by Brian Tomasik
First written: summer 2007; last update: 14 Apr. 2015

## Summary

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.

Thus, while a nice idea in theory, the implicit leverage of out-of-the-money call options does not seem to be a feasible way for risk-neutral altruists to achieve high expected returns. It's also worth noting that certain groups are opposed to options and leverage in general -- especially Islamic scholars but also some Western commentators worried about financial instability.

## Introduction

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. Altruistic donors should generally be less risk-averse than selfish donors, so it might make sense to invest that money in high-risk securities. 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.)  Small-company stocks 12.4% Large-company stocks 11.0% Long-term government bonds 5.3% Treasury bills 3.8% Inflation 3.1% 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 7%) would be larger than the expected return on your debt (say 4%), 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").

## Options as leverage

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).

Here is an Excel workbook that allows the user to calculate theoretical expected options returns based on two different formulas. The first is an instantaneous expected rate of return that uses the formula

(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. (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.

One investing-advice article reports that far out-of-the-money LEAP (long-term) call options have extremely high risk but possibly also extremely high expected returns.

## Theory meets data

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. 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.)

 Strike price - stock price (\$) -15 to -10 -10 to -5 -5 to 0 0 to 5 5 to 10 Expected weekly return (%) 1.48 1.19 1.85 2.00 4.13 Median weekly return (%) 0.0 -1.99 -4.46 -9.55 -17.39 Minimum weekly return (%) -80.67 -86.51 -89.33 -92.85 -92.31 Maximum weekly return (%) 141.82 190.24 256.63 426.65 619.41

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.

Pages 42-46 of the paper include the following data:

 Strike price / Stock price <= 0.85 0.85 to 0.95 0.95 to 1.05 1.05 to 1.15 >1.15 Average one-month return of holding call option to expiration using bid-ask-midpoint prices (%) 2.31 2.50 1.98 -10.15 -36.86 Average two-month return of holding call option to expiration using bid-ask-midpoint prices (%) 2.88 4.70 6.34 1.68 -21.88 Average three-month return of holding call option to expiration using bid-ask-midpoint prices (%) 2.64 4.20 5.55 0.39 -23.20 Average one-month return of holding call option to expiration using ask prices (%) -0.97 -2.25 -6.30 -22.54 -46.52

According to "Returns from Trading Call Options" (2013):

I find a general and consistent result that call option returns are low on average and decreasing in the strike price. Only in-the-money options held for a month exhibit positive returns. Deep out-the-money options deliver large negative returns on average, consistent with risk-seeking investing on the part of buyers.

including options in the portfolio most often results in underperformance relative to the benchmark portfolio. However, a portfolio that incorporates written options can outperform the benchmark on a raw and risk-adjusted basis.

A reader of this article wrote to me with some of his/her thoughts. The remainder of this section mostly quotes from this reader's message, with some modifications on my part.

What got me interested in this topic was the book Lifecycle Investing by Ian Ayres and Barry Nalebuff; they advocate the use of leverage for retirement investing. I found the overall idea was good, although they don't go into much detail about the mechanics of leverage. They do suggest deep in-the-money calls of the S&P500 as one option to obtain leverage.

The authors do go into more detail in "Diversification Across Time". They show that for 1-year S&P500 calls, you have to go deep in the money ("leverage" ratio of 2:1) to get an effective rate of interest that will make the strategy work. There's not enough detail in what they publish to implement the strategy.

I found Enhanced Indexing Strategies by Tristan Yates to be of some use. He discusses several strategies, and the strategy of yearly buying deep in-the-money 2-year call options and then selling them after one year made sense to me.

In a taxable account, my guess is that using margin to obtain leverage would be more successful. In a tax advantaged account, a call option strategy might work. But after reading the literature, I'm not entirely convinced of that. The advantage of margin in a taxable account is its tax deductibility. Of course, that depends on one's marginal tax rate. With the appropriate tax laws, it's possible to lose money on a pretax basis using margin, but make money on a post-tax basis. That's because your interest may be fully deductible against income, but your capital gains can be deferred and is often taxed at a lower rate than income; sometimes dividends are also taxed at a lower rate.

More references on options returns:

## Conclusion

Without knowing more, it seems to me that using ordinary margin borrowing is the more sure way to achieve leverage. That form of leverage is transparent and doesn't rely on theoretical formulas for how derivatives (options or futures) should be priced. In the meantime, it would be interesting to explore alternatives beyond margin borrowing in case some of them prove to offer higher expected returns in practice and not just in theory.