen anyone sort on Sharpe or Sortino? Quick and easy way to take advantage of momentum and low volatility

Eric Falkenstein said...

Sharpes for stocks? As you might expect, cross-sorting stocks by momentum and vol generates better sharpes than using either alone. But absolute returns for a momentum strategy do go down.

Anonymous said...

Here's another paper on the bias:

I had at one point calculated option and straddle returns for portfolios created by sorting on book-to-market, mkt cap and momentum of the underlying. Theoretically the option price should not depend on the variables that predict mu, but it does. You see some interesting patterns.

gary said...

"I think from a retail perspective, buying options is really stupid, selling them just stupid."

Can you explain this? If buying the highest out-of-the-money options lose 37% per month, wouldn't writing that option gain 37%? What am I missing?

Eric Falkenstein said...

Gary, I think this is because these options are all way out of the money, so they have bid-asks like 0.10-0.20. At that level, the bid-ask spread is greater than 37%.

Anonymous said...

So making markets on OOM options (being the ask) is the least stupid of all.