Dividend Forecast Biases in Index Option Valuation

Review of Derivatives Research, 4, 285–303, 2000. c 2001 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.
Dividend Forecast Biases in Index Option Valuation DON M. CHANCE [email protected] First Union Professor of Financial Risk Management, Pamplin College of Business, Virginia Tech, Blacksburg, Virginia 24061 RAMAN KUMAR Professor of Finance, Pamplin College of Business, Virginia Tech, Blacksburg, Virginia 24061 [email protected] DON RICH [email protected] Associate Professor of Finance, College of Business Administration, 413 Hayden Hall-CBA, Northeastern University, Boston, MA 02115 Abstract. Since the early days of option pricing theory, the assumption that the dividends on the underlying stock or index over the life of the contract are known has not been challenged. We examine the sensitivity of index option prices to the assumption of dividend uncertainty. We consider a number of issues related to the forecasting of dividends and build a dividend forecasting model that passes several rigorous tests for unbiasedness. We then generate option prices using contemporary market levels and interest rates. We find that prices generated with the actual dividends are unbiased with respect to those generated using the forecasted dividends. The magnitudes of the forecast errors, however, are sufficiently large to suggest a concern, but the percentage errors are consistently small, typically amounting to less than two percent of the option price. We conclude that the convenient assumption that the stream of future dividends is known is probably innocuous. Keywords: index options, option pricing, dividend forecasting JEL classification: G13, C5 1. Introduction The uncertainty of future dividends is a standard assumption in the theory of finance. Nonetheless, it is common in derivative pricing to assume that the dividends paid on a stock over the life of the derivative are known. Assuming a known dividend stream implies that both the timing and magnitude of the dividends are known in advance. Most academics treat these requirements as trivial and innocuous and proceed to conduct research using the advantage of hindsight to collect dividend information ex post and use it as if it were available ex ante. Practitioners, however, must price and hedge derivatives ex ante, and, consequently, must forecast dividends. Thus, prices observed in the market are derived using dividend forecasts. The issue of dividend uncertainty and its effect on the prices of derivatives has been ignored in the literature. In this paper we examine the question of how dividend uncertainty can induce biases in the prices of index options, exploring the technical issues of forecasting dividends, tracking the effects of forecast errors through pricing models and providing a reality check of whether the assumption of known dividends is a safe one. 286 CHANCE, KUMAR AND RICH Black (1975) first suggested that the valuation of European options on dividend-paying stocks could be performed with the Black-Scholes formula by “subtracting the present value of the dividends likely to be paid before maturity from the stock price” and using “this adjusted stock price instead of the actual stock price in the formula.” Roll (1977) described this procedure as “escrowing” the dividends and invoked it to derive a closedform solution for the American call. Merton (1973) used the assumption of a constant but known dividend yield to derive a closed-form solution for European puts and calls. These classic papers established a well-accepted and rarely questioned precedent: dividends are either known or can be expressed as a known proportion of the value of the underlying stock. Since that time virtually every theoretical model and empirical study in option and futures/forward pricing has employed one assumption or the other.1 The inconsistency is obvious if one considers the dividend discount model in which the stock price is the present value of all future dividends. Since stock prices are clearly stochastic, at least a few components of the future dividend series should also be stochastic.2 The standard assumption, of course, is that the dividends over the life of the derivative are known and the remaining dividends are stochastic; however, there is no economic rationale for such an assumption. Even if this assumption were reasonable, there is no basis for dichotomizing certain and uncertain future dividends at the derivative’s expiration. There are significant implications of dividend uncertainty for the question of how to price derivative contracts and how to conduct empirical tests, which use ex post data. Yadav and Pope (1990, 1994) look at the sensitivity of futures prices to dividend uncertainty and find that there is no material impact from small errors in dividends on futures prices. The cost of carry futures pricing formula is, however, simple and linear in the dividends; a formula such as the Black-Scholes model is relatively complex and non-linear in the dividends. Moreover, dividends are a much smaller percentage of a futures price than of an option price. Harvey and Whaley (1992) provide evidence that for American index option pricing, the use of non-stochastic yields, which generate stochastic dividends, as a proxy for actual discrete dividends can induce significant errors. Though that study firmly establishes the need to use discrete dividends rather than a yield, it begs the question of whether the actual dividends themselves could have been predicted with reasonable accuracy and whether these dividend predictions lead to reasonably accurate option prices. This paper is organized in the following manner: Section 2 describes the issues associated with pricing options under dividend uncertainty. Section 3 describes the data set and statistical properties of our S&P 500 dividends. Section 4 discusses how we derive and test a time series forecasting model of S&P 500 dividends. Section 5 contains the primary set of tests of the accuracy of option prices generated from the forecasted and actual series. Section 6 contains our summary and conclusions. 2. Issues in Option Pricing Under Dividend Uncertainty 2.1. Pricing Models A number of important issues are raised with dividend uncertainty. The most fundamental question is whether arbitrage-based models require the assumption of dividend certainty and DIVIDEND FORECAST BIASES IN INDEX OPTION VALUATION 287 whether the consequences of dividend uncertainty are sufficient to invalidate such models. We do not attempt to address questions of theoretical pricing issues here, but one way to get around this problem is to tie the derivativeís payoff to the total return—dividends plus capital gains—on the stock or index. This is a common feature in the equity swaps market whereby the party going long effectively receives the dividends as well as capital gains. Another solution is suggested in a totally different context by Brennan (1998), who advocates the stripping of dividend streams from the stock index. The dividends could then be bought and sold in the same manner as U. S. Treasury strips. While obviously one cannot strip the entire perpetual stream of dividends, the removal of the dividends spanning the life of the option would allow its payoff to be tied strictly to the growth value of the index and eliminate this problem. Absent a solution of this sort, however, practitioners routinely trade options using the Black-Scholes and binomial models, often tweaking them in such a manner as to seemingly take care of violations of the assumptions. For example, the volatility smile, the incorporation of stochastic volatility, and the handling of writer default are issues that are commonly dealt with by simply pricing the options slightly higher or lower than justified by the model. Since option pricing models typically assume that a known dividend stream is discounted at the risk-free rate, it is our belief that a common practice for practitioners is to forecast dividends and discount them at the risk-free rate. Consequently, they violate the assumptions of the model and the characteristics of the data. Accordingly, this study examines the consequences of these actions. 