INTRODUCTION

According to standard asset-pricing theory, options should exhibit expected returns that are in line with the systematic risk their holder is exposed to. Much of the recent research in options has focussed on valuation and hedging aspects, whereas option return characteristics, though of course closely related to studies of valuation models, have been addressed very rarely. Moreover, while there is an overwhelming literature on empirical CAPM studies, only few papers address option returns empirically (see, eg, Coval and Shumway,1 Bakshi and Kapadia,2 Jones3).

To the best of our knowledge, no comprehensive investigation of option returns has been carried out so far for the European equity option market that is widely concentrated at the Eurex (European Exchange), the world's largest derivatives exchange. Therefore, based on a study framework developed in Coval and Shumway,1 we aim at filling a conspicuous gap in the empirical literature by conducting an analysis of Eurex index equity options with respect to compatibility with asset-pricing theory and potential trading strategies based thereon. For the period between January 2004 and December 2005, we find that option returns exhibit inconsistencies even with very general propositions of asset-pricing theory. Quantitatively, option returns deviate significantly from expected values in a Black–Scholes/CAPM framework, with results being sensitive to option type, moneyness, and time-to-maturity. With regard to exploitable trading opportunities against the CAPM background, short positions in certain zero-beta straddle portfolios are shown to yield remarkable positive (excess) returns on average that can readily cover usual transaction costs.

The remainder of the paper is organised as follows. In the next section, basic asset-pricing results applied to options are outlined briefly. The further section describes the composition of the data base. The final section presents and discusses the empirical results.

ASSET-PRICING RESULTS APPLIED TO OPTIONS

General asset-pricing framework

As proven in Coval and Shumway,1 any (plain-vanilla) call option on a security will have a positive expected return that is increasing in the strike price. For any put option, the expected return will lie below the risk-free rate and will be decreasing with the strike price. A straightforward interpretation is that the leverage effect should be ‘priced’. A sufficient precondition for these relations to hold is that the stochastic discount factor is negatively correlated with the price of the underlying over its entire domain — a violation of this condition would not be in line with any of the existing asset-pricing theories. From an empirical perspective, these results thus allow a qualitative insight into the nature of option returns under very weak assumptions.

Black–Scholes/CAPM framework

Imposing the more strict assumptions required for the Black and Scholes4 model and the continuous-time CAPM model, in particular, asset prices following a Geometric Brownian Motion, allows a quantitative assessment of option returns. According to the CAPM Equation E[r i ]=r+E[r m −r]β i with r as the risk-free (short-term) interest rate and r m as the return on the market portfolio, a security's expected return E[r i ] is linked to its systematic risk (given by the beta factor β i ) in an affine-linear way. For a call option in a Black–Scholes world, the beta is found to equal5

where S denotes the current price of the underlying, K the option's strike price, σ the (constant) expected volatility, τ the option's time-to-maturity, and δ the continuous dividend yield of the underlying. Φ(x) stands for the c.d.f. of the standard normal distribution, while β s equals the beta of the underlying.6

In the special case of a zero beta, an option position is expected to yield the risk-free rate. Such zero-beta option portfolios can be constructed by combining two otherwise identical call and put options with market betas β c and β p , respectively. In order to satisfy the relation

in such a straddle strategy, a weight function

is applied. The return of a straddle portfolio is thus equal to the weighted sum of call and put returns. These relations will serve as a basis for the quantitative analysis of option returns.

DATA

We refer to exchange-traded plain-vanilla options on the German blue-chip stock index DAX, one of the leading equity indices in Europe. The index options are European-style with maturities of up to five years. They are traded liquidly at the Eurex, usually most actively for shorter times-to-maturity and near-the-money strikes. We employ daily settlement prices that refer to the close of the trading day at 5:30pm.7 In investigating the data from January 2004 to December 2005, we restrict our attention to records with option prices of at least ten times the minimum tick size of €0.10, that is, at least €1.00, and those with at least seven calendar days to maturity in order to avoid bias associated with very small option (time) values. Moreover, obviously erroneous records that violate distribution-free arbitrage boundaries are removed from the database.

The DAX quotes are obtained from the electronic XETRA trading system at the Frankfurt Stock Exchange at 5:30pm.8 The market environment in the reference period is characterised by a rather low DAX volatility of approximately 15 per cent p.a. with an index ranging between approximately 3,650 and 4,250 points in 2004, followed by a likewise steady increase to about 5,450 points during 2005. Risk-free interest rates are interpolated based on Euribor for intervals up to 12 months and German (AAA) government bonds for periods between two and five years. Any market frictions such as taxes are ignored. We end up with a total of 334,878 option records. For purposes of beta estimation (assuming β s DAX=1), we employ the VDAX that serves as a reference measure of implied volatility in the Eurex DAX option market.9 We use discrete option returns in order to bound negative returns at −100%.

