Christian Gilles

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 Bessels inequality).

A model of the federal funds market

SummaryThis paper develops a stochastic general equilibrium model of the federal funds market that incorporates non-Fisherian effects on interest rates stemming from both supply and demand shocks to reserves. Such a model may reconcile the widespread belief in a liquidity effect of money supply shocks with the difficulty many researchers have had in finding empirical support for such an effect. The model also cautions against interpreting the observed negative correlation between the federal funds rate and innovations to nonborrowed reserves as empirical confirmation of the ability of the Federal Reserve to lower short-term real interest rates.

Charges as equilibrium prices and asset bubbles

Abstract In economies with L ∞ for commodity space, equilibrium prices in the dual of the commodity space may involve a finitely additive part (a pure charge) which we can sometimes interpret as indicating the presence of asset bubbles. In this paper, we show that a certain kind of patience on the part of the consumer or certain kinds of technological pathologies are sufficient to guarantee the presence of bubbles.

Bubbles as payoffs at infinity

SummaryWe define rational bubbles to be securities with payoffs occurring in the infinitely distant future and investigate the behavior of bubbles values. We extend our analysis to a setting of uncertainty. In an infinite horizon arbitrage-free model of asset prices, we interpret the money market account as the value of a particular bubble; a similar interpretation holds for other assets related to the state-price deflator and to payoffs on bonds maturing in the distant future. We present three applications of this characterization of bubbles.

Consumption and Asset Prices with Recursive Preferences

We analyze consumption and asset pricing with recursive preferences given by Kreps--Porteus stochastic differential utility (K--P SDU). We show that utility depends on two state variables: current consumption and a second variable (related to the wealth--consumption ratio) that captures all information about future opportunities. This representation of utility reduces the internal consistency condition for K--P SDU to a restriction on the second variable in terms of the dynamics of a forcing process (consumption, the state--price deflator, or the return on the market portfolio). Solving the model for (i) optimal consumption, (ii) the optimal portfolio, and (iii) asset prices in general equilibrium amounts to finding the process for the second variable that satisfies this restriction. We show that the wealth--consumption ratio is the value of an annuity when the numeraire is changed from units of the consumption good to units of the consumption process, and we characterize certain features of the solution in a non-Markovian setting. In a Markovian setting, we provide a solution method that is quite general and can be used to produce fast, accurate numerical solutions that converge to the Taylor expansion.

The liquidity premium in average interest rates

This paper studies recent models of the liquidity effect of money on interest rates to determine if a systematic relationship between liquidity shocks and the economy could affect the average real interest rate.

Consumption and Asset Prices with Homothetic Recursive Preferences

When preferences are homothetic, utility can be expressed in terms of current consumption and a variable that captures all information about future opportunities. We use this observation to express the differential equation that characterizes utility as a restriction on the information variable in terms of the dynamics of consumption. We derive the supporting price system and returns process and thereby characterize optimal consumption and portfolio decisions. We provide a fast and accurate numerical solution method and illustrate its use with a number of Markovian models. In addition, we provide insight by changing the numeraire from units of consumption to units of the consumption process. In terms of the new units, the wealth-consumption ratio (which is closely related to the information variable) is the value of a coupon bond and the existence of an infinite-horizon solution depends on the positivity of the asymptotic forward rate.

Arbitrage, martingales and bubbles

Abstract Viability of security prices implies linear valuation of payoffs but, if there exist an infinite number of securities or trading dates, does not imply the existence of a risk-neutral probability since countable additivity may fail. An example is given.