Valuation of general contingent claims: Existence,uniqueness, and comparisons of solutions

The Black-Scholes equation for the price u(x,t) of a call option for a single share of common stock with dividend policy d(x,t) is $\text{1}\text{2}\text{σ}{}^2\text{x}{}^2\text{u}{}_{\mathrm{xx}}\text{+(rx?d(x,t))u}{}_{\mathrm{x}}\text{?ru?u}{}_{\mathrm{t}}\text{=0, 0<x, 0<t<T}$, with boundary conditions u(x,0)=max(0,x?E), 0≤x, u(0,t)=0, 0≤t≤T.The coefficients are unbounded and the equation is not uniformly parabolic. We prove an existence (and recall a uniqueness) theorem for a class of equations with boundary conditions that includes the Black-Scholes equation. These may be used to show that an American option must, or will not, sell for the same price as a European option.     

A note on relative efficiency and risk aversion

Stochastic dominance rules have recently been suggested as criteria for efficient choice under uncertainty. However, the aspect of relative efficiency has been neglected, i.e., given any specific utility function u₀, find the conditions imposed on F and G such that $\text{E}{}_{\mathrm{F}}\text{u≧E}{}_{\mathrm{G}}\text{u}$ for all u?U (u₀), where U(u₀)={u| ? u′/u′ ≥ ? u″₀/u′₀}. This note suggests a necessary and sufficient condition for relative efficiency in the case that F and G cross other only once. Some applications are discussed.     

R2

Even with hindsight, the ability to explain stock price changes is modest. $R^2$s were calculated for the returns of large stocks as explained by systematic economic influences, by the returns on other stocks in the same industry, and by public firm-specific news events. The average adjusted $R^2$ is only about .35 with monthly data and .20 with daily data. There is little relation between explanatory power and either the firm's size or its industry. There is little improvement in $R^2$ from eliminating all dates surrounding news reports in the financial press. However, the sample kurtosis is quite different when such news events are eliminated, thereby revealing a mixture of return distributions. Non-news dates also indicate the presence of a distributional mixture, perhaps due to traders acting on private information.     

Asset-pricing Tests under Alternative Distributions

Given the normality assumption, we reject the mean-variance efficiency of the Center for Research in Security Prices value-weighted stock index for three of the six consecutive ten-year subperiods from 1926 to 1986. However, the normality assumption is strongly rejected by the data. Under plausible alternative distributional assumptions of the elliptical class, the efficiency can no longer be rejected. When the normality assumption is violated but the ellipticity assumption is maintained, many tests tend to be biased toward overrejection and both the accuracy of estimated beta and $R^2$ are usually overstated.     

Predictive Regressions: A Present‐Value Approach

We propose a latent variables approach within a present‐value model to estimate the expected returns and expected dividend growth rates of the aggregate stock market. This approach aggregates information contained in the history of price‐dividend ratios and dividend growth rates to predict future returns and dividend growth rates. We find that returns and dividend growth rates are predictable with $R^2$ values ranging from 8.2% to 8.9% for returns and 13.9% to 31.6% for dividend growth rates. Both expected returns and expected dividend growth rates have a persistent component, but expected returns are more persistent than expected dividend growth rates.     

Assessing Goodness-of-Fit of Asset Pricing Models: The Distribution of the Maximal R2

The development of asset pricing models that rely on instrumental variables together with the increased availability of easily-accessible economic time-series have renewed interest in predicting security returns. Evaluating the significance of these new research findings, however, is no easy task. Because these asset pricing theory tests are not independent, classical methods of assessing goodness-of-fit are inappropriate. This study investigates the distribution of the maximal $R^2$ when k of m regressors are used to predict security returns. We provide a simple procedure that adjusts critical $R^2$ values to account for selecting variables by searching among potential regressors.     

Patterns of Productivity in the Finance Literature: A Study of the Bibliometric Distributions

This study finds a bibliometric regularity in the finance literature that the number of authors publishing n papers is about $1/n^c$ of those publishing one paper. We find that the finance literature conforms very well to the inverse square law $(c=2)$ if data are taken from a large collection of journals. When applied to individual finance journals, we find that values of c range from 1.95 to 3.26. We also find that top-rated journals have higher concentrations among their contributors. This implies that the phenomenon “success breeds success” is more common in higher quality publications.     

A Theoretical Framework for Evaluating the Impact of Universal Reserve Requirements

This paper provides an appropriate framework to evaluate the impact of the universal reserve requirements called for by the new DIDMC Act of 1980. We derived the optimal reserve ratios for a dual banking system under the objective of controlling the monetary aggregates and the level of output. Then optimal reserve requirements were calculated from illustrative money market and macroeconomic parameters since the usual comparative statics were not useful. The results, generally, suggested optimal reserve ratios which were significantly higher than the old dual or the new universal reserve regimes for all targets. However, the calculation of values for the loss functions under various reserve regimes suggests that attainment of $r_1^{*},r_2^{*}$ and $t^{*}$ may not be imperative, since the discrepancy between losses for optimal and various nonoptimal reserve schemes were not large. A major result of this paper, observed for both monetary and real targets, was that the differences in the instability of the targets for the old dual reserve ratios and the Fed's new universal reserve scheme were small. This result clearly suggests that although the DIDMC Act may solve the Federal Reserve's membership problem, it will not significantly enhance the Fed's effectiveness in controlling monetary or real sector aggregates.     

