Do Call Options Have High Expected Returns?

Abstract. In theory, far out-of-the-money call options should offer extremely high expected returns, sometimes annualized rates of 100%, 200%, or more. At least one study has confirmed such return magnitudes on index options, at least when bid-ask spreads and transactions costs are ignored. However, another study on index futures options found that call options, especially far out-of-the-money ones, offer low or negative expected returns, possibly on account of "favorite-longshot bias." A third study of individual-stock options similarly found that calls had negative expected returns that became more negative at higher strike prices.

Suppose you raise $15,000 for a charitable cause. You plan to invest the money in the capital markets for a few years and then give it away to, say, Population Services International, which has been estimated to save lives at a cost of $650 - $1000 each (not counting ancillary benefits). (Better yet would be donating to a utilitarian research organization like the Singularity Institute for Artificial Intelligence.) For small amounts of money donated, the expected benefit of your donation is essentially proportional to dollars donated (a donation of $d saves, conservatively, 0.001 * d lives). Thus, you should, if you are donating out of altruism, aim to maximize the expected value of your return, presumably by putting your $15,000 into a risky investment.

One way to do this is just to invest in regular risky stocks. Ibbotson Associates reports the following geometric average returns over the period 1925-2000. (Note that these figures do not consider capital-gains taxes.)

To do better than 12.4%, an investor could take a leveraged position in a small-company stock. Stock brokers allow investors to open margin accounts through which to borrow money to buy more stocks. Investors are allowed to borrow up to 50% of the purchase price, so if you have $5,000, you could borrow another $5,000 and buy $10,000 worth of stock. Since the expected return on your stock (say 12.4%) would be larger than the expected return on your debt (say 10%), you would, on average, improve your returns. The Federal Reserve enforces a minimum maintenance margin of 25% on margin accounts, so the power of leverage is not unlimited.

In practice, it may be better to put one's money into leveraged funds that are able to borrow much closer to the risk-free rate, though it's also important to consider the expense ratios and other relevant features of such funds (see pp. 8-9 of Dean P. Foster, Sham M. Kakade, and Orit Ronen, "Early Retirement Using Leveraged Investments").

Another way to achieve leverage, in theory at least, is by buying call options, especially far out-of-the-money ones. As the concept of a replicating portfolio makes clear, calls are theoretically equivalent to leveraged stock positions, with the amount of leverage increasing with the strike price. Leverage can also be achieved with futures. Unlike leverage through margin accounts, the leverage implicit in these investments must be based on an interest rate near the risk-free rate, or else arbitrage would be possible (Foster, Kakade, and Ronen, p. 19).

(annualized expected instantaneous option return - risk-free rate) = (expected annualized stock return - risk-free rate) * (stock price / option price) * (delta of option).

See, e.g., p. 687 of Robert L. McDonald, Derivatives Markets, 2nd edition, 2006, or this link.

The second formula is taken from Mark Rubinstein, "A Simple Formula for the Expected Rate of Return of an Option over a Finite Holding Period," Journal of Finance 39:5 (1984): pp. 1503-1509. (Available here as a Word document. The formula is boxed on page 3. I assume the investor's estimate of volatility is the same as the market's estimate.)

Here's an example calculation using the workbook. Assume a stock is currently at $40, has an expected annual arithmetic return of 11%, has an arithmetic dividend yield of 2%, and has a volatility of logarithmic returns of 0.3. Suppose we buy a European call option on Aug. 21, 2007, with strike price $50, set to expire on Dec. 21, 2007. We plan to sell the option on Oct. 21, 2007. Our annualized instantaneous expected return is then 80%, and our annualized expected return using the Rubinstein formula is 109%. (Note that annualization is valid because the expected value of a product is the product of the expected values for independent random variables.) "Expected" returns here are not the same as "typical" returns; usually, option prices decrease over time. If, instead of selling the option on Oct. 21, we held it to maturity, there's only a 13% chance it would pay out anything at all.

Are such high expected returns confirmed by the actual data? In at least one study they were. The following figures are taken from p. 993 of "Expected Option Returns" by Joshua D. Coval and Tyler Shumway, The Journal of Finance, 56.3 (2001): pp. 983-1009 (JSTOR link). They represent weekly returns for European S&P 500 call options between January 1990 and October 1995; expiration times of the options were roughly one month. (The authors also computed average daily returns on S&P 100 calls between 1986 and 1995; the results were comparable and, if anything, even higher.)

It's not clear from the paper whether these weekly returns assume five full trading days per week or whether they represent average returns from week to week, including non-trading days. If the former, we can assume 252 trading days per year and compute a naive yearly expected return of (1.0413)^(252/7) = 4.29, or 329%. In the latter case, we have (1.0413)^(365.24/7) = 8.26, or 726%. These returns are based on options prices calculated as an average of bid and ask quotes, so actual returns should be somewhat lower. The figures also ignore broker transaction costs. And of course, as a glance at the median weekly returns shows, the risk involved is substantial.

At least one other study found drastically different results. In "The Favorite-Longshot Bias in S&P 500 and FTSE 100 Index Futures Options: The Return to Bets and the Cost of Insurance," Stewart D. Hodges, Robert G. Tompkins, and William T. Ziemba examined S&P 500 and FTSE 100 American index-futures options from 1985 to 2002 and found that monthly/quarterly returns were, on the whole, unimpressive or even negative (p. 10). At a few strike-price levels, calls had average returns of 5%, 7%, or 13%, but others had returns of -3%, -7% or -14%. Worst were the farthest out-of-the-money options, with average returns of -96% or lower. The authors found similar results for options on British Pound/US Dollar futures, except (surprisingly) for the most out-of-the-money ones, which averaged a 329% return (p. 14).

A 2007 paper by Sophie Xiaoyan Ni, "Stock Option Returns: A Puzzle," addresses directly the conflict between theory and evidence. She notes in the Abstract:

Under very weak assumptions, the expected returns of European call options must be positive and increasing in the strike price. This paper investigates the returns to call options on individual stocks that do not have an ex-dividend day prior to expiration. The main findings are that over the 1996 to 2005 period (1) out-of-the-money calls have negative average returns and (2) average returns of high strike calls are lower than those of low strike calls. Preliminary evidence is presented that is consistent with investor risk- seeking contributing to the puzzling call returns.

If readers have further insights on this topic, or know of relevant research not mentioned here, please email me: <webmaster [put "at" here] utilitarian-essays.com>. Thanks!