Knowledge that Transforms

I am pleased to offer a comment on this very interesting article by an author who is always in the forefront of the research on empirical financial models. This article presents data analysis that estabishes the stylized facts about stock market volatility around market crashes. He concludes that volatility is high during periods of stock market decline and that it gradually returns to more normal levels. In the case of 1987, the peak was higher than usual and the decline was more rapid. The article uses 28,000 daily observations but does not really estimate a usable model; instead, it explores the data by estimating highly overparameterized models that reveal important features of the data. I suggest that this be considered an exploratory investigation and that in the face of more parsimonious models, rather interesting and somewhat different conclusions are revealed. The basic model estimated by Schwert is a 22-order autoregression of daily returns with a heteroskedastic error standard deviation which is itself assumed to be a 22-order autoregression in the absolute errors. Even with 28,000 observations, there is apparently a lot of noise in the coefficients. To allow for a risk premium, the mean is related to the variance, and in this case it is therefore related to 22 lagged absolute residuals. This part of the model uses 66 parameters. An alternative model is a first-order generalized autoregressive conditionally heteroskedastic model with variance influencing the mean [GARCH (l, l)-m], with a first-order moving average to correct for non-synchronous trading as used in Engle, Lilien, and Robins (1987), French, Schwert, and Stambaugh (1987), or Chou (1988), following the earlier work of Engle (1982). This requires only four coefficients! In the context of the parsimonious model, the parameter regulating the risk-return trade-off can be interpreted as the median agent’s taste for risk or his coefficient of relative-risk aversion. One naturally asks whether this parameter is constant over time, and we then recognize that the Schwert parameterization cannot answer the question.

The short-run interdependence of prices and price volatility across three major international stock markets is studied. Daily opening and closing prices of major stock indexes for the Tokyo, London, and New York stock markets are examined. The analysis utilizes the autoregressive conditionally heteroskedastic (ARCH) family of statistical models to explore these pricing relationships. Evidence of price volatility spillovers from New York to Tokyo, London to Tokyo, and New York to London is observed, but no price volatility spillover effects in other directions are found for the pre-October 1987 period.

We find that conditional means and variances of consumption growth vary through time, and this variation appears to be associated with the business cycle. A pricing model with fluctuating means and variances of consumption growth provides implications about conditional moments of returns for both short and long investment horizons, and these implications are explored empirically. The U-shaped pattern of first-order autocorrelations of returns, as well as business cycle patterns in the price of risk, appears to be consistent with the model, but our exploration suggests that other implications about conditional return moments are at odds with the data.

New evidence is provided on the determinants of stock-return variances. First, when the Tokyo Stock Exchange is open on Saturday, the weekend variance increases; weekly variance is unaffected, however, despite an increase in weekly volume. Second, the listing of U.S. stocks in Tokyo substantially increases the number of trading hours, but Tokyo volume is negligible for these U.S. stocks and their 24-hour variance is unaffected. The overall results are consistent with the predictions of private-information-based rational trading models, but inconsistent with both the irrational trading noise and public-information hypotheses.

A binomial approximation to a diffusion is defined as “computationally simple” if the number of nodes grows at most linearly in the number of time intervals. It is shown how to construct computationally simple binomial processes that converge weakly to commonly employed diffusions in financial models. The convergence of the sequence of bond and European option prices from these processes to the corresponding values in the diffusion limit is also demonstrated. Numerical examples from the constant elasticity of variance stock price and the Cox, Ingersoll, and Ross (1985) discount bond price are provided.

When the price process for a long-lived asset is of a mixed jump–diffusion type, pricing of options on that asset by arbitrage is not possible if trading is allowed only in the underlying asset and a riskless bond. Using a general equilibrium framework, we derive and analyze option prices when the underlying asset is the market portfolio with discontinuous returns. The premium for the risk of jumps and the diffusion risk forms a significant part of the prices of the options. In this economy, an attempted replication of call and put options by the Black–Scholes type of trading strategies may require substantial infusion of funds when jumps occur. We study the cost and risk implications of such dynamic hedging plans.

We test whether banks’ investment and financing policies can be explained by tax status. We document changes in bank holdings of municipal bonds in response to changes in tax rules relating to deductibility of interest expense. We also document an association between banks’ marginal tax rates and their investment and financing decisions, which is consistent with the existence of tax clienteles. However, banks do not sort themselves perfectly into investment and financing clienteles because of adjustment costs. We posit specific types of transaction-cost impediments to tax planning, and document that banks apparently trade off these costs against tax-planning benefits.

The procedure for opening stocks on the NYSE appears to affect price volatility. An analytical framework for assessing the magnitude of the structurally induced volatility is presented. The ratio of variance of open-to-open returns to close-to-close returns is shown to be consistently greater than one for NYSE common stocks during the period 1982 through 1986. The greater volatility at the open is not attributable to the way in which public information is released since both the open-to-open return and the close-to-close return span the same period of time. Instead, the greater volatility appears to be attributable to private information revealed in trading and to temporary price deviations induced by specialist and other traders. The implied cost of immediacy at the open is significantly higher than at the close. Other empirical evidence in this article documents the volume of trading at the open, the time delays between the exchange opening and the first transaction in a stock, the difference in daytime volatility versus overnight volatility, and the extent to which volatility is related to trading volume.

No analytic solution exists for the valuation of American options written on futures contracts and foreign currencies for which early exercise may be optimal. This article formulates the American option valuation problem in economically and mathematically meaningful ways. This enables us to derive valuation formulas for American options. The properties associated with the optimal exercise boundary are examined, and a numerical technique to implement the valuation formulas is presented.

In an adverse selection model of a securities market with one informed trader and several liquidity traders, we study the implications of the assumption that the informed trader has more information on Monday than on other days. We examine the interday variations in volume, variance, and adverse selection costs, and find that on monday the trading costs and the variance of price changes are highest, and the volume is lower than on Tuesday. These effects are stronger for firms with better public reporting and for firms with more discretionary liquidity trading.