2.2. The Potential Significance of Dividend Uncertainty On the surface it might appear that any error induced by dividend uncertainty could be small and insignificant. During the period examined, dividends amount to only 2–4 percent of the value of the S&P 500 and on many days no dividends are paid. Closer inspection suggests, however, that the errors induced by dividend forecasts could be economically significant. Consider European options. The stock price is reduced by the present value of the dividends over the life of the contract. If this present value estimate is incorrect, it is equivalent to using the wrong stock price to value an option on a stock without dividends. Would traders and researchers feel comfortable knowing that the error, however small, was in the stock price?3 Obviously any such error is transmitted to the option price by its delta. Moreover, the delta as well as the gamma, vega, and the other Greeks are in error, making hedging of derivative positions problematic. Another issue is that S&P 500 dividends are highly skewed, a result we document later. Large dividends occur with surprising regulatory. The maximum dividend is more than eight times the average. Values of 2–3 times the average are routinely observed. Consequently, focusing on the average dividend, while ignoring the extremes, can lead to false conclusions about the effects of dividends on the prices of options. For an American option, an even greater potential problem exists. Ceteris paribus, early exercise of calls is induced by large dividends. Dividend forecasts with errors can either incorrectly induce early exercise or fail to trigger exercise early when they should. Since the early exercise decision is made just before the ex-dividend date, and the dividend is 288 CHANCE, KUMAR AND RICH known just before the ex-date, any incorrect decisions will be due to valuation effects that are attributable to forecast errors of subsequent dividends. For American puts, large dividends tend to discourage early exercise. Large dividends can, therefore, result in failing to exercise puts early when one otherwise should. These effects are almost—but not completely—independent of the present value effect. Given two dividend streams with different present values, the stream with the larger present value could have more frequent occurrence of early exercise of calls. Whether it does depends on the magnitude and timing of the dividends relative to other factors such as the degree of moneyness and the volatility. 2.3. Issues in Forecasting Dividends For options on individual stocks, it is fairly common to simply forecast the individual dividends. For index options, there are two alternatives. One can forecast the individual dividends of the component stocks and then aggregate those forecasts. Alternatively, one can forecast the aggregate index dividends. The former method is preferred by some practitioners, possibly due to the fact that index dividend series have not been around as long as index derivatives themselves. It is tempting to assume that, using a naive forecast model, the forecast errors for individual dividends will be random, but this is not true. Dividend payments exhibit distinct seasonal patterns. Consequently, forecast errors will not be independent. These seasonal patterns, as we show later, are quite useful in building an index dividend forecasting model. Forecasting accuracy is often supported by evidence of unbiasedness. Unbiased forecasts of dividends, however, will not necessarily lead to unbiased option prices for two reasons. One is that some of the forecasts, such as the nearby dividend forecasts, might be biased in one direction and other forecasts, such as the deferred dividend forecasts, might be biased in the other direction, thereby offsetting and leading to the conclusion that the overall forecasts are unbiased. If that were the case, then transforming the forecasts by the present value function means that each forecast is not equally weighted, which could lead to bias in the overall present value of the forecasted dividends. Another reason option prices could be biased in spite of unbiased dividend forecasts is that an option price is a non-linear transformation of the magnitude and timing of the individual dividends and their present values. An unbiased forecast remains unbiased if the transformation is linear but is not necessarily following a non-linear transformation. Suppose, however, that option prices generated with forecasted dividends were unbiased with respect to those generated with actual dividends. Even then, an unbiased set of prices may have offsetting large positive and negative errors. Clearly the impact of dividend uncertainty has been ignored in the literature. Dividends have the potential to induce significant problems, particularly if they cannot be forecasted with a high degree of accuracy. Accordingly, we undertake our study in two main parts. First we attempt to forecast dividends on a major stock index, the S&P 500. Then we compare option prices generated from forecasted dividends to those generated when the actual known ex post dividends are used. The results should reveal insights into whether dividend certainty is a reasonable assumption and, if it is, how well dividends can be forecasted and used as a proxy for the presumably known dividends. Our focus is on short-term stock index options of the type traded on options exchanges. A large amount DIVIDEND FORECAST BIASES IN INDEX OPTION VALUATION 289 of data are required to build a reliable time series model, and separate data are required for out-of-sample testing. Given that the dividend series we use started only in 1988 (as discussed below), we were required to focus on short-term options. 3. The S&P 500 Dividend Data and Test Methodology 3.1. An Overview of the Methodology This research proceeds in two steps. The first is to build and test a predictive model for the time series of S&P 500 daily dividends. We do this by incorporating knowledge about the seasonal patterns and any other factors that might be relevant such as serial correlations and forecast adjustments to previous forecast errors. We try numerous combinations of the variables and examine the quality of alternative models by generating out-of-sample dividend forecasts. We divide the data set into two equal parts, one period for forecasting and the other for out-of-sample testing.4 After settling on a single best predictive model, in the second stage of the research we generate separate European and American option prices using the predicted dividends and the actual dividends and examine how those series differ. 