ImplementingOption Pricing Models When Asset Returns Are Predictable

The predictability of a n asset's returns will affect the prices of options on that asset, even though predictability is typically induced by the drift, which does not enter the option pricing formula. For discretely-sampled data, predictability is linked to the parameters that do enter the option pricing formula. We construct an adjustment for predictability to the Black-Scholes formula and show that this adjustment can be important even for small levels of predictability, especially for longer maturity options. We propose several continuous-time linear diffusion processes that can capture broader forms of predictability, and provide numerical examples that illustrate their importance for pricing options. THEREIS NOW A substantial body of evidence that documents the predictability of financial asset returns.' Despite the lack of consensus as to the sources of such predictability-some attribute i t to time-varying expected returns, perhaps due to changes in business conditions, while others argue that predictability is a symptom of inefficient markets or irrational investors-there is a growing consensus that predictability is a genuine feature of many financial asset returns. In this article, we investigate the impact of asset return predictability on the prices of an asset's options. A comparison between the polar cases of perfect predictability (certainty) and perfect unpredictability (the random walk) suggests that predictability must have an effect on option prices, although what that effect might be is far from obvious. However, in the *Both authors are from the Sloan School of Management, Massachusetts Institute of Technology. We thank Petr Adamek, Lars Hansen, John Heaton, Chi-fu Huang, Ravi Jagannathan, Barbara Jansen, Re& Stulz, and especially Bruce Grundy and the referee for helpful suggestions, and seminar participants at Boston University, Northwestern University, the Research Triangle Econometrics Workshop, the University of Texas a t Austin, the University of Chicago, Washington University, the University of California a t Los Angeles, the Wharton School, and Yale University for their comments. Financial support from the Laboratory for Financial Engineering is gratefully acknowledged.A portion of this research was conducted during the first author's tenure as an Alfred P. Sloan Research Fellow. See, for example, Bekaert and Hodrick (19921, Bessembinder and Chan (19921, Breen, Glosten, and Jagannathan (19891, Campbell and Ammer (19931, Campbell and Hamao (19921, Chan (19921, Chen (19911, Chen, Roll, and Ross (19861, Chopra, Lakonishok, and Ritter (19921, DeBondt and Thaler (19851, Engle, Lilien, and Robbins (19871, Fama and French (1988a, 1988b, 19901, Ferson (1989, 19901, Ferson, Foerster, and Keim (19931, Ferson and Harvey (1991a, 1991b), Ferson, Kandel, and Stambaugh (19871, Gibbons and Ferson (19851, Harvey (1989b1, Jegadeesh (19901, Keim and Stambaugh (19861, King (19661, Lehmann (19901, Lo and MacKinlay (1988, 1990, 19921, and Poterba and Summers (19881. 88 The Journal of Finance continuous-time no-arbitrage pricing framework of Black and Scholes (1973) and Merton (1973), and in the martingale pricing approach of Cox and Ross (1976) and Harrison and Kreps (1979), option pricing formulas are shown to be functionally independent of the drift of the price process. Since the drift is usually where predictability manifests itself-it is, after all, the conditional expectation of (instantaneous) returns-this seems to imply that predictability is irrelevant for option prices.2 The source of this apparent paradox lies in the attempt to link the properties of finite holding-period returns, such as predictability, to the properties of infinitesimal returns, such as the instantaneous volatility a , without properly fixing the appropriate quantities. In particular, while it is true that changes in predictability arising from the drift cannot affect option prices under the Black-Scholes assumption that a is fixed, fixing a implies that the unconditional variance of finite holding-period returns will change as predictability changes. But since the unconditional variance of returns is usually fixed for any given set of data irrespective of predictability-for example, the historical annual standard deviation of the return on the S&P 500 Index is 19.9 percent-fixing a and varying predictability can yield counterfactual implications for the data.3 The resolution of the paradox lies in the observation that if we fix the unconditional variance of the "true" (finite holding period) asset return process, i.e., the data, then as more predictability is introduced via the drift, the population value of the diffusion coefficient must change so as to keep the unconditional variance constant. Therefore, although the option pricing formula is unaffected by changes in predictability, option prices do change. In this respect, ignoring predictability in the drift is tantamount to committing a specification error that can lead to incorrect prices just as any other specification error can (e.g., Merton (1976b)). But why should the unconditional variance be fixed? One answer is provided by the fact that the marginal distribution of asset returns is a more fundamental or primitive object than the joint distribution of asset returns and other economic variables. Therefore, a logical sequence of investigation is to first match the marginal distribution of returns, and then focus on the implications for the joint and conditional distributions. This is the approach typically taken in studies of the predictability of asset returns. When regresPredictability can also manifest itself in the diffusion coefficient, in the form of stochastic volatility with dynamics that depend on predetermined economic factors. However, since predictability is more commonly modeled as part of the conditional mean, we shall focus solely on the drift. In fact, we argue more generally below that all of the unconditional moments of the marginal distribution of returns are "fixed," in the sense that empirical estimates of their values are readily obtained from the data, hence any economic or statistical model of predictability must be calibrated to these values to be of empirical relevance. But there is a compelling reason for focusing first on the unconditional variance of returns: any sensible comparative static analysis of predictability must keep fixed the unconditional variance of the variable to be predicted, since this is the benchmark against which the predictive power of a forecast is to be measured. 