On the Effects of Barriers to International Investment

A simple model is presented in which it is costly for domestic investors to hold foreign assets. The implications of the model for the composition of optimal portfolios at home and abroad are derived. It is shown that all foreign assets with a beta larger than some beta ${\beta }^{*}$ plot on either one of two security market lines. Some foreign assets with a beta smaller than ${\beta }^{*}$ are not held by domestic investors even if their expected return is increased slightly.     

A new approach to assessing model risk in high dimensions

A central problem for regulators and risk managers concerns the risk assessment of an aggregate portfolio defined as the sum of d individual dependent risks $X_i$. This problem is mainly a numerical issue once the joint distribution of $\left(X_1\text{,}{X}_2\text{,}\dots \text{,}{X}_d\right)$ is fully specified. Unfortunately, while the marginal distributions of the risks $X_i$ are often known, their interaction (dependence) is usually either unknown or only partially known, implying that any risk assessment of the portfolio is subject to model uncertainty.Previous academic research has focused on the maximum and minimum possible values of a given risk measure of the portfolio when only the marginal distributions are known. This approach leads to wide bounds, as all information on the dependence is ignored. In this paper, we integrate, in a natural way, available information on the multivariate dependence. We make use of the Rearrangement Algorithm (RA) of Embrechts et al. (2013) to provide bounds for the risk measure at hand. We observe that incorporating the information of a well-fitted multivariate model may, or may not, lead to much tighter bounds, a feature that also depends on the risk measure used. In particular, the risk of underestimating the Value-at-Risk at a very high confidence level (as used in Basel II) is typically significant, even if one knows the multivariate distribution almost completely.Our results make it possible to determine which risk measures can benefit from adding dependence information (i.e., leading to narrower bounds when used to assess portfolio risk) and, hence, to identify those situations for which it would be meaningful to develop accurate multivariate models.     

Assessing Specification Errors in Stochastic Discount Factor Models

In this article we develop alternative ways to compare asset pricing models when it is understood that their implied stochastic discount factors do not price all portfolios correctly. Unlike comparisons based on ${\chi }^2$ statistics associated with null hypotheses that models are correct, our measures of model performance do not reward variability of discount factor proxies. One of our measures is designed to exploit fully the implications of arbitrage-free pricing of derivative claims. We demonstrate empirically the usefulness of our methods in assessing some alternative stochastic factor models that have been proposed in asset pricing literature.     

Portfolio Inefficiency and the Cross-section of Expected Returns

The Capital Asset Pricing Model implies that (i) the market portfolio is efficient and (ii) expected returns are linearly related to betas. Many do not view these implications as separate, since either implies the other, but we demonstrate that either can hold nearly perfectly while the other fails grossly. If the index portfolio is inefficient, then the coefficients and $R^2$ from an ordinary least squares regression of expected returns on betas can equal essentially any values and bear no relation to the index portfolio's mean-variance location. That location does determine the outcome of a mean-beta regression fitted by generalized least squares.     

A Theoretical Analysis of Real Estate Returns

In this paper, we consider two hypotheses for the recent performance of real estate returns. The first is the random event argument that real estate is positively correlated with unanticipated inflation but that structural change in expected returns due to a change in the perceived sensitivity of returns to unanticipated inflation has not taken place. The second is the hedge demand argument that formulates the structural shift hypothesis. The paucity of real estate and other expectations data as well as the general identification problem make it extremely difficult to distinguish between these hypothesis. Our tests consist of estimates of inflation betas for various asset categories overtime as well as estimates of the hedge vector, $S^{-1}C$. Although some support for the hedge argument is found, the results are not strong enough to reject the random event argument and conclude that a decline in the required return on real estate due to a relative increase in inflation beta drove returns during the 1970's.     

The Panzar–Rosse revenue test and market power in banking

The Panzar–Rosse H statistic is a commonly used measure of market power in banking. It is widely believed that $H>0$ is inconsistent with significant market power. This study rigorously disproves that perception. Instead, the possibility of $H>0$ under conditions of substantial market power turns out robust to the timing of banks’ actions, relative costs, choice of strategic variable, degree of product differentiation, strategy (static or dynamic), and degree of heterogeneity in banks’ conduct (collusive versus fringe), and hence may be common in practice.     

The characteristics of takeover targets q and other measures

This study attempts to assess differences in the financial characteristics of target and non-target firms using logit analysis and a case-control methodology in which control groups are matched by size or industry. The results indicate that unregulated non-financial target firms are characterized by low $\text{q}$ ratios (market/replacement values) and to a lesser extent high current financial liquidity. Measures of financial leverage were not found to be significant.