On the Rate of Convergence of Discrete‐Time Contingent Claims

This paper characterizes the rate of convergence of discrete‐time multinomial option prices. We show that the rate of convergence depends on the smoothness of option payoff functions, and is much lower than commonly believed because option payoff functions are often of all‐or‐nothing type and are not continuously differentiable. To improve the accuracy, we propose two simple methods, an adjustment of the discrete‐time solution prior to maturity and smoothing of the payoff function, which yield solutions that converge to their continuous‐time limit at the maximum possible rate enjoyed by smooth payoff functions. We also propose an intuitive approach that systematically derives multinomial models by matching the moments of a normal distribution. A highly accurate trinomial model also is provided for interest rate derivatives. Numerical examples are carried out to show that the proposed methods yield fast and accurate results.     

The Role of Beta and Size in the Cross-Section of European Stock Returns

This paper examines the ability of beta and size to explain cross-sectional variation in average returns in 12 European countries. We find that average stock returns are positively related to beta and negatively related to firm size. The beta premium is in part due to the fact that high beta countries outperform low beta countries. Within countries high beta stocks outperform low beta stocks only in January, not in other months. We reject the hypothesis that differences in average returns on size- and beta-sorted portfolios can be explained by market risk and exposure to the excess return of small over large stocks (SMB). Consistent with recent US evidence, we find that after controlling for size, there is no association between average returns and exposure to SMB.     

IMPROVED INTERNATIONAL COMPARISONS OF REAL PRODUCT AND ITS COMPOSITION: 1950–1980

A set of international comparisons is developed for 124 countries over the three post World War II decades, 1950-80. A Data Table is presented which gives, for most countries and most years, real product estimates for three different national income concepts and for the major subaggregates consumption, investment, and government. Detailed comparative price level estimates are provided as well.     

Options and Bubbles

The Black-Scholes-Merton option valuation method involves derivingand solving a partial differential equation (PDE). But thismethod can generate multiple values for an option. We providenew solutions for the Cox-Ingersoll-Ross (CIR) term structuremodel, the constant elasticity of variance (CEV) model, andthe Heston stochastic volatility model. Multiple solutions reflectasset pricing bubbles, dominated investments, and (possiblyinfeasible) arbitrages. We provide conditions to rule out bubbleson underlying prices. If they are not satisfied, put-call paritymight not hold, American calls have no optimal exercise policy,and lookback calls have infinite value. We clarify a longstandingconjecture of Cox, Ingersoll, and Ross. (JEL G12 and G13)     

NEW INSIGHTS INTO THE STRUCTURE OF THE WORLD ECONOMY*

The United Nations' newly completed study of purchasing power parities covering 34 countries varied in region, income level, and form of economic organization shows the systematic differences between the usual view of the structure of the world economy arising out of international comparisons based upon foreign exchange rate conversions and the structure one sees when actual prices are available. The real per capita GDP of developing countries is understated relative to developed countries when exchange rates are used in converting countries' national income accounts to a common currency, with the degree of understatement for any two countries being inversely related to the per capita income difference between them. The reason for this is that relative prices in the non-traded goods sector are lower relative to traded goods prices in low income countries. The systematic pattern observed in the 1975 data of the 34 countries has been extrapolated over time and space to get estimates of GDP for other years and countries. In the absence of detailed price data, the real shares of final expenditures devoted to particular components of the total can only be estimated as the proportion of own currency total expenditure devoted to the components. The observed differences in the pattern of prices of poor countries relative to rich for different components makes this clearly wrong for international comparisons, and in systematic ways. For example, (i) the relative price of services compared with commodities in poor countries is lower than in rich; so the apparent tendency of the share of services to rise as a country's income rises disappears when real quantities are considered; similarly, (ii) the relative price of capital goods is greater in poor countries compared with rich ones, so the difference in investment ratios out of GDP between rich and poor countries is understated.     

INTERNATIONAL COMPARISONS OF REAL PRODUCT AND ITS COMPOSITION: 1950–77

Structural relationships estimated from data obtained in a benchmark study of the expenditures and prices of 16 countries are used to develop a table of real gross domestic product and shares of gross domestic product devoted to private and public consumption and investment for each of over 100 countries in the years 1950 and 1960 through 1977. Price level estimates for total product and the three components are also provided.     

The Differential-Productivity Hypothesis and Purchasing-Power Parties: Some New Evidence*

The structure of prices of goods entering into international trade relative to those that do not plays a key role in the Balassa-Samuelson explanation of why countries ‘exchange rates differ systematically from their currencies’ purchasing power. the B-S analysis leads to the proposition that the tradable-nontradable price difference is lower for rich countries than for poor. This paper examines the gap, using prices collected by the International Comparison Programme. A variety of regressions were run to see if indeed the difference between tradable and nontradable price parities moved with income in the way B-S expected. They did.