Primitive Security Research Articles

Pricing by Arbitrage Under Arbitrary Information

A substantial applications literature on pricing by arbitrage has effectively restricted information to that arising solely from securities markets; return distributions are then governed solely by past prices. We reconsider pricing by arbitrage in markets rendered incomplete by arbitrary information, which, moreover, may influence required returns. We show that contingent claims depending solely on securities’ normalized price histories are priced by arbitrage if and only if all risk‐adjusted probabilities agree upon the law of primitive securities’ normalized prices. We show how existing diffusion‐based results can be preserved, and resolve an issue relating to Merton's (1973) stochastic interest rate model.

Primitive Securities

Primitive Securities

On the arbitrage pricing theory

The Arbitrage Pricing Theory relates the expected rates of return on a sequence of primitive securities to their factor exposures, suggesting that factor risk is of critical importance in asset pricing. However, we show that if the sequence of primitive returns is replaced by a sequence of returns on portfolios formed from the primitive securities, then the factor subspace is arbitrary. The implication is that the theorems relating expected returns to factor risk require substantial reinterpretation. Our reinterpretation consists of a demonstration that exact and approximate factor pricing do not constitute substantive characterizations of asset pricing. Instead, they are implications of the characterization of the returns space as a Hilbert space (exact factor pricing corresponds to the Riesz representation theorem and approximate factor pricing is a consequence of Bessel's inequality).

Spanning and Completeness with Options

The role of ordinary options in facilitating the completion of securities markets is examined in the context of a model of contigent claims sufficiently general to accommodate the continuous distributions of asset pricing theory and option pricing theory. In this context, it is shown that call options written on a single security approximately span all contingent claims written on this security and that call options written on portfolios of call options on individual primitive securities approximately span all contingent claims that can be written on these primitive securities. In the case of simple options, explicit formulas are given for the approximating options and portfolios of options. These results are applied to the pricing of contingent claims by arbitrage and to irrelevance propositions in corporate finance.

Prices of State-Contingent Claims Implicit in Option Prices

The time-state preference approach to general equilibrium in an economy as developed by Arrow (1964) and Debreu (1959) is one of the most general frameworks available for the theory of finance under uncertainty. Given the prices of primitive securities (a security that pays $1.00 contingent upon a given state of the world at a given date, and zero otherwise, is a primitive This paper implements the time-state preference model in a multiperiod economy, deriving the prices of primitive securities from the prices of call options on aggregate consumption. These prices permit an equilibrium valuation of assets with uncertain payoffs at many future dates. Furthermore, for any given portfolio, the price of a $1.00 claim received at a future date, if the portfolio's value is between two given levels at that time, is derived explicitly from a second partial derivative of its calloption pricing function. An intertemporal capital asset pricing model is derived for payoffs that are jointly lognormally distributed with aggregate consumption. It is shown that using the Black-Scholes equation for options on aggregate consumption implies that individuals' preferences aggregate to isoelastic utility.

Default Risk in a General Equilibrium Asset Economy with Incomplete Markets

IN THE ARROW [1], Debreu [2] economy with contingent commodities, or derivable primitive securities (i. e., a security that pays one unit of account if a certain state of the world occurs), the analytical apparatus of certainty general equilibrium theory can be applied directly with suitable reinterpretation. Nevertheless, many writers (e. g., Arrow [1], Radner [11]) have criticized the complete market model as unrealistic, in requiring an embarrassingly rich set of forward insurance markets. Arrow has argued that transaction costs and moral hazard problems would be sufficient to destroy the existence of complete markets and introduce an incomplete market theory.2 There are some simplified incomplete market models in existence. One of the first formulations was based upon the mean-variance formulation of portfolio theory. This model was developed by Sharpe [12], Lintner [7] and other writers, and is the basis for much subsequent work in finance theory. 3 Another model was introduced by Diamond [4], who used the contingent commodity framework, but restricted trades to linear combinations or contingent commodities. Diamond's model is more general for analytical purposes than the mean-variance formulation, because it does not restrict preferences of subjective probability distributions. In these types of models there has been some confusion over the role of default risk, when agents borrow or short-sell. Diamond assumes no-bankruptcy, but does not include this assumption explicitly in his analysis, nor does he explore its implications. A similar situation applies for the mean-variance formulation.4 In this paper we will attempt to clarify the default risk issue, and at the same time generalize the Diamond model in a number of directions. Because the model has the same properties as the Debreu model (except for the possibility of unboundedness below for consumption sets when there are short sales) it is easy to show the existence of an equilibrium for a suitably restricted asset economy. The paper is divided into two sections: Section 1 provides the formal model and proofs; Section 2 is concerned with interpretations, extensions and limita-