3.2. S&P 500 Daily Dividends and their Statistical Properties Daily dividends were collected from the S&P 500 Information Bulletin, a monthly publication of Standard & Poor’s that provides data on the S&P 500 Index. S&P began publishing this dividend series on June 1, 1988. Our data consist of these daily dividends for the period June 1, 1988 through November 17, 1994, a total of 1,637 trading days. Figure 1 illustrates our dividend series. The series is quite volatile with distinct peaks suggesting a possible seasonal pattern. Evidence on the seasonality of stock index dividends has been noted in Harvey and Whaley (1992), Gastineau and Madansky (1983), Kipnis and Tsang (1989) and Cornell and French (1983). Table 1 provides descriptive statistics and evidence on seasonal patterns in the dividend data. Obviously these statistics are unconditional estimates, but later we incorporate conditional estimates by a daily updating of their time series properties. We observe that dividends are positively skewed so the standard deviation is somewhat less relevant but is still informative. The skewness is apparent from the large difference between the means and the medians. The overall average daily dividend, as seen in Panel A, is .047 with a median of .023 and a maximum of .401. About ten percent of the days have no dividend. Panel B shows that there are distinct day-of-the-week patterns. Mondays are clearly a heavy dividend day with an average of .084 and a median of .072. About 4.8 percent of the Mondays are zero dividend days. The second largest dividend day is Tuesday, of which only 2.7 percent are zero dividend days. The smallest dividend day is Wednesday with an average of .025 and a median of .012 and of which 14.4 percent are zero dividend days. The most zero dividend days are on Thursday with 16 percent. Panel C shows the results by month. February, May, August and November are the heaviest months with May being the 290 CHANCE, KUMAR AND RICH Figure 1. Daily Dividends on the S&P 500. The data were collected from monthly issues of the S&P 500 Information Bulletin, a publication of Standard & Poor’s that provides data and information on the S&P 500 Index. Standard & Poor’s constructed its dividend series starting June 1, 1988. This series reflects dividends paid on the component stocks in the index, with values converted to index points. The time period shown above is June 1, 1988 through November 17, 1994, a total of 1,637 trading days. largest. No zero days were observed in February and August. The lowest dividend months are January, July, October and April. Almost one-fourth of the days in those months have no dividends. Panel D shows the breakdown by year from the beginning of the sample period in 1988 to the end in 1994. The annual average dividend has risen steadily from .038 to .052, the median has risen from .021 to .026 and the standard deviation has risen similarly from .047 to .062. The percentage of zero dividend days has also risen from around 10 percent to the 15-16 percent range. The results by quarter are not shown but are similar from one quarter to the other with the second quarter being only slightly higher on average, and the percentage of zero dividend days across the quarters being about the same at 10 percent. These results suggest two important points: there is substantial uncertainty in the dividends of the S&P 500 index, but there are distinct patterns to the timing and magnitude of dividend payments. On the first point, note the maximum dividend statistics, which are five to ten times the average. The minimum dividends are, naturally, zero and occur unconditionally about once every two weeks. The seasonal patterns are helpful, however, in reducing the uncertainty about future dividends. Indeed a predictive model can exploit these seasonalities to improve the forecasting accuracy. 4. A Predictive Model of S&P 500 Daily Dividends As indicated in Section 3.1, we divide the data set into two parts, each consisting of 759 trading days. This provides approximately three years of daily data with which to build the forecasting model. The first period runs from June 1, 1988 through May 31, 1991. The sec- 291 DIVIDEND FORECAST BIASES IN INDEX OPTION VALUATION Table 1. Summary Statistics and Seasonal Patterns for S&P 500 Daily Dividends. Summary descriptive statistics for the entire sample of S&P 500 daily dividends broken down into various seasonal subperiods. Dividend data are taken from the S&P 500 Information Bulletin from June 1, 1988 through November 17, 1994. This series was first constructed by Standard & Poor’s starting June 1, 1988 and reflects dividends paid on component stocks in the index, with values converted to index points. The sample consists of 1,637 trading days of dividends. The column labeled “Percentage of Zero Days” is the number of days out of the total for each category in which the dividend was zero. Sample Number of Days Mean Standard Deviation Median Maximum Percentage of Zero Days .047 .058 .023 .401 10.1 .084 .053 .025 .034 .043 .070 .051 .039 .052 .059 .072 .032 .012 .016 .016 .401 .251 .219 .340 .267 4.8 2.7 14.4 16.0 12.5 .028 .073 .040 .030 .080 .040 .028 .065 .047 .029 .070 .043 .038 .062 .054 .040 .073 .054 .039 .058 .060 .043 .071 .062 .012 .065 .018 .013 .069 .019 .013 .054 .021 .010 .048 .020 .179 .282 .258 .211 .355 .219 .230 .337 .299 .205 .401 .320 23.4 0.0 7.6 22.0 0.8 5.3 24.0 0.0 7.0 22.9 5.1 2.4 .038 .044 .048 .048 .049 .050 .052 .047 .049 .059 .059 .062 .063 .062 .021 .024 .023 .023 .023 .024 .026 .236 .214 .401 .327 .340 .337 .355 10.7 9.1 5.5 7.1 7.9 16.2 15.2 A. Full Sample 1,637 B. By Day of the Week Monday 311 Tuesday 334 Wednesday 334 Thursday 331 Friday 327 C. By Month of the Year January 128 February 114 March 132 April 123 May 127 June 151 July 146 August 157 September 143 October 153 November 137 December 126 D. By Year 1988 149 1989 252 1990 253 1991 253 1992 254 1993 253 1994 223 ond period runs from June 1, 1991 through June 1, 1994. Since we wish to predict dividends for 120 trading days afterwards, we extend the second period to November 17, 1994. 4.1. Model Development As noted in Section 2.3, the daily time series of cash dividends on the S&P 500 can be forecasted in two ways. We could forecast the cash dividends on each of the 500 stocks in the S&P 500 and aggregate the forecasts. Alternatively, we could use the seasonal, autoregressive, and moving average patterns in the daily time series of cash dividends to 292 CHANCE, KUMAR AND RICH forecast the future aggregate dividends. Predicting the dividend amounts and dates for the individual stocks in the S&P 500 would require the forecasting of 500 distinct dividend processes, which would also require periodic updating. Moreover, given that a maximum of four daily observations in a year will be non-zero for most stocks, it would be difficult if not impossible to estimate an appropriate time series econometric model to forecast the dates and the amounts of the cash dividends on the individual stocks. It is true that the forecasting errors made for each stock will be non-systematic and will, therefore, be mitigated to a certain extent in the process of aggregation. Nonetheless, the errors, especially on the timing of the individual firms’ dividends, are likely to be so large that even after aggregation, would produce forecasts that are not as accurate as the alternative procedure and would most certainly be more costly.5 Therefore, it makes little sense to attempt to forecast the components of the series and then reconstruct the series by aggregation. Furthermore, to disaggregate a series, forecast it and reaggregate the forecasts adds an unnecessary step. We demonstrate in the following pages that the aggregate series has desirable time series properties that facilitate forecasting. The procedure of estimating aggregate cash dividends on the S&P 500 can be done using a time series econometric model that exploits the seasonal, autoregressive, and moving average patterns in the time series of daily cash dividends. The accuracy of the forecasts obtained using this method would depend on the effectiveness of the estimated model. We shall demonstrate in the following sub-sections that we have estimated a very effective time series model of aggregate dividends on the S&P 500 that produces reliable forecasts. Figure 2 shows the serial correlations for up to 400 lags over the June 1, 1988–November 17, 1994 period. There are distinct patterns in which the serial correlation reaches a high of about .4 corresponding to every sixty trading days. These results reflect the quarterly payment patterns that firms are known to follow. The other small positive and negative autocorrelations are mainly a result of the day of the week patterns and will disappear when these patterns are accounted for. We know that dividend payments have weekday, monthly and autoregressive patterns. The autoregressive pattern can be isolated by noting that with an average of 253 trading days in a year, the highest dividends occur every 253/4 = 63.25 days. Using this as a guide, we might expect the dividend patterns to repeat on lag days t − 63 and t − 64, t − 126 and t − 127, t − 189 and t − 190 and t − 253 and t − 254. This is precisely what one finds if the autocorrelations are examined more closely. We know, however, that if we attempt to predict based on these factors alone that we will occasionally miss high or low dividend days by a day or two. Errors from estimating will be negatively correlated with errors over the next several days. Thus, we should incorporate a moving average error term. After extensive testing we found that three lagged errors were sufficient. After trying numerous combinations, we found that the following overall model provided the greatest accuracy: Dt = b0 + b1 Dt−63 + b2 Dt−64 + b3 Dt−126 + b4 Dt−189 + b5 Dt−253 + b6 Dt−254 + b7 W1 + b8 W2 + b9 W3 + b10 W4 + b11 M1 + b12 M2 + b13 M3 + b14 M4 + b15 M5 + b16 M6 + b17 M7 + b18 M8 + b19 M9 + b20 M10 + b21 M11 + b22 µt−3 + b23 µt−2 + b24 µt−1 + µt (1) DIVIDEND FORECAST BIASES IN INDEX OPTION VALUATION 293 Figure 2. Serial Correlations for Lags 1–400 of daily dividends on the S&P 500. The data were collected from monthly issues of the S&P 500 Information Bulletin, a publication of Standard & Poor’s that provides data and information on the S&P 500 Index. Standard & Poor’s constructed its data series beginning June 1, 1988. This series reflects dividends paid on the component stocks in the index, with values converted to index points. The time period is June 1, 1988 through November 17, 1994, a series of 1,637 trading days. where Dt− j is the dividend for lag j, Wk is a weekday dummy where W1 = Monday, . . ., W4 = Thursday (Friday is the omitted class), Mi is a monthly dummy where M1 = January, . . ., M11 = November (December is the omitted class) and µt−q is the forecasting error for q = 0, 1, 2 and 3 days prior.6 To verify the adequacy of the model, we tested for white noise in the residuals and were unable to reject the hypothesis of white noise with the Portmanteau test. In addition about 75 percent of the residual autocorrelations up to a lag of 25 were within one standard error and 100 percent were within two standard errors. We used the Akaike Information Criterion as a primary indicator of the effectiveness of the model compared to alternative specifications with respect to in-sample forecasts. Our purpose in developing the model, however, is to generate out-of-sample dividend forecasts. 4.2. Generating Out-of-Sample Dividend Forecasts To replicate the process in which a firm or trader would forecast dividends, we generate conditional forecasts of dividends using the current information set. As that information set is updated, we generate new forecasts. We begin by assuming that the model has been sufficiently identified as presented in the previous section. We initially estimate the 294 CHANCE, KUMAR AND RICH coefficients of the model using data over the first 759 trading days. We then generate dividend forecasts over the next 120 trading days. Using trading day 2 through trading day 760, we update the model. These new estimated coefficients are then used to generate a forecast of the dividends over the next 120 trading days. We continue this process, moving forward one trading day, using 759 trading days to estimate the model, and forecasting the dividends for the next 120 trading days. Consequently, for each trading day we obtain a prediction of the daily dividends over the next 120 trading days, conditional on the information set available on that day, which is comprised of the dividends over the last 759 trading days. Each forecasted dividend is paired with the actual dividend on the given day. 4.3. Model Testing with Out-of-Sample Dividend Forecasts Before testing for the unbiasedness of the out-of-sample forecasts, we examine the significance of the correlation between the dividend forecasts and the forecasting errors for each of the 759 sets of 120-day ahead forecasts. A significant correlation between the forecasts and the forecasting errors would suggest that the forecasts can be improved. None of the 759 estimated correlations between the forecasts and the forecasting errors are significant at the 5 percent level. Only one of the 759 correlations is significant at the ten percent level.7 We conduct three tests of the quality of our dividend forecasts. The obvious test is to compare the actual with the forecasted dividends. In derivative pricing, however, the present value of the dividends plays such an important role that it is also worthwhile to compare the present values of the actual and forecasted dividends. In fact as noted above, for European options the present value of the dividends is the only way in which the dividends affect the price of the derivative. As we previously discussed, even if the individual dividend forecasts are unbiased there is still the possibility that the present value of the dividends will be biased, a result of biases in components of the sample that cancel out in the aggregate and are unequally weighted in the present value transformation. Tests for unbiasedness of the dividend forecasts, and later of the option prices themselves, can be conducted using standard linear regression techniques. For example, let the dividend be written as an expectation conditional on the available information set,  , and a mean zero error orthogonal to  : D = E[D|] + ε where E[ε|] = 0. From this we can use the well-known regression form, D = α+β D()+e, where D() is the dividend conditional on the information set  , which in our case is the dividend derived from the forecasting procedure. If the forecast is unbiased, then the coefficient on α should be 0.0 and the coefficient on β should be 1.0. Following the classic approach of Theil (1966), we specify for the first test a set of 759 regressions of the form: Dta = α + β[Dte |( j)] + ε jt ; t = j + 1, . . . , j + 120 [for j = 759, . . . , 1517] (2) where Dta is the actual dividend observed on trading day t, Dte |( j) is the forecast made on trading day j, conditional on the information set available on trading day j, of the dividend DIVIDEND FORECAST BIASES IN INDEX OPTION VALUATION 295 on trading day t where t begins the next trading day and goes out for 120 trading days. These tests make up a set of 759 regressions, one for each of the trading days in the second half of the sample period. As noted above, unbiasedness is indicated by failure to reject the hypotheses α = 0 and β = 1. We use Newey-West (1987) estimates of the standard errors to address the problem of an inconsistent covariance matrix resulting from overlapping observations. The results strongly support unbiasedness. The R-squares range from 30 to 58 percent. Out of the 759 regressions, the intercept was significantly different from zero at the .05 level only seven times and the slope was significantly different from one at the .05 level only six times. At the .05 level, one would expect about 38 rejections. Though these regressions are not independent, the evidence suggests that the forecasts are unbiased, but further testing is required. Our second test is a set of five regressions of the form: a D j+n = α + β[Dne |( j)] + ε jn ; j = 760, . . . , 1518 [for n = 1, 30, 60, 90, or 120] (3) a where D j+n is the actual dividend n trading days later than the current trading day j where n = 1, 30, 60, 90 or 120. This is a rolling regression of the actual dividend exactly n trading days ahead on the forecasted dividend n trading days ahead. Letting n = 30, for example, the first observation is the actual and forecasted dividend 30 trading days ahead, i.e., trading day 789, for the 759th trading day of the sample, where the forecast is generated using the first 759 trading days of data. The second observation is positioned on trading day 760 and is the 30-day-ahead actual and forecasted dividend, i.e., trading day 790, using estimates obtained from data over the previous 759 trading days. This continues until 759 observations are generated. Separate regressions were run for varying values of n as indicated above. Here we also use the Newey-West (1987) estimators. Table 2 presents these results. None of the intercepts are significantly different from zero and none of the slopes are significantly different from one. The R-squares range from about 56 percent for one-day ahead forecasts to almost 42 percent for 120 trading-day-ahead forecasts. So far we have looked at the forecasts two different ways and found no evidence to suggest that the forecasts are not unbiased. As we noted earlier, however, the present value of the dividends is an important factor in option pricing. We might expect, tautologically, that if the individual dividend forecasts are unbiased, their present value should be unbiased. It would, therefore, be tempting to stop here and conclude that we have an unbiased model. We feel, however, that it is extremely important that our model pass every reasonable hurdle. As we mentioned previously, there remains the possibility that some dividend forecasts may be biased in one direction and some biased in another. With unequal weighting by the present value operator, we might still end up with bias in the present values of the dividend forecasts. We conducted tests on the present value of the dividends over the next 30, 60, 90 and 120 trading days. For each day we obtained the rate on a T -bill with maturity of 1, 2, 3, 4, 5 and 6 months. The present values were computed by discounting the dividends using the rate on the Treasury bill on that day with the maturity closest to that of the dividend 296 CHANCE, KUMAR AND RICH Table 2. Regressions of Actual on Predicted Dividends. This table provides statistics for the regression D aj+n = α +β[Dne |( j)]+ε jn where D aj+n is the actual dividend on trading day j +n ( j = 759, . . ., 1517; n = 1, 30, 60, 90 or 120) on the S&P 500 and [Dne |( j)] is the forecasted dividend for n trading days ahead, conditional on the information set  available on trading day j. The first observation is the actual and forecasted dividends n trading days ahead on the 759th day, where the forecast is generated using an autoregressive moving average model with seasonal dummy variables based on the previous 759 trading days. The second observation is the actual and forecasted dividends n trading days ahead on the 760th trading day, where the forecast is generated using the autoregressive moving average model with seasonal dummy variables based on the previous 759 trading days. This continues until 759 observations are generated. Due to the possible autocorrelation, the Newey-West method is used to adjust the standard errors, s(α) and s(β). The corrected t-statistics are also presented. Unbiasedness is indicated by failure to reject the hypotheses that α = 0 and β = 1. *denotes that the null hypothesis of α = 0.0 and β = 1.0 are rejected at a significance level of .05 level. Model parameters are estimated over the period June 1, 1988 through May 31, 1991, and forecasts are evaluated over the period June 1, 1991 through June 1, 1994. α s(α) t (α) β s(β) t (β) R 2 (%) F 1 day ahead 30 days ahead 60 days ahead 90 days ahead 120 days ahead −0.00073 0.00182 −0.40110 1.04244 0.04168 1.01823 56.35 977.355 0.00081 0.00190 0.42632 0.98830 0.04098 −0.28551 43.46 581.817 0.00075 0.00194 0.38660 0.98941 0.04208 −0.25166 42.61 562.037 −0.00046 0.00206 −0.22330 1.01642 0.04357 0.37686 41.53 537.708 0.00006 0.00197 0.03046 1.00617 0.04092 0.15075 41.62 539.782 day. These tests were constructed identically to the second set of tests as indicated above. That is, in this test we examined the association between the present value of the actual dividends for up to n trading days ahead and the present value of the forecast of the next n daily dividends. In the interest of brevity, we do not report these results in tabular form, but a table containing them is available upon request. In no cases were the intercepts or slopes significantly different from zero or one, respectively. Further analysis of descriptive statistics shows that the average difference between the actual and forecasted present values ranges from .005 for the 30-day case to .023 for the 120 trading day case, the median difference ranges from .002 to .028 and the mean absolute difference ranges from .05 to .082. So while it is important to realize that we are making unbiased forecasts, typical differences are not zero. Whether this will matter in derivative pricing is a subject we explore in the next section. It is extremely unlikely that a naive forecasting model could outperform our model. A model based on historical mean dividends, however, would be a special case of zero values for all of our coefficients. A model based on a simple extrapolation of previous dividends might occasionally beat our model, but it is inconceivable that it could be superior over the long run to a model that incorporates conditional updating of time series and seasonality properties. It is possible that a better forecasting model that incorporates different estimates or different variables could be developed, but it seems unlikely that significant improvements could be made. DIVIDEND FORECAST BIASES IN INDEX OPTION VALUATION 297 5. Tests of Options Pricing Models Under Dividend Uncertainty An obvious test is whether the forecasted dividends explain the actual market prices of derivatives. Unfortunately such a test would be difficult to interpret. Market prices, even in the form of high quality trade-by-trade data, would be influenced by bid-ask spreads and other transaction costs, measurement errors in the parameters of the model and more importantly model bias itself. The risk-free rate and volatility would have to be estimated and the effects of the volatility smile would distort the results. Although these are all important considerations when investigating the quality of derivative pricing models, they can seriously confound the effects of our tests. Moreover, we need to compare European with American options. There is no underlying index on which the European and American options both trade in U.S. markets. Consequently we need a controlled experiment, one in which we can observe rational European and American option prices and where the only biases we might possibly introduce derive from dividend forecasts. We, thus, choose to undertake a combination of simulation and market-based tests. We generate simulated derivative prices from standard pricing models using actual index levels and interest rates from the periods corresponding to the periods over which the dividends were paid. This allows us to vary the moneyness, volatility and expiration for options.8 Corresponding to the procedure described previously, we start on the 759th day and used the previously described forecasts of dividends over the next 120 trading days. Recall that these forecasts were based on coefficients obtained using data from the first 759-day period. These forecasts are then used with other inputs to generate option prices. We then move forward one day, updating the model so that each day we have a set of 120 trading-day-ahead forecasted and actual dividends. Then a new set of option prices is generated. The inputs used with the dividends to generate the option prices are a combination of concurrent market information and several variables whose values we control. For the current market information, we use the actual level of the index and the maturity-matching Treasury bill rate on the given day as previously described, the latter as a proxy for the riskfree rate. We generate several sets of prices with different degrees of moneyness, including at-the-money and 10 and 20 percent in- and out-of-the-money, and times to expiration of 30, 60, 90 and 120 calendar days. Rather than attempt to extract the market’s volatility and given our interest in the possible effects of higher or lower levels of volatility, we use alternative standard deviations of .1, .2 and .3, which span the range of reasonable S&P 500 volatility levels. Both the European and American option prices are produced using the binomial model with 360 time steps.9 The different combinations of input variables lead to 336 possible sets of option prices, which are derived from seven different strike prices, three different volatilities, four different expiration dates, two different option types (call or put) and two different exercise styles (American or European). 5.1. Tests for Unbiasedness The first question we address is whether the derivative prices generated by the forecasted and actual dividends are unbiased with respect to each other. All of our arguments for use of the regression approach to unbiasedness of the dividends apply to unbiasedness of the 298 CHANCE, KUMAR AND RICH option prices. As in the dividend test, we choose the option prices using the actual dividends as the dependent variable and the prices using the forecasted dividends as the independent variable. As previously stated, failure to reject the hypothesis that the intercept, α, is 0.0 and the slope, β, is 1.0 suggests unbiasedness. We again use the Newey-West (1987) corrected standard errors.10 On about five percent of the regressions, all cases of deep in-the-money options, there is so little variation in prices that we were unable to invert the matrix to run the regressions. To conserve space the 336 regression results are not presented but are summarized in Table 3, which shows the frequency of significant regression slopes and intercepts for various classifications. We examine the singular hypotheses that the intercept is significantly different from zero and the slope is significantly different from one as well as the joint hypothesis that both are significant. As the table shows, significance in any case is rare.11 5.2. A Further Look at the Price Differentials The tests for unbiasedness show that option prices generated using actual dividends are unbiased estimates of option prices generated using forecasted dividends. This is a useful result for researchers conducting empirical tests of option pricing models and for practitioners who must price options with forecasted dividends. For researchers who test models and market efficiency by looking for evidence of arbitrage opportunities, however, it should be apparent that errors, unbiased though they be, can suggest that arbitrage opportunities exist. Thus, it is important that we examine the size of the differences between option prices generated using forecasted dividends versus those generated using actual dividends. We examined the properties of the differences between the option prices generated using actual dividends and those using forecasted dividends. Table 4 shows results for a representative set of these 336 comparisons. We show the case for a volatility of .2 only because we found that the differences across volatilities are negligible. Also, we show the differences only for the case of ten percent out-of-the-money, ten percent in-the-money and at-the-money and for 30- and 90-day American calls.12 The mean differences appear to be quite small. For calls, the means are very close to zero, with the exception of the 30-day in-the-money American call, which has a mean of .02, and the 90-day European call, which has a mean of −.014. The mean values, however, reflect a canceling of positive and negative errors. Since we interested in pricing accuracy, we should focus on the extremes and the average deviations. For both European and American calls, the maximums are further apart and the mean absolute deviations are larger for in-the-money than at-the-money calls and larger for atthe-money than out-of-the-money calls. These values are also larger for 90-day calls than for 30-day calls. For at-the-money calls, these ranges are from about 31 cents for 30-day American calls to almost 49 cents for 90-day European calls. For at-the-money calls, the range is from about 16 cents for 30-day American calls to almost 30-cents for 90-day European calls. The mean absolute deviations for out-of-the-money options are less than one cent for 30-day calls and slightly more than one cent for 90-day calls. For at-the-money options, the mean absolute deviations are 2.2 and 2.5 cents for 30-day calls and 3.3 and 3.7 cents Option Classification European Call Options American Call Options European Put Options American Put Options σ = .1 σ = .2 σ = .3 Moneyness = 0.80 (% of S) Moneyness = 0.90 (% of S) Moneyness = 1.00 (% of S) Moneyness = 1.10 (% of S) Moneyness = 1.20 (% of S) Expiration = 30 calendar days Expiration = 60 calendar days Expiration = 90 calendar days Expiration = 120 calendar days Number of regressions 81 81 78 78 96 110 112 38 46 48 48 42 76 80 80 82 Number of p-values rejecting H0 : α = 0 at 5% level 2 7 0 0 5 4 0 4 1 0 2 2 4 3 0 2 Number of p-values rejecting H0 : β = 1 at 5% level 2 5 0 0 4 3 0 2 1 0 2 2 4 1 0 2 Number of p-values rejecting H0 : α = 0 and β = 1 at 5% level 2 5 0 0 4 3 0 2 1 0 2 2 4 1 0 2 DIVIDEND FORECAST BIASES IN INDEX OPTION VALUATION Table 3. Summary of Regression Tests for Unbiasedness of Option Prices for all Maturities. This table provides a summary of the frequency of rejection of the hypothesis that α = 0 or β = 1 or both at the .05 level in regressions of option prices generated by using actual dividends on option prices generated using forecasted dividends. The forecasted dividends for up to 120 trading days are generated using an autoregressive moving average model with seasonal dummy variables over the previous 759 trading days. Both European and American option prices are produced using the actual level of the S&P 500 and the appropriate risk-free rate proxied by the maturity-matching Treasury bill rate on the given day, with moneyness and volatility varied as indicated in the table. Both European and American option prices are obtained using the binomial model. The forecasts are rolled forward one trading day by dropping the oldest trading day and the coefficients are re-estimated daily. The moneyness factor is the ratio of the index level to the exercise price. Due to the overlapping nature of the observations, serial correlation up to the 29th order (for 30 calendar day options) and 89th order (for 90 calendar day options) is possible. Thus, the standard errors are estimated using the correction of Newey and West (1987). Due to a singular matrix, a small number of deep in-the-money regressions could not be run. Model parameters are estimated over the period June 1, 1988 through May 31, 1991, and forecasts are evaluated over the period June 1, 1991 through June 1, 1994. 299 300 CHANCE, KUMAR AND RICH Table 4. Descriptive Statistics of Differences Between Call and Put Option Prices for Volatility of 20%. This table provides statistics on the differences between the call and put option prices generated using the actual dividend series and those generated using the forecasted dividend series for a volatility of .2 and various degrees of moneyness. The forecasted dividends for up to 120 trading days are generated using an autoregressive moving average model with seasonal dummy variables over the previous 759 trading days. Both European and American option prices are produced using the actual level of the S&P 500 and the appropriate risk-free rate proxied by the maturity-matching Treasury bill rate on the given day, with moneyness and volatility varied as indicated in the table. Both European and American option prices are obtained using the binomial model. The forecasts are rolled forward one day by dropping the oldest trading day and the coefficients are re-estimated daily. The option maturities are in calendar days. The moneyness factor is the ratio of the index level to the exercise price. Thus an option listed as 10 % in-the-money is an option in which the index level is 10 percent higher than the exercise price. MAD is the mean absolute deviation. Model parameters are estimated over the period June 1, 1988 through May 31, 1991, and forecasts are evaluated over the period June 1, 1991 through June 1, 1994. 30 Day European Calls 30 Day American Calls 90 Day European Calls 90 Day American Calls Mean Min Max MAD Mean Min Max MAD 10 % ATM 10 % ITM OTM −.005 −.002 −.000 −.208 −.108 −.010 .178 .089 .009 .048 .025 .002 10% ATM 10% ITM OTM .020 .003 −.000 −.146 −.084 −.008 .161 .080 .009 .039 .022 .002 10 % ATM 10% ITM OTM −.014 −.008 −.003 −.256 −.153 −.053 .232 .144 .052 .061 .037 .014 30 Day European Puts 30 Day American Puts 90 Day European Puts 10 % ATM 10 % ITM OTM .004 .002 .000 −.176 −.095 −.006 .204 .105 .007 .047 .024 .001 10% ATM 10% ITM OTM .026 .007 .000 −.094 −.092 −.003 .162 .101 .007 .040 .023 .001 10 % ATM 10% ITM OTM .013 .007 .002 −.214 −.124 −.036 .243 .145 .042 .057 .033 .009 10 % ATM 10 % ITM OTM .007 −.001 −.002 −.136 −.100 −.041 .200 .138 .052 .047 .033 .013 90 Day American Puts 10 % ATM 10 % ITM OTM .030 .016 .003 −.105 −.090 −.028 .163 .126 .040 .046 .030 .009 for 90-day calls. For in-the-money options, they are 3.9 and 4.8 cents for 30-day calls and 4.7 and 6.1 cents for 90-day calls. They are somewhat smaller for American calls than for European calls. Remarkably similar results are obtained for puts. The mean differences are quite small, being close to zero in most cases and of a similar order of magnitude to their call counterparts. The maxima and minima are likewise quite close to those of the corresponding calls. Similar comments apply to the mean absolute deviations. Note in particular that the mean absolute deviations are somewhat smaller for American puts than for European puts. The result that the effect of pricing errors is smaller for American options than for European options may be due to the fact that an American option has a lower expected maturity than an otherwise identical European option. Thus, forecast errors beyond the expected life of the American option have less effect on its price than are the same errors in the case of a European option. This effect is also consistent with the fact that the effect of forecast errors is greater on longer maturity options than on shorter maturity options. While we have characterized the properties of these errors, it remains to be seen whether they are economically significant. Discussions with a representative of a firm that actively trades listed index options revealed that a typical spread for at-the-money index options is DIVIDEND FORECAST BIASES IN INDEX OPTION VALUATION 301 about 2 points, quoted in terms of volatility. Calculations spanning a range of reasonable values for the input parameters showed that the differences observed here would be less than the bid-ask spread with the exception of some options in- or out-of-the-money more than 10 percent with volatility less than 15 percent and expirations of less than two months. These options, however, would have spreads wider than the 2 point spreads for at-the-money options. After tripling the spread, it still appears that in a few cases the differences could exceed the spread. We computed the percentage errors, defined as (actual price—estimated price)/estimated price, to examine the relative order of magnitude of the differences. To conserve space, we omit the details, but the results strongly suggest that the errors are very small relative to the option prices. For 30-day European calls, the median errors were less than .2 percent and the mean absolute deviation percentage errors were far less than one percent regardless of moneyness or time to expiration. Where unusually large percentage errors occurred, they were due to extremely small option prices. For American calls, the percentage errors were comparable to those of the European calls. Similar results hold for European and American puts. We also computed statistics for the difference between the actual and estimated early exercise premiums. To conserve space we do not show those results, but we summarize them with the following statements: (1) the median difference between the actual and estimated early exercise premiums was nearly always less than 5 cents, (2) the differences were largest for in-the-money options, (3) the differences were larger, the lower the volatility, and (4), the differences were larger, the longer the time to expiration, and (5) the differences were slightly larger for puts than for calls. 