美国顶级经济学论文

Do Call Prices and the Underlying Stock Always Move in the Same Direction? Gurdip Bakshi University of Maryland Charles Cao Pennsylvania State University Zhiwu Chen Yale University This article empirically analyzes some properties shared by all one-dimensional diffusion option models. Using S&P 500 options, we find that sampled intraday (or interday) call (put) prices often go down (up) even as the underlying price goes up, and call and put prices often increase, or decrease, together. Our results are valid after controlling for time decay and market microstructure effects. Therefore one-dimensional diffusion option models cannot be completely consistent with observed option price dynamics; options are not redundant securities, nor ideal hedging instruments—puts and the underlying asset prices may go down together. Much of the extant knowledge about option pricing is based on the assumption that the underlying asset price follows a one-dimensional diffusion process. Examples of such option pricing models include the classic Black–Scholes (1973), Merton (1973), the Cox–Ross (1976) constant elasticity of variance, the ones studied in Derman and Kani (1994), Rubinstein (1994), Bergman, Grundy, and Wiener (1996), Bakshi, Cao, and Chen (1997, 2000), and Dumas, Fleming, and Whaley (1998). All models in the one-dimensional dif- fusion class share three basic properties. First, call prices are monotonically increasing and put prices are monotonically decreasing in the underlying asset price (the monotonicity property). Second, as the underlying asset price is the sole source of uncertainty for all of its options, option prices must be perfectly correlated with each other and with the underlying asset (the perfect We would like to thank Kerry Back, Cliff Ball, David Bates, Steve Buser, Tarun Chordia, Alex David, Ian Domowitz, Phil Dybvig, Mark Fisher, Eric Ghysels, John Griffin, Frank Hatheway, Bill Kracaw, Craig Lewis, Dilip Madan, Jim Miles, Shashi Murthy, Ron Masulis, Louis Scott, Lemma Senbet, Hans Stoll, Rene Stulz, Guofu Zhou, and especially the anonymous referee, Bernard Dumas, and Ravi Jagannathan for their comments. This article has benefited from comments by seminar participants at the Federal Reserve Board, Pennsylvania State University, Seventh Annual Conference on Financial Economics and Accounting at SUNY-Buffalo, Vanderbilt University, and the 1998 Western Finance Association meetings. Any remaining errors are our responsibility alone. Address correspondence to Charles Cao, Smeal College of Business, Pennsylvania State University, University Park, PA 16802, or e-mail: charles@loki.smeal.psu.edu. The Review of Financial Studies Fall 2000 Vol. 13, No. 3, pp. 549–584 © 2000 The Society for Financial Studies The Review of Financial Studies / v 13 n 3 2000 correlation property). Third, options can be replicated using the underlying and a risk-free asset, and are hence redundant securities (the option redun- dancy property). These model predictions have been the foundation of stan- dard options textbooks. But are they consistent with observed option-price dynamics? From the perspective of pricing, hedging, and/or model internal consistency, many existing studies have examined the empirical performance of the Black–Scholes and other members in the one-dimensional diffusion class [see, e.g., Rubinstein (1985, 1994), Bakshi, Cao, and Chen (1997, 2000), Dumas, Fleming, and Whaley (1998), and Bates (2000)]. However, none has focused directly on the above model predictions. To be exact, in our inquiry, we address an important empirical question: Do call prices and the underlying stock always move in the same direction, and do put prices and the underlying stock always move in the opposite direction? If they do not, how often does it occur? In addition, if these predictions are violated, to what extent are they related to market microstructure factors and option time decay. Are additional state variables required to characterize empirical option pricing dynamics? This article serves to fill each one of these gaps. Specifically, for our study we use bid-ask midpoint prices of S&P 500 index options, sampled at various intraday intervals (e.g., every half hour, 1 hour, 2 hours, and so on). The S&P 500 option market is one of the most active, and this index is also the basis for the most actively traded equity index futures contract. Furthermore, we focus on intraday sampling intervals, as they help minimize the impact of time decay in option premium on our results. This consideration is important because, unless one assumes a parameterized option pricing formula, it is not possible to decompose an option price change in a given period into a time-decay and a non-time-decay component. Overall, our findings can be summarized as follows. First, depending on the intraday sampling interval, bid-ask midpoint call prices move in the opposite direction with the underlying asset between 7.2% and 16.3% of the time. We refer to such violations as type I violations. This is true whether the spot S&P 500 index or the lead-month S&P 500 futures are used as a proxy for the underlying asset. Thus this type of violation of the model predictions cannot be a consequence of stale S&P 500 component stock prices. When put option prices are used in place of call prices, similar violation rates are documented. Second, when the sampling interval changes from intraday to interday, the occurrence rate actually decreases, suggesting a role played by time decay in option premium. Third, the violation occurrence rate differs across options’ maturity. Of a given moneyness, long-term calls are the most likely to move in the opposite direction, followed by medium- term and short-term calls. In general, there is no clear association between the moneyness of the option and its tendency to move in the opposite direction to the underlying stock. Finally, call and put prices with the same strike and the same expiration often move in the same direction, regardless of the changes in the underlying and irrespective of the intraday sampling interval. In fact, 550 Do Call Prices and Stock Always Move in the Same Direction? when prices are sampled every 3 hours, call and put prices go up or down together as often as 16.9% of the time. Whether the underlying asset goes up or down, it is more likely for the call and put prices to go down together than up together. These observed option price movements are contrary to standard textbook predictions. Most of these occurrences cannot be treated as “outliers” since one cannot imagine throwing away as much as 17% of the observations. As these price quotes are usually binding at least up to 10 contracts, neither can they be treated as insignificantly misquoted prices. Our empirical exercise also documents those occurrences in which the underlying asset price has changed during a given interval, but the option price quote has not (type II violations). Such occurrences are between 3.5% and 35.6%. Our analysis points to a class of violations in which the call/put prices have changed even though the spot index has not (type III violations), and the market makers overadjust option quotes in response to a change in the underlying stock price (type IV violations). The frequency of the lat- ter occurrence is as much as 11.7% for calls and 13.7% for puts. But our study shows that type II and type IV violations are essentially due to market microstructure-related effects and minimum tick size restrictions, and type III violation occurrence frequency is relatively rare. In our quest to comprehend observed option price dynamics, we focus our efforts on four candidates of interest: (1) market microstructure factors, (2) the violations of put-call parity, (3) the impact of time decay, and (4) a two-factor stochastic process for the underlying stock price. Maintaining a single-factor setting, our analysis reveals that market microstructure factors are important for some type of violations. For instance, the type II (type IV) violation occurrence rate is monotonically declining (increasing) in the dol- lar bid-ask spread. However, there is no association between type I violation rate and each of the market microstructure factors. When we adopt deviation from one round of transaction costs as a benchmark, about 3% of the intra- day option sample violates put-call parity. Nonetheless, our results establish that type I violation frequency is robust to the inclusion or the exclusion of such observations. In evaluating the impact of time decay, we notice that the magnitude of time decay is comparatively larger for interday samples, but negligible for intraday samples. As a consequence, time decay explains more of type I violations for daily samples, but less for intraday samples (the samples stressed in our work). Each empirical result holds across calendar time, and is stable under alternative test designs. In summary, it is evident from our study that some contradictory option- price movements are attributable to market microstructure factors and time decay. Still, it is difficult to explain why some types of violations occur in the first place. Regardless of the minimum tick size or bid-ask spread, market makers can at the minimum choose not to move call prices (bid or ask) up, for example, when the underlying price is going down. We are then led to reexamine model specifications beyond which much of the 551 The Review of Financial Studies / v 13 n 3 2000 extant option pricing knowledge is based. To search for an option pricing model that can explain the documented price dynamics (beyond what can be accounted for by market microstructure factors), one may need to incor- porate, besides the underlying asset price, additional state variables. In this direction, we stipulate that if one is to introduce another state variable that affects option prices, this second stochastic process should not be perfectly correlated with the underlying. Furthermore, since call prices often go down (up) when the underlying goes up (down), this state variable should either affect call prices differently than the underlying, or be negatively correlated with the underlying price. Among the known ones, the stochastic volatility (SV) model of Heston (1993) possesses such features. For this reason we investigate the extent to which the SV model can explain the documented option price dynamics. Our simulation results show that about 11% of call prices generated from the SV model move in the opposite direction with the underlying price. On the other hand, the regression results indicate the SV model quantitatively falls short of completely explaining the observed option price movements. Using implied volatilities and SV model parameters, we find that about 47% of the type I violations become consistent with the pre- dictions of the SV model. This analysis also uncovers the finding that type II and type IV violations are mostly outside the scope of the SV model. Thus a more plausible story of violation patterns should not leave out the role of market microstructure and time decay-related factors. Our empirical results have important implications for investment manage- ment practice. They suggest that certain standard hedging strategies might not perform as well as one might expect. For instance, as students learn from textbooks, a typical hedge for a stock position involves shorting calls (or, buying puts), with the understanding that the call and the stock will move up or down in tandem. But as often as 7% of the time call prices go up even when the underlying goes down, a conventional hedge may actually double, rather than reduce, the hedger’s loss. Another lesson from the textbooks is that when applying such models as the Black–Scholes to create a dynamic hedge (e.g., portfolio insurance), one should revise the hedge as often as the market condition changes. The reasoning is that, absent market frictions, hedging errors should converge to zero as the hedge revision interval shrinks to zero. Given our evidence that call (put) prices often move in the oppo- site (same) direction with the underlying, however, it is likely that beyond a certain point a higher frequency of hedge rebalancing will actually lead to higher hedging errors. Using the Black–Scholes formula as an example, we show that this is indeed the case: as revision takes place more frequently, the hedging errors decrease initially but increase after a certain point. This is true even without taking transaction costs into account. The rest of the article is organized as follows. Section 1 develops the theoretical implications of one-dimensional diffusion option pricing models. In Section 2, we describe the S&P 500 option and futures data. Section 3 552 Do Call Prices and Stock Always Move in the Same Direction? presents the main empirical results. Section 4 sheds light on a stochastic volatility option pricing model. In Section 5, we discuss the robustness of our findings. Concluding remarks are offered in Section 6. 1. Properties of Option Prices in a Diffusion Setting In this section we discuss several properties possessed by all option pric- ing models in which the underlying asset price follows a one-dimensional diffusion (and hence Markov) process. We refer to each such model as a one-dimensional diffusion (option pricing) model. 1.1 Basic properties The problem at hand is to determine the price of a European call with strike price K and τ years to expiration, written on some non-dividend-paying asset whose time t price is denoted by S(t). To solve this problem, we need to specify (i) the process followed by S(t) and (ii) the valuation rule of the econ- omy, granted that {S(t) : t ≥ 0} is a well-defined stochastic process on some probability space. Assume that S(t) follows a one-dimensional diffusion: dS(t) = µ[t, S] dt + σ [t, S]S(t)dW(t), t ≥ 0, (1) with S(0) > 0, where the drift, µ[t, S], and the volatility, σ [t, S], are both functions of at most t and S(t) and satisfy the usual regularity conditions, and W(t) is a standard Brownian motion. This process contains many of those assumed in existing option pricing models. For example, in Black and Scholes (1973), σ [t, S] = σ , for some constant σ ; in the Cox and Ross (1976) constant elasticity of variance model, σ [t, S] = σS(t)α , for some constants σ and α; and in the models empirically investigated by Dumas, Fleming, and Whaley (1998), σ [t, S] is the sum of polynomials of K and τ . The class of models covered by Equation (1) is also the focus of Bergman, Grundy, and Wiener (1996). For the valuation rule of the economy, assume, as is standard in the literature, that interest rates are constant over time and that the financial markets admit no free lunches.1 Then there exists an equivalent martingale measure with respect to which any asset price today equals the expected risk-free discounted value of its future payoff. For example, letting C(t, τ,K) denote the time t price of the call option under consideration, we have C(t, τ,K) = E∗(e−rτ max{S(t + τ)−K, 0}) (2) where the expectation operator, E∗(·), is with respect to a given equivalent martingale measure, and r is the constant spot interest rate. Note that under 1 The interest rates can be stochastic as in Bakshi, Cao, and Chen (1997) and others. But as found by Bakshi, Cao, and Chen (1997), stochastic interest rates may not be that important for pricing or hedging options. Given our focus on intraday option price changes, the impact of stochastic interest rates should also be negligible. 