89 Option Pricing When Asset Returns Are Predictable sors are added to or subtracted from a forecasting equation, the conditional moments of returns change, affecting the joint distribution of returns and predictors, but the unconditional moments of the marginal distribution of returns, e.g., mean, variance, skewness, kurtosis, etc., remain the same as long as the data are fixed.4 Of course, when choosing among several competing specifications of the data, we hope to select the specification that matches most closely all of its properties, i.e., its finite-dimensional distribution^.^ But since our most basic understanding of and intuition for the data comes from its marginal distribution, at the very least we shall require that any plausible specification must match the marginal distribution's unconditional moment^.^ This is tantamount to fixing the mean, variance, skewness, etc. at the "true" values. Alternatively, from a purely empirical standpoint, the unconditional sample moments of the data are fixed at a given point in time since we have only one historical realization of each asset return series. But the conditional moments of the data depend on the conditioning information, which changes as we learn more about the underlying economic structure of the data. The specification searches that we undertake can almost always be viewed as an attempt to fit a statistical model to these fixed sample moments. Finally, yet another symptom of the link between predictability and option prices is the observation that implied volatilities will generally be biased estimates of the sample volatility of finite holding-period asset returns in the presence of predictability (see, for example, equation (24) below). The nature and magnitude of such biases depend on the nature and magnitude of predictability. By fixing the unconditional moments of the data and specifying the form of predictability in asset returns, we show that changes in predictability affect the population value of the diffusion coefficient, and this in turn will affect option prices. The particular effect on option prices depends critically on how predictability is specified in the drift. For example, if the drift depends only on exogenous time-varying economic factors, then an increase in predictability unambiguously decreases option values. But if the drift also depends upon lagged prices, then an increase in predictability can either increase or decrease option values, depending on the particular specification of the drift. This is also the approach taken in the growing "calibration" literature begun by Mehra and Prescott (1985). More recent examples include Abel (19921, Cecchetti, Lam, and Mark (19931, Heaton and Lucas (1992, 19941, Kandel and Stambaugh (1988, 19901, and Weil(1989). Although the finite-dimensional distributions do not completely determine a continuous-time stochastic process, for our purposes they shall suffice. More rigorously, the concepts of separability and measurability must be introduced to complete the definition of continuous-time processes s e e , for example, Doob (1953, Chapter 11.2). For convenience, we shall refer to the unconditional moments of the marginal distribution of returns as simply the "unconditional moments." These moments are not to be confused with unconditional "co-moments," which are moments of the joint distribution of returns, not of the marginals. 90 The Journal o f Finance We derive explicit pricing formulas for options on assets with predictable returns, and show that even small amounts of predictability can have a large impact on option prices, especially for longer maturity options. For example, under the standard Black-Scholes assumption of a geometric random walk for stock prices, the price of a one-year at-the-money call option on a $40 stock with a daily return volatility of 2 percent per day is $6.908. However, under a trending Ornstein-Uhlenbeck (0-U) price process-which yields serially correlated returns-we show that a daily first-order autocorrelation coefficient of -0.20 and a daily return-volatility of 2 percent per day would yield an arbitrage-free option price of $7.660, an increase of about 11 percent (see Section 1I.C and Table I). Of course, the particular adjustment to option prices depends on the specification of the drift, and we propose several specifications that can account for a broad variety of predictability in asset returns and illustrate the importance of these adjustments with several numerical examples. In Section I we provide a brief review of the Black-Scholes option pricing model to clarify the role of the drift and to emphasize the distinction between the data-generating process and the "risk-neutralized" process for the underlying asset's price. The implications of this distinction for option prices are developed in Section 11, where we present an adjustment for the volatility parameter a that accounts for the most parsimonious form of predictability: autocorrelation in asset returns. To account for more general forms of predictability, we propose two classes of linear diffusion processes in Sections I11 and IV, the bivariate and multivariate trending 0-U processes, respectively. In Section V we show how the parameters of these predictable alternatives can be estimated with discretely sampled data by recasting them in statespace form and using the Kalman filter to obtain the likelihood function. We consider several extensions and qualifications in Section VI, and we conclude in Section VII. I. The Black-Scholes Option Pricing Formula and the Drift The fundamental insight of the option pricing models of Black and Scholes (1973) and Merton (1973) is the existence of a dynamic investment strategy involving the underlying asset and riskless bonds that replicates the option's payoff exactly. In particular, if the underlying asset's price process P( t ) satisfies the following stochastic differential equation: d log P ( t ) E dp(t) = p(.) dt + adW, (1) where a is the diffusion coefficient, p(.) the drift coefficient, W(t) a standard Wiener process, and trading is frictionless and continuous, then the no-arbitrage condition yields the following restriction on the call option price C: 91 Option Pricing When Asset Returns Are Predictable where r is the instantaneous risk-free rate of r e t ~ r n . ~ Given the two boundary conditions for the call option, C(P(T), T )= Max[P(T) K, 01 and C(0, t) = 0, there exists a unique solution to the partial differential equation (2), the celebrated Black-Scholes formula: