Unbiased estimation of the Black/Scholes formula

The Black/Scholes model gives the price of an option as a function of the true variance rate of the underlying stock and other parameters. Because the true variance rate is unobservable, an estimate of the variance rate is used in empirical tests. But, because the Black/Scholes formula is non-linear in the variance, option price estimates using an estimated variance are biased, even if the variance estimate itself is unbiased. This paper develops an unbiased estimator of the Black/Scholes formula from a Taylor series expansion of the formula and the properties of the pdf of the estimated variance.     

Removing biases in computed returns

This paper presents a straightforward method for asymptotically removing the well-known upward bias in observed returns of equally-weighted portfolios. Our method removes all of the bias due to any random transient errors such as bid-ask bounce and allows for the estimation of short horizon returns. We apply our method to the CRSP equally-weighted monthly return indexes for the NYSE, Amex, and NASDAQ and show that the bias is cumulative. In particular, a NASDAQ index (with a base of 100 in 1973) grows to the level of 17,975 by 2006, but nearly half of the increase is due to cumulative bias. We also conduct a simulation in which we simulate true prices and set spreads according to a discrete pricing grid. True prices are then not necessarily at the midpoint of the spread. In the simulation we compare our method to calculating returns based on observed closing quote midpoints and find that the returns from our method are statistically indistinguishable from the (simulated) true returns. While the mid-quote method results in an improvement over using closing transaction prices, it still results in a statistically significant amount of upward bias. We demonstrate that applying our methodology results in a reversal of the relative performance of NASDAQ stocks versus NYSE stocks over a 25 year window.     

TRADE SIZES AND THEORIES OF THE BID-ASK SPREAD

Relations between trade-size characteristics and the bid-ask spread are developed to distinguish among major theories of the spread. These trade-size characteristics are determinants of the spread for NASDAQ/NMS stocks. They explain much of the cross-sectional variation in the spread commonly associated with volume, volatility, and share price. Evidence shows that order-processing costs are dominant relative to inventory effects for low-price, small-capitalization, and low-volume stocks, but that the opposite is true for high-price, large-capitalization, and high-volume stocks. Inventory effects are more important relative to asymmetric information costs when stock price or capitalization is lower.     

Stock Returns in Thinly Traded Markets

We examine the share price behavior of thinly traded NASDAQ National Market System stocks during periods when financial markets are open but the individual stocks do not trade. The absence of trade allows us to isolate the effect of nontrading from that of market closure. We find that nontrading stocks have negative mean returns and lower variances regardless of whether markets are open or closed. Two-day returns that include one nontrading day have a mean daily return of -0.226% compared to +0.164% for two-day returns over consecutive trading days. Two-day returns that include one nontrading day have only 3.8% higher variance than one-day returns. We conclude that the relation between transaction arrival, mean returns, and volatility depends on whether a stock is trading and not simply on whether the market is open.     

Does Risk Seeking Drive Stock Prices? A Stochastic Dominance Analysis of Aggregate Investor Preferences and Beliefs

We use various stochastic dominance criteria that account for(local) risk seeking to analyze market portfolio efficiencyrelative to benchmark portfolios formed on market capitalization,book-to-market equity ratio and price momentum. Our resultssuggest that reverse S-shaped utility functions with risk aversionfor losses and risk seeking for gains can explain stock returns.The results are also consistent with a reverse S-shaped patternof subjective probability transformation. The low average yieldon big caps, growth stocks, and past losers may reflect investors’twin desire for downside protection in bear markets and upsidepotential in bull markets.     

Testing asset pricing models with changing expectations and an unobservable market portfolio

When the assumption of constant risk premiums is relaxed, financial valuation models may be tested, and risk measures estimated without specifying a market index or state variables. This is accomplished by examining the behavior of conditional expected returns. The approach is developed using a single risk premium asset pricing model as an example and then extended to models with multiple risk premiums. The methodology is illustrated using daily return data on the common stocks of the Dow Jones 30. The tests indicate that these returns are consistent with a single, time-varying risk premium.     

Size effect,book-to-market effect,and survival

Previous studies find that small stocks have higher average returns than large stocks, and the difference between the returns can not be accounted for by the systematic risk, β. In my analysis of Compustat and CRSP data from 1976 to 1995, and simulation experiments based on the data, I find the size effect can be largely explained by data truncation that is caused by survival. Small stocks’ returns are more volatile, and small stocks are more likely to go bankrupt and less likely to meet the stock exchanges’ minimum capitalization requirements for listing. As a result, they are more likely to drop out of the sample. Including small stocks that do well and excluding those that do poorly, ex post, gives rise to higher returns for small-size portfolios. I conclude that the size effect is largely a spurious statistical inference resulting from survival bias, not an asset pricing ‘anomaly’.     

Estimation of global systematic risk for securities listed in multiple markets

In this era of rapid globalization of financial markets there has been a substantial increase in cross-listings of stocks in foreign and regional capital markets. As many as a third to a half of the stocks in some major exchanges are foreign listed. The multiple listings of stocks has major implications for the concept of systematic risk. This paper demonstrates that the estimator for systematic risk and the methodology itself changes when stocks are listed in multiple markets. The paper suggests general procedures, using maximum information from the multiple markets, to obtain the estimator of beta under a variety of assumptions about the error terms of the market models in the different capital markets. The assumptions pertain both to the volatilities of the abnormal returns in each market, and to the relationship between the markets.     

THE OVER-THE-COUNTER MARKET AND NEW YORK STOCK EXCHANGE TRADING HALTS

This paper examines the over-the-counter (OTC) market activities for stocks temporarily suspended by the New York Stock Exchange (NYSE). Unlike previous studies, we use transaction-to-transaction data on the NASDAQ during NYSE trading halts to investigate the price adjustment process between market equilibria. The evidence indicates that while being halted by the NYSE, the same stocks have exhibited significantly greater volatility in the OTC market. Since the volatile price movement is mainly random and provides no arbitraging opportunities for the OTC market traders, we do not find support for the proposal that trading halts should be mandatory for all trading locations.     

Average Returns,B/M,and Share Issues

The book‐to‐market ratio (B/M) is a noisy measure of expected stock returns because it also varies with expected cashflows. Our hypothesis is that the evolution of B/M, in terms of past changes in book equity and price, contains independent information about expected cashflows that can be used to improve estimates of expected returns. The tests support this hypothesis, with results that are largely but not entirely similar for Microcap stocks (below the 20^th NYSE market capitalization percentile) and All but Micro stocks (ABM).     

Microstructure and reverse stock splits

In this paper we show that, similar to NYSE/AMEX stocks, NASDAQ stocks exhibit significant ex date returns for reverse stock splits. Although the 10-day cumulative return after the ex date is close to –10%, this does not violate market efficiency, because the average bid-ask spread for the reverse split stock is at least double this return. We also document that these large negative returns are mostly due to a drop in the ask price while bid prices barely change at all. Furthermore, the ex date returns are negatively related to trading volume.These results suggest that there is abnormal selling and a significant buildup of market makers' inventories near the ex date. To reduce the inventory buildup, market makers lower ask prices to induce buying by investors, resulting in the observed negative returns. Lowering bid prices, an alternative strategy for reducing inventories, is not attractive to market makers due to competitive factors and the reduction of commissions associated with a smaller number of transactions. Notably, selling investors have no incentives to sell their stocks early to avoid the observed negative ex date return, since this return is largely an ask price phenomenon and does not represent realized returns to sellers.     

THE DAY-OF-THE-WEEK ANOMALY: THE TORONTO STOCK EXCHANGE EXPERIENCE

This paper studies the day-of-the-week effect employing Canadian stock returns from January 1, 1975 to June 30, 1989. The study finds that, as opposed to large capitalization stocks, low capitalization (thinly-traded) stocks tend to have a larger negative return on Tuesday rather than on Monday - possibly due to lags in the price adjustment of these stocks following the release of negative information. Two main issues are investigated in an attempt to explain the day-of-the-week effect and its persistence over time: (a) the role of dividends, and (b) the role of information flows. The study finds that firms are much more likely to go ex-dividend on Monday than on any other day of the week; however, after correcting for the dividend effect, Monday's returns are still significantly negative. With respect to information flows, we find evidence consistent with an information-flows-related explanation of the day-of-the-week effect, particularly with the idea that macro announcements cause negative Monday returns.     

One Security,Many Markets: Determining the Contributions to Price Discovery

When homogeneous or closely-linked securities trade in multiple markets, it is often of interest to determine where price discovery (the incorporation of new information) occurs. This article suggests an econometric approach based on an implicit unobservable efficient price common to all markets. The information share associated with a particular market is defined as the proportional contribution of that market's innovations to the innovation in the common efficient price. Applied to quotes for the thirty Dow stocks, the technique suggests that the preponderance of the price discovery takes place at the New York Stock Exchange (NYSE) (a median 92.7 percent information share).     

Price Limits and Beta

Price limits, which restrict daily price changes of a stock within a pre-specified range, make the stochastic properties of observed returns deviate from those of true returns, and hence lead to a biased estimates of the market model parameters. To investigate the impacts of price limits on the market model parameters, especially on beta, the restricted regression analysis is performed as well as the two-pass regression analysis used in examining the intervalling effect bias on beta. Empirical results suggest that, when prices are observed within a pre-specified bound, the estimates of beta using ordinary least squares substantially understate the true beta and suffer more from the intervalling effect bias. However, the delay effect of price limits on the adjustment of a security's price does not last too long, that is, remaining information is reflected on the subsequent day's stock prices very rapidly.     

Nonlinearity and Flight‐to‐Safety in the Risk‐Return Trade‐Off for Stocks and Bonds

We document a highly significant, strongly nonlinear dependence of stock and bond returns on past equity market volatility as measured by the VIX. We propose a new estimator for the shape of the nonlinear forecasting relationship that exploits variation in the cross‐section of returns. The nonlinearities are mirror images for stocks and bonds, revealing flight‐to‐safety: expected returns increase for stocks when volatility increases from moderate to high levels while they decline for Treasuries. These findings provide support for dynamic asset pricing theories in which the price of risk is a nonlinear function of market volatility.     

Did institutions herd during the internet bubble?

We examine the trading behavior of institutional investors during the internet bubble and crash of 1998–2001, and its impact on stock prices. Similar to some recent findings concerning the trading behavior of hedge funds and NASDAQ 100 stocks, we find that during the bubble all types of institutions herded with great intensity into internet stocks for a comprehensive sample of institutional investors and internet stocks. In addition to this, we present three entirely new results. First, institutional herding was much greater than what can be explained by momentum trading. Second, institutions as a group continued to increase their holdings of internet stocks for two quarters past the market peak during the first quarter of 2000, and three quarters past the peak for individual stock prices, suggesting that institutions were unable to time the price peaks. Finally and most importantly, we find positive abnormal returns contemporaneous with institutional herding and negative abnormal returns (reversals) at the point that herding ceased. This finding suggests that institutions’ trading created temporary price pressures, and may have contributed to the bubble.     

A SUGGESTED SIMPLE PROCEDURE TO OBTAIN BLUE ESTIMATOR OF GLOBAL BETA

Common stocks are typically traded on numerous exchanges both domestically and internationally. The price behaviour of stocks traded in multimarket settings provides investors with additional information regarding the risk characteristics of those shares. In this paper we provide a simple procedure to obtain the best linear unbiased (BLUE) estimator of relevant systematic risk (beta) of a stock by incorporating as much information as is possibly available in the market place.     

Idiosyncratic volatility and equity returns: UK evidence

The proposition that idiosyncratic volatility may matter in asset pricing is currently a topic of research and controversy. Using data from the UK market we examine the predictive ability of various measures of idiosyncratic risk and provide evidence which suggests that: (a) it is the idiosyncratic volatility of small capitalization stocks that matters for asset pricing and (b) that small stocks idiosyncratic volatility predicts the small capitalization premium component of market returns and is unrelated to either the market or the value premium. The predictive power of the aggregate idiosyncratic volatility of small stocks remains intact even after we control for the possible proxying effects of business cycle fluctuations and liquidity and is robust across time and different econometric specifications.     

Is there Value-Added Information in Liquidity and Risk Premiums?

Size has become a significant factor in explaining returns. According to the size effect, smaller capitalization stocks on average outperform larger capitalization stocks over long periods of time. This paper first documents the traditional size effect on the French market for the 1986–1998 period. It introduces a new proxy for size, free float, which is argued to be the appropriate measure of size and liquidity for most non‐US markets. Evidence is presented of a negative link between historical returns and free float. The link is significant even outside of the month of January, a notable divergence from results obtained on the NYSE. The rest of the paper is an attempt to take advantage of this 'ex‐post' phenomenon on an 'ex‐ante' basis, with an empirical study of the link between expected return, risk, and liquidity in a sample consisting of the main 150 stocks quoted on the Paris Bourse between January 1986 and January 1998. Liquidity premiums are estimated for portfolios from both a univariate and a multivariate perspective. The paper shows how risk and liquidity premiums can be used separately or in tandem for market timing and asset allocation. In all cases, the use of both premiums together leads to superior performance. Results confirm our measurements of liquidity and liquidity premiums and supply evidence that liquidity premiums together with risk premiums are useful in active asset management.