6. Summary and Conclusions It is common in derivative pricing to assume that a stock will pay a known dividend stream over the life of the derivative. Most academics and practitioners treat this assertion as a trivial and innocuous assumption. This paper provides empirical evidence that verifies that this assumption is upheld. That is, dividend risk is not economically relevant. Nonetheless, one must still forecast dividends, and this paper reveals that doing so is a complex exercise. We undertook dividend forecasts at the aggregate level. Our finding that dividend risk does appear to be trivial suggests there is no reason to examine individual stock dividend forecasts. Such an exercise would, at best, only bear that dividend risk is again insignificant in the pricing of stock index options. We also hasten to add that our results are indirect since we used simulated prices and not actual market prices. Moreover, our results apply only to exchange-listed derivatives, which typically have relatively short maturities. Over-the-counter derivatives, which can have much longer lives, could exhibit characteristics that might make dividend uncertainty more relevant. Indeed we found evidence that errors from dividend uncertainty are larger the longer the time to expiration. In addition, the yield on the S&P 500 index is today much lower than it was during the time period covered by this study, which could further reduce the magnitudes of the errors. Although we cannot predict with certainty what a more contemporaneous analysis would find, we conjecture that dividend uncertainty would be 302 CHANCE, KUMAR AND RICH found to be even less important today. On the other hand, the dividend yield on the S&P 500 varied from 3.8% to 5.2% from one year to the next in our study. It is reasonable to believe that yields can become much higher than they are today or were in the time period covered in this study and could potentially make dividend uncertainty a more critical concern. Further research is certainly required before one can assert that dividend uncertainty has essentially no material effect on the pricing of stock index options. Acknowledgments The authors thank participants at the finance seminars at Strathclyde, Alabama, Virginia Tech, Tennessee, Rice, Oklahoma State, and the Commodity Futures Trading Commission. This paper was also presented at the International Association of Financial Engineers and Washington Area Finance Association conferences. We appreciate the comments of Joanne Hill, Menachem Brenner (the editor), Sharon Brown-Hruska, Abon Mozumdar, Jeff Fleming, George Wang, Mike Brennan, Steve Kim and an anonymous referee. We are also grateful to Bill Sydor for assistance with computational issues and IBM for providing Virginia Tech with the SP2 system used to expedite the computations. We also thank the Virginia Tech Center for the Study of Futures and Options Markets for support. Kumar acknowledges support from an R. B. Pamplin College of Business Summer Research Grant. Notes 1. In the only published exception, Geske (1978) derives a formula for an option on a stock with a lognormally distributed stochastic dividend yield. That paper, however, has been largely ignored, and the model has not been empirically tested. In related work, Gibson and Schwartz (1993) price futures options on oil, which has a stochastic convenience yield. Empirical tests of index derivatives typically assume either constant known dividends or estimate a yield based on future or recent historical dividends. For representative empirical studies on index derivatives that use actual dividends, see Sheikh (1991) for options and Chung (1991) for futures. 2. Technically a stochastic price could be generated with non-stochastic dividends and stochastic discount factors but such a scenario is not of much interest here and highly unlikely. 3. We concede that prices are often measured with some error due to such things as asynchronous prices and spreads, but in this case we are considering the potential for a systematic, consistent and potentially measurable bias. 4. The decision to divide the dividend series into equal parts was partially arbitrary. Multiple years of data are necessary, of course, to estimate the time series parameters to confirm that seasonal patterns repeat. This means we could have used a two-year estimation period and a four-year out-of-sample-testing period or three years and three years. We decided on the three-year estimation period and the three-year testing period so we would have more data for the time series estimation. The patterns documented are similar for both sub-periods. 5. An obvious exception to this would be the pricing of short-dated index options where many of the underlying firms have declared their dividend payments. 6. Obviously our forecasting model will occasionally predict a negative dividend. We decided to truncate these forecasts to zero. There were 6,086 such cases out of 91,080 or slightly less than seven percent of the forecasts. This had only a slight effect on the mean forecasted dividend, changing it from .0488 to .0494, which naturally moves it closer to the actual mean. The average negative forecast was very close to zero at −.009. 7. We also calculate the mean square error (MSE) of the out-of-sample forecasts. The average MSE for the 759 sets of 120-day ahead forecasts is 0.0023. DIVIDEND FORECAST BIASES IN INDEX OPTION VALUATION 303 8. The S&P 500 was chosen for this study because its daily dividend series is readily available. Actual exchangelisted options on the S&P 500 are strictly European-style, save for the exchange-traded American-style flex options. This is not a major consideration, however, since the results are simulations. Moreover, over-thecounter dealers commonly write American-style S&P 500 options. 9. The European prices could have been generated using the Black-Scholes model but we wished to keep the results as comparable as possible so we simply use the binomial model on the European options as well. We did, however, conduct a number of robustness tests. We checked the binomial option values against the Black-Scholes model and rarely found more than a one cent difference. Moreover, the use of 360 time steps was not done arbitrarily. We conducted several of the tests using 500 time steps but they gave virtually the same results. We also used a constant four time steps per day implying that each option maturity has a different number of time steps but the results were again unaffected. Consequently, the choice of 360 time steps is clearly sufficient. 10. The Newey-West adjustment is especially critical in these tests. As we show later, the frequency of rejection of the null hypothesis α = 0 and β = 1 occurs rarely when using the Newey-West standard errors but occurs much more often when using OLS standard errors. 11. As noted above the Newey-West standard errors were necessary because of the overlapping observations. 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