Academic journal article Quarterly Journal of Finance and Accounting

Options Expected Returns: Variation by Moneyness and Maturity

Academic journal article Quarterly Journal of Finance and Accounting

Options Expected Returns: Variation by Moneyness and Maturity

Article excerpt

Introduction

Coval and Shumway (200l) show that under standard asset pricing assumptions expected returns to options are increasing in the strike price (i.e., higher for out-the-money (OTM) calls than in-the-money (ITM) calls and higher for ITM puts than OTM puts). Empirical work on options returns has reported departures from theoretical expected returns, most strikingly that out-the-money call and put options deliver large losses on average. (20) The goal of this paper is to expand the literature on options expected returns theory to provide a richer context for interpreting the results from the growing empirical literature. This paper confirms and expands expected returns variation by moneyness and adds to the literature with details on expected returns variation by time-to-maturity for calls and puts. A further goal is to reconcile the differences between expected returns theory and empirically documented average returns by studying which assumptions of expected returns theory must be relaxed inorder for theoretical returns to match the empirically documented returns. I find that normality in the returns of the underlier is the key assumption that must be relaxed, with skewness in the returns of the underlying asset able to reconcile expected returns theory and empirically reported returns.

I examine option returns by means of a simulation that relies on randomly generated prices of the underlying asset to produce the option returns. I begin with a base case that follows all the standard Black and Scholes (1973) assumptions, including normally distributed returns in the underlying asset. The underlier furthermore is assumed to have a positive expected return. The goal is to examine the variation in option returns by the amount of time remaining until expiry and to decompose this result by moneyness. (21)

The simulation with normally distributed returns and constant volatility in the underlier produces returns consistent with the theoretical derivation of expected option returns in Coval and Shumway (2001). Specifically, returns are large and positive for call options and increasing in the strike price. This is true for options of all maturities. (I consider options with 200, 120, 60, and 30 days remaining until expiry.) Put option returns are low--less than the risk-free rate of return--and also increasing in the strike price. Furthermore, a new finding is that expected returns of call options are decreasing in the time to maturity, while expected returns are increasing in the time to maturity for put options. In other words, expected returns to 30-day call options are higher than those for 200-day call options, while the reverse is true for expected put option returns (although put option returns are negative, therefore more specifically long-dated put returns are less negative than short-dated put returns).

I follow this initial analysis, which can be considered an effort to establish a baseline based on well-accepted theory and assumptions, with a simulation using non-normal returns for the underlier. I randomly sample from the realized returns of the S&P 500 Index over the January 1991--November 2010 period to generate the returns for the underlying asset in the simulation. The S&P 500 Index returns over this period have the convenient property of exhibiting two distinct sub-periods. From January 1991--December 1999 the index delivered strong positive returns with positive skewness (bull market), while the January 2000--November 2010 period saw the index deliver weaker positive returns with negative skewness (bear market). The S&P 500 Index returns over this period therefore present a great opportunity to study the general effect of the performance of the underlying asset on option returns, specifically the effect of skewness in the underlying asset returns. The potential impact of skewness in the returns of the underlier on option returns has been noted in Ni (2009). …

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