Duration is a value-weighted measure of average maturity which is commonly associated with portfolios of fixed-income securities. However, the concept finds application in option pricing theory also. This article shows that if options are valued by the Black (1976) formula and a comparative-statics methodology is employed, then the interest rate sensitivity of a portfolio of European options is equal to its duration. If the options are instead valued through the Black-Scholes (1973) formula, then the interest rate sensitivity is equal to only the ‘bond-equivalent duration’ inherent in a dynamic replication strategy for the option portfolio.

The new ‘supershare’ securities proposed by Hakansson (1977, 1976) are subject to the same sort of rickless-hedge combinations as are other forms of secondary securities such as stock options. In consequence, the prices of supershares must, even in the absence of distributional assumptions, obey certain pricing relationships with each other and with the underlying primary security. When the primary security is assumed in addition to follow a geometric Brownian motion process, exact supershare valuation formulae of the Black-Scholes (1973) type are obtained. The ‘hedge portfolio algebra’ of Garman (1976) is employed to make the analysis concise.

It is assumed that a collection of market agents can be treated as a statistical ensemble. Their market activities are depicted as the stochastic generation of market orders according to a Poisson process. The objective is to effectively describe the ‘temporal microstructure’, or moment-to-moment trading activities in asset markets. Two basic models, ‘dealership’ vs. ‘auction’ markets (and their variants) are put forth. Implications are drawn from each model. The implications include several testable hypotheses regarding the aggregate behavior of markets and market-makers as well as some qualitative insight into the transaction-to-transaction nature of realistic exchange processes.

In an arbitrage‐free economy, there will always exist a set of linear operators which map future contingent dividends of securities into their current prices. It happens that such operators will also form an “evolution semigroup” as a consequence of intertemporal analysis of the no‐arbitrage restriction. This paper summarizes some of the major implications of the semigroup properties, but avoids almost all of the technical discussion which underlies them. Instead, several practical examples are presented. Some well‐known continuous‐time results are replicated by this alternative method, and certain new developments are explored.

In an arbitrage-free economy, there will always exist a set of linear operators which map future contingent dividends of securities into their current prices. It happens that such operators will also form an “evolution semigroup” as a consequence of intertemporal analysis of the no-arbitrage restriction. This paper summarizes some of the major implications of the semigroup properties, but avoids almost all of the technical discussion which underlies them. Instead, several practical examples are presented. Some well-known continuous-time results are replicated by this alternative method, and certain new developments are explored.

Modern contingent pricing theory (CPT) dates its genesis from the pioneering work of Arrow [1] and Debreu [9] in the context of complete markets. Beja [2, 3] demonstrated the application of contingent pricing concepts to incomplete markets. The approach has been applied to the valuation of options (Cox and Ross [7]; Rubinstein [30]) and a variety of other financial instruments (e.g., Ross [28])- Tne fundamental insight of CPT is that in arbitrage-free markets complex securities may always be viewed as additive combinations of simple “state-claims” having positive value which, in effect, pay off one unit if and only if a given state is attained at a given date. Concurrently, the continuoustime viewpoint pioneered by Black and Scholes [4] and Merton [22] has grown in significance. The basic simplification of the continuous-time approach is that relevant valuation quantities may all be expressed in terms of the first two moments, i.e., mean and variance, of the state variable distributions employed. When CPT adopts a continuous-time format, it has been shown (Garman [13]) that a basic differential equation holds for all securities; that differential equation involves, of course, the state-claim values, the distributional parameters of state variable evolution, and the prices and dividends of securities. Alternatively, somewhat stronger assumptions which lead to the existence of a rational consensus investor allow thedifferential equation to be expressed in terms of marginal utilities (Cox, Ingersoll, and Ross [8]). This paper applies the techniques of continuous-time CPT to the foreign exchange market. Since we wish to substantively treat inflationary and productive sources of risk in two countries, four state variables are necessarily involved. In a sense, therefore, this is an ambitious attempt since the mostcomplex continuous-time models to date (e.g.. Brennan and Schwartz [5]), have substantively treated only two state variables. Such complexity is simplified through the use of some compact notation, but not by the use of ad hoc modeling. Indeed, it should be emphasized that the present treatment is a full-equilibrium approach, and that while the compact quality of the notation might be made to incorporate a significant amount of possible additional structure, nothing here is inconsistent with a complete equilibrium.

Valuation, Valuation model, Arbitrage free Economy, Information production

This paper analyzes the equilibrium valuation of risky assets in the case where transactions costs are present. The methodology involves applying ‘theorems of the alternative’ (Farkas' Lemma) as a consequence of arbitrage-free markets. Under relevant assumptions, it is found that the price of an asset having transactions costs is the corresponding price that would obtain in a perfect market, plus a ‘fudge factor’. This latter factor is provided explicit bounds.