NO‐ARBITRAGE IN A NUMÉRAIRE‐INDEPENDENT MODELING FRAMEWORK

The classic approach to modeling financial markets consists of four steps. First, one fixes a currency unit. Second, one describes in that unit the evolution of financial assets by a stochastic process. Third, one chooses in that unit a numéraire, usually the price process of a positive asset. Fourth, one divides the original price process by the numéraire and considers the class of admissible strategies for trading. This approach has one fundamental drawback: Almost all concepts, definitions, and results, including no‐arbitrage conditions like NA, NFLVR, and NUPBR depend by their very definition, at least formally, on initial choices of a currency unit and a numéraire. In this paper, we develop a new framework for modeling financial markets, which is not based on ex‐ante choices of a currency unit and a numéraire. In particular, we introduce a “numéraire‐independent” notion of no‐arbitrage and derive its dual characterization. This yields a numéraire‐independent version of the fundamental theorem of asset pricing (FTAP). We also explain how the classic approach and other recent approaches to modeling financial markets and studying no‐arbitrage can be embedded in our framework.     

LONG‐RUN RISING SUPPLY PRICE AND THE NUMÉRAIRE

When a set of industries is kept in long‐run equilibrium, it is never possible to change just one price at a time. But when various (or all) prices are changing, the direction of change of any one price can depend on the numéraire adopted. What does it mean, then, to say that a long‐run supply curve is upward (or downward) sloping? Can this qualitative property be independent of the numéraire in terms of which the product price is being measured? In general, it cannot.     

Cover's universal portfolio,stochastic portfolio theory,and the numéraire portfolio

Cover's celebrated theorem states that the long‐run yield of a properly chosen “universal” portfolio is almost as good as that of the best retrospectively chosen constant rebalanced portfolio. The “universality” refers to the fact that this result is model‐free, that is, not dependent on an underlying stochastic process. We extend Cover's theorem to the setting of stochastic portfolio theory: the market portfolio is taken as the numéraire, and the rebalancing rule need not be constant anymore but may depend on the current state of the stock market. By fixing a stochastic model of the stock market this model‐free result is complemented by a comparison with the numéraire portfolio. Roughly speaking, under appropriate assumptions the asymptotic growth rate coincides for the three approaches mentioned in the title of this paper. We present results in both discrete and continuous time.     

THE NUMÉRAIRE PROPERTY AND LONG‐TERM GROWTH OPTIMALITY FOR DRAWDOWN‐CONSTRAINED INVESTMENTS

We consider the portfolio choice problem for a long‐run investor in a general continuous semimartingale model. We combine the decision criterion of pathwise growth optimality with a flexible specification of attitude toward risk, encoded by a linear drawdown constraint imposed on admissible wealth processes. We define the constrained numéraire property through the notion of expected relative return and prove that drawdown‐constrained numéraire portfolio exists and is unique, but may depend on the investment horizon. However, when sampled at the times of its maximum and asymptotically as the time‐horizon becomes distant, the drawdown‐constrained numéraire portfolio is given explicitly through a model‐independent transformation of the unconstrained numéraire portfolio. The asymptotically growth‐optimal strategy is obtained as limit of numéraire strategies on finite horizons.     

BENCHMARKED RISK MINIMIZATION

This paper discusses the problem of hedging not perfectly replicable contingent claims using the numéraire portfolio. The proposed concept of benchmarked risk minimization leads beyond the classical no‐arbitrage paradigm. It provides in incomplete markets a generalization of the pricing under classical risk minimization, pioneered by Föllmer, Sondermann, and Schweizer. The latter relies on a quadratic criterion, requests square integrability of claims and gains processes, and relies on the existence of an equivalent risk‐neutral probability measure. Benchmarked risk minimization avoids these restrictive assumptions and provides symmetry with respect to all primary securities. It employs the real‐world probability measure and the numéraire portfolio to identify the minimal possible price for a contingent claim. Furthermore, the resulting benchmarked (i.e., numéraire portfolio denominated) profit and loss is only driven by uncertainty that is orthogonal to benchmarked‐traded uncertainty, and forms a local martingale that starts at zero. Consequently, sufficiently different benchmarked profits and losses, when pooled, become asymptotically negligible through diversification. This property makes benchmarked risk minimization the least expensive method for pricing and hedging diversified pools of not fully replicable benchmarked contingent claims. In addition, when hedging it incorporates evolving information about nonhedgeable uncertainty, which is ignored under classical risk minimization.     

Generic Existence and Robust Nonexistence of Numéraires in Finite Dimensional Securities Markets

A numéraire is a portfolio that, if prices and dividends are denominated in its units, admits an equivalent martingale measure that transforms all gains processes into martingales. We first supply a necessary and sufficient condition for the generic existence of numéraires in a finite dimensional setting. We then characterize the arbitrage‐free prices and dividends for which the absence of numéraires survives any small perturbation preserving no arbitrage. Finally, we identify the cases when any small, but otherwise arbitrary, perturbation of prices and dividends preserves either the existence of numéraires, or their nonexistence under no arbitrage.     