RESULTS

We calculate 331,911 daily option returns as described statistically in detail in Table 1, further divided by five moneyness and four time-to-maturity classes.10 Qualitatively, we first note that for some classes of (deep) out-of-the-money call options, the basic property of positive expected option returns is violated. For instance, very short-term out-of-the-money calls (class [A, −1]) exhibit a statistically significant mean daily return of −5.75 per cent during the examination period. Besides this phenomenon, we would expect an increase in the mean return from deep in-the-money through deep out-of-the-money call options due to the higher leverage. This holds across deep in-the-money, in-the-money, and at-the-money calls for all terms, while inconsistencies are obvious across the remaining moneyness classes. For put options, we would expect mean returns below the risk-free rate, decreasing with the moneyness. (Deep) in-the-money short-term puts (classes [B, +1] and [B, +2]) with mean returns of approximately 0.5 per cent per day are found to violate this proposition to a great extent. Besides this and with the exception of very short-term options, put returns are found to be decreasing for decreasing strikes, as expected.11

Table 1 Statistics of daily option returns (in %) and betas for different moneyness and time-to-maturity classes
Full size table

Applying the CAPM framework that allows a quantitative assessment of returns, among the classes of call options with positive mean returns, nearly all options clearly exhibit returns that are, on average, too high given the CAPM forecast. For very short-term at-the-money calls, for example, the mean daily return in 3,446 observations amounted to approximately 3.3 per cent and thus to an impressive annualised value of more than 700 per cent. With a risk-free rate around 3 per cent p.a. and an equity risk premium usually between 4 and 6 per cent p.a., a quick check reveals that average realised returns more than compensate for the option inherent risk, given by the mean beta of approximately 49. This carries over to all other call option classes as well. While we observe an intuitive return-risk pattern across many call option classes — the higher the mean beta, the higher the mean return —, however, this relation does not hold in all cases. For put options, mean realised returns in those option classes with average returns below the risk-free rate (ie, option classes that are, on average, in line with the very general proposition outlined at the beginning of the paper) are lower than expected. Short-term at-the-money puts with a mean daily return of −2.52 per cent and a mean beta of −36.5 are an example: The average losses from these puts are much higher than the CAPM would suggest. In contrast to the call options, the picture of return-risk patterns for the puts is far more heterogenous. The lower the mean (signed) beta, the lower the mean return does hardly hold across moneyness classes for the different terms.

Turning to zero-beta portfolios, we analyse daily returns of 141,903 straddle strategies, for which descriptive statistics are provided in Table 2. The moneyness classification refers to the calls employed in the straddle portfolios.

Table 2 Statistics of daily zero-beta straddle returns (in %) for different moneyness and time-to-maturity classes
Full size table

We note that for very short-term and short-term options, straddle returns decrease from deep out-of-the-money through deep in-the-money classes, thereby changing sign from plus to minus. Strategies with at-the-money calls and puts in these term classes yield significantly negative returns of approximately −1.0 and −0.5 per cent per day, respectively. Selling straddles created from short-term at-the-money options thus promise remarkable (excess) returns that could easily cover usual transaction costs encountered in the option market. For medium-term options, the picture is less clear, while for long-term option strategies, we report straddle returns that do not significantly deviate from zero and that are thus conform with the CAPM framework.

Reflecting these results, we would like to point out that our findings are more or less robust against small changes in beta estimates. Compared to the study of Coval and Shumway,1 who present an in-depth analysis of short-term S&P 100 and S&P 500 index options and find that expected option returns for both calls and puts are too low compared to their systematic risk, our results are similar for puts though different for calls. From a practical perspective, one might argue that settlement prices used in this study do not always reflect real trading opportunities. While this can indeed be the case for deep in-the-money or deep out-of-the-money and/or very long-term options, Eurex DAX options considered here trade, for instance, very actively for near-the-money strikes and shorter times-to-maturity. That means that straddle strategies in classes [A, 0] and [B, 0], for example, should be rather easily implementable. Beyond doubt, usual limitations regarding the CAPM and its assumptions apply. Moreover, using realised (ie, ex post) returns to verify expected (ie, ex ante) returns is frequently subject to criticism in empirical asset-pricing studies. There is also certain evidence in the empirical literature that claimed extraordinary returns can at least be partly attributed to other risk factors such as volatility and jump risk and the associated risk premia (see, for instance, Bakshi and Kapadia,2 Jones3). Nevertheless, results in this study might yield a promising starting point for actual trading strategies and encourage further research in the area of option returns, especially in the European derivatives market.