553 The Review of Financial Studies / v 13 n 3 2000 the equivalent martingale measure, the underlying asset price obeys the fol- lowing stochastic differential equation: d S(t) S(t) = rdt + σ [t, S]dW(t), (3) that is, the expected rate of return on the asset is, under the martingale measure, the same as the risk-free rate r . By Ito’s lemma, the call price dynamics are determined as follows: dC = { Ct + 1 2 σ 2[t, S]S2CSS } dt + CS dS, (4) where the subscripts on C stand for the respective partial derivatives. The result below is adopted from Bergman, Grundy, and Wiener (1996) (a proof is provided therein). Proposition 1. Let the underlying price S(t) follow a one-dimensional dif- fusion as described in Equation (1). Then the option delta of any European call written on the asset must always be nonnegative and bounded from above by one: 0 ≤ CS ≤ 1. (5) The delta of any European put, denoted by PS , must be nonpositive and bounded below by −1: −1 ≤ PS ≤ 0. (6) That is, provided that everything else is fixed, the value of a European call (put) should be nondecreasing (nonincreasing) in the underlying asset price. We refer to the above as the monotonicity property. Before discussing its implications further, we note that in a one-dimensional diffusion model the sole source of stochastic variations for all options is the underlying asset, and hence all option prices, regardless of moneyness and maturity, must covary perfectly with each other and with the underlying asset. This perfect correla- tion property imposes a simple, but potentially stringent, restriction on option price dynamics. It also implies another property of one-dimensional diffusion models, that is, option contracts are redundant securities, in the sense that they can be exactly replicated by dynamically mixing the underlying asset with a risk-free bond. 554 Do Call Prices and Stock Always Move in the Same Direction? 1.2 Testable Predictions The monotonicity and the perfect correlation properties are both directly testable using properly sampled option data. As these properties are shared by all one-dimensional diffusion option pricing models, any rejection applies to the entire class. To formulate the testable predictions more directly, note that in our empirical exercises to follow we measure contemporaneous price changes in the underlying asset and its options by using sampling intervals ranging from 30 minutes to 1 day. That is, if we use �t to denote the time length of the sampling interval, the highest value for �t is a day and the lowest is 30 minutes. Under the assumption that { Ct + 12σ 2[t, S]S2CSS } �t in Equation (4) is small and negligible for small �t (this assumption will be empirically justified in Section 3.5), intraday changes in an option price are mostly in response to contemporaneous changes in the underlying market. This together with Proposition 1 leads to the following predictions: 1. Over any intraday interval, price changes in the underlying asset and in any call written on it should share the same sign: �S�C ≥ 0, where �C denotes changes in the call price. 2. Over any intraday interval, price changes in the underlying asset and in any put on it should have opposite signs: �S�P ≤ 0, where �P denotes changes in the put price. 3. Over any intraday interval, �C/�S ≤ 1 and �P/�S ≥ −1, provided �S = 0. 4. Over any intraday interval, contemporaneous price changes in call and put options with the same strike price and the same maturity should be of opposite signs: �C�P ≤ 0. These predictions mainly examine the left-hand and right-hand inequali- ties in Equations (5) and (6). In order to derive exact relationships between changes in a call (or a put) and the underlying asset price, one would need to parameterize the underlying price process and the valuation framework in fur- ther detail, which is not the main purpose of the present article. Instead, our focus is on the empirical validity of the above model-independent predictions. 2. Intraday Index Option and Futures Data The dataset employed in this study includes all intraday observations on (i) the S&P 500 spot index, (ii) lead-month S&P 500 futures prices, and (iii) bid-ask midpoint prices for S&P 500 index options. The use of bid- ask midpoint prices for options is to help eliminate the impact of bid-ask bounces. The sa