PRODUCTIVITY ACCOUNTING BASED ON PRODUCTION PRICES

In this paper we propose a method of productivity accounting based on production prices. By using input–output tables from four major OECD countries between 1970 and 2000, we compute the associated wage–profit frontiers and the net national products, and from these we derive two measures of productivity growth based on production prices and a chosen numéraire. Our findings support the general conclusions in the existing literature on the productivity slowdown and later rebound, and supply new important insights to the extent and timing of these events.     

Financial models with defaultable numéraires

Financial models are studied where each asset may potentially lose value relative to any other. Conditioning on nondevaluation, each asset can serve as proper numéraire and classical valuation rules can be formulated. It is shown when and how these local valuation rules can be aggregated to obtain global arbitrage‐free valuation formulas.     

Affine multiple yield curve models

We provide a general and tractable framework under which all multiple yield curve modeling approaches based on affine processes, be it short rate, Libor market, or Heath–Jarrow–Morton modeling, can be consolidated. We model a numéraire process and multiplicative spreads between Libor rates and simply compounded overnight indexed swap rates as functions of an underlying affine process. Besides allowing for ordered spreads and an exact fit to the initially observed term structures, this general framework leads to tractable valuation formulas for caplets and swaptions and embeds all existing multicurve affine models. The proposed approach also gives rise to new developments, such as a short rate type model driven by a Wishart process, for which we derive a closed‐form pricing formula for caplets. The empirical performance of two specifications of our framework is illustrated by calibration to market data.     

MEASURES OF PRODUCTION PRICE‐LABOUR VALUE DEVIATION AND INCOME DISTRIBUTION IN ACTUAL ECONOMIES: A NOTE

Many empirical studies indicate that the deviations of actual prices of production from labour values are not too sensitive to the type of measure used for their evaluation. This paper attempts to theorize this rather ‘stylized fact’ by focusing on the relationships between the traditional and the numéraire‐free measures of deviation. On the empirical side, it provides an illustration of these relationships using input–output data from the Japanese economy.     

CONVERGENCE OF BARRIER OPTION PRICES IN THE BINOMIAL MODEL

In this paper, we study the rate of convergence of the European barrier call option price given by the CRR binomial model to the Black–Scholes price as the number of periods n tends to infinity. In general the error is of order and we give explicit formulas for the coefficients of and 1/n in the asymptotic expansion of the error. These coefficients depend on the positions of the barrier and strike in the binomial lattice and enable us to give a rigorous explanation of the observed fact that the error is of order 1/n when n is chosen in an appropriate way.     

Erratum to: Competitive price-matching guarantees under imperfect store availability

This paper analyzes the role of verification of product availability in the context of competitive price-matching guarantees (PMGs). PMGs involve a retailer matching any lower price offered by competition for an identical item. Until now, researchers focused on the scenario where customers can receive the lower price with a simple proof (e.g., weekly flyers). However, in reality, retailers reserve the right to verify the availability at the competitor location; if the product is not available there, then the price-match request might be declined. We develop a price competition model to investigate the effects of verification of availability on price decisions and profit levels of competing retailers. In our model, demand is driven by the availability levels of retailers as well as the price-search cost and store-switching cost incurred by customers. Price-search cost leads to two customer segments: uninformed ones who have no knowledge about prices, and informed ones who are knowledgeable about prices. On the other hand, store-switching cost determines how many customers search for the product at an alternative location because of high price or unavailability at their preferred retailers. Our analysis reveals, among others, that the outcome of availability verification is linked to three factors: price-search cost, store-switching cost, and the reservation prices of customer segments. Verification of availability allows retailers to price discriminate customers who could not be discriminated otherwise, specifically those belonging to the informed customer segment. Furthermore, it is a significant profit-enhancing mechanism only if there are switching customers in the market. Interestingly, even though customers view the verification of availability as a hassle, it can actually result in them paying lower retail prices by increasing the level of retail price competition.     

Interest rate equalization and nonequalization by trade in Leontief-Sraffa models

A half-dozen theorems and some corollaries concerning equalization of factor prices in a time-phased Leontief-Sraffa system with steady-state profit rates are carefully stated and discussed. An example is the new Mainwaring theorem, that a 1-technique world will either have the normalized price ratio vector independent of the profit rate or uniquely determined by it. Another example is that, when intermediate goods can be imported, unchanged profit rates and changed tastes can negate the nonsubstitution theorem and alter price ratios, since now the world has more than one primary (i.e. geographical) labor.     

CHOICE OF TECHNIQUES AND THE THEOREM OF OKISHIO

This paper discusses the sufficiency of the »viability condition« of Okishio's theorem for signalling profit improving choice of techniques. Some counter-examples are given for linear technologies with joint production. Technological change which conform to the prediction of the theorem - choices made on the basis of the viability condition lead to higher average rate of profit - is called regular, and sufficient conditions for regularity are specified.     

HAS THE OKISHIO THEOREM BEEN REFUTED?

This paper evaluates the alleged refutation of the Okishio theorem by the Temporal Single‐System Interpretation (TSSI) of Marxian value theory. The TSSI's refutation of the Okishio theorem requires not only changes in the definition of the rate of profit but also an arbitrary assumption about the time paths of labour productivity and prices. An ad hoc assumption about price changes used to disprove the Okishio theorem is meaningless because the aim of Marx's economics was to explain capitalist exploitation and accumulation without resorting to exogenously given changes on the market.     

Local costs of distribution,international trade costs and micro evidence on the law of one price

This paper connects trade flows to deviations from the law of one price (LOOP) in a structural model of trade and retailing. It accounts for the observed cross-country dispersion in prices of goods, based on retail price survey data, by focusing on two sources of goods market segmentation — (i) international trade costs, and (ii) non-traded input costs of distribution. I find that a multi-sector Ricardian trade model, ala Eaton–Kortum, augmented with a distribution sector, can account for the average price dispersion for a basket of goods fully and generates 70% of the variation in price dispersion across goods within the basket. While tradability of goods is important in explaining the average price dispersion for the basket of goods, distribution costs are important in explaining why, within the basket, some goods show more price dispersion than others.     

MICROECONOMIC FOUNDATIONS OF MACROECONOMICS IN THE POST-KEYNESIAN APPROACH

The purpose of this paper is to explore further some issues within the ⇐ mark-up ⇒ models. Among the numerous alternative formulations which have been put forward in the literature, two of them are particularly analysed in this paper. Kalecki's microanalysis, and especially the price determination as from the Theory of Economic Dynamics, is discussed in section II. Eichner's approach to the determination of the ⇐ mark-up ⇒ in oligopoly is examined in section III. These two different models yield different dynamic implications: section IV and section V contain a comparison between these different approaches in terms of the relation between the mark-up and the growth rate especially. Finally, in section V, we suggest a direction of research to provide a theoretical justification of price setting according to the ⇐ mark-up ⇒. Indeed, we show that under certain assumptions, this pricing behaviour can arise from a process of learning.     

Redistribution in a Neo‐Kaleckian Two‐country Model

We investigate the interaction between demand‐driven growth and income distribution in open economies, by combining expenditure‐switching and demand spillover effects in a neo‐Kaleckian two country model. First, we specify elasticities of wage share and real exchange rate to the money wage relative to labor productivity, in order to precisely describe the distributive pass‐through from money wages to the labor share and the real exchange rate. Second, we analyze the demand effects of an increase in the money wage for given labor productivity (a redistribution towards labor) in both Home and Foreign country, as well as globally. We derive closed form results for two identical countries. These results indicate that redistribution towards labor at Home: (i) always increases growth globally if Home is wage‐led, but can lead to lower growth at Home relative to Foreign; and (ii) will always imply lower growth at Home relative to Foreign if Home is profit‐led, but can still be growth‐enhancing at Home. Thus, to the extent that countries are concerned with their relative economic performance, a fallacy of composition can emerge. Numerical simulations suggest that these fallacies could indeed occur. As a consequence, ‘returns to coordination’ over international labor policies might be substantial.     

A MODEL OF THE FIRM'S DEMAND FOR MONEY UNDER UNCERTAINTY

Recently, economists have centered their explanation of the demand for money balances on the transactions motive. The roles of precautionary and speculative motives remain ambiguous. The purpose of this paper is to fill this void. We develop a static model of the individual firm's demand for money in which prices and the interest rate are assumed to be random variables. In this model, the entrepreneur must decide his money holdings ex ante, and both capital and labor inputs are determined ex post. The demand for money corresponding to the various motives is a consequence of analyzing the ex ante profit function. Implications for monetary policy are derived from this analysis.     

ANALYTIC APPROXIMATIONS FOR MULTI‐ASSET OPTION PRICING

We derive general analytic approximations for pricing European basket and rainbow options on N assets. The key idea is to express the option’s price as a sum of prices of various compound exchange options, each with different pairs of subordinate multi‐ or single‐asset options. The underlying asset prices are assumed to follow lognormal processes, although our results can be extended to certain other price processes for the underlying. For some multi‐asset options a strong condition holds, whereby each compound exchange option is equivalent to a standard single‐asset option under a modified measure, and in such cases an almost exact analytic price exists. More generally, approximate analytic prices for multi‐asset options are derived using a weak lognormality condition, where the approximation stems from making constant volatility assumptions on the price processes that drive the prices of the subordinate basket options. The analytic formulae for multi‐asset option prices, and their Greeks, are defined in a recursive framework. For instance, the option delta is defined in terms of the delta relative to subordinate multi‐asset options, and the deltas of these subordinate options with respect to the underlying assets. Simulations test the accuracy of our approximations, given some assumed values for the asset volatilities and correlations. Finally, a calibration algorithm is proposed and illustrated.