Market clearing, utility functions, and securities prices

Summary. We prove the existence of equilibrium in a continuous-time finance model; our results include the case of dynamically incomplete markets as well as dynamically complete markets. In addition, we derive explicitly the stochastic process describing securities prices. The price process depends on the risk-aversion characteristics of the utility function, as well as on the presence of additional sources of wealth (including endowments and other securities). With a single stock, zero endowment in the terminal period, and Constant Relative Risk Aversion (CRRA) utility, the price process is geometric Brownian motion; in essentially any other situation, the price process is not a geometric Brownian motion.JEL Classification Numbers: D52.This paper is part of my Dissertation (UC Berkeley). I am very grateful to my advisor Professor Robert M. Anderson. I also would like to thank Steve Evans, Roger Purves, Jacob Sagi, Chris Shannon and the participants of the 2002 NBER General Equilibrium Conference at the University of Minnesota (Minneapolis) for very helpful discussions and comments. This work was supported by Grant SES-9710424 from the National Science Foundation.     

Identification of consumers’ preferences when their choices are unobservable

Summary. We provide conditions under which the heterogenous, deterministic preferences of consumers in a pure exchange economy can be identified from the equilibrium manifold of the economy. We extend those conditions to consider exchange economies, with two commodities, where consumers preferences are random. For the latter, we provide conditions under which consumers heterogenous random preferences can be identified from the joint distribution of equilibrium prices and endowments. The results can be applied to infer consumers preferences when their demands are unobservable.Received: 8 May 2003, Revised: 14 September 2004, JEL Classification Numbers: D12, D51.I am very grateful to an anonymous referee, Donald Brown, and Daniel McFadden for their detailed comments and insightful suggestions. Section 2 of this paper is joint work with Donald J. Brown; it is included here for publication with his permission. Those results were presented at the 1990 Workshop on Mathematical Economics at the University of Bonn, the 1992 SITE Workshop on Empirical Implications of General Equilibrium Models at Stanford University, and, more recently, at the June 2000 Conference in Honor of Rolf Mantel, in Buenos Aires, Argentina. The comments of the participants at those conferences and workshops are much appreciated. The research presented in this paper was supported by NSF Grants SES-8900291, SBR-9410182, and SES-0241858. This paper is dedicated to Marcel K. Richter, who has inspired much of my research.     

Maturity and volatility effects on UK smiles

The smile effect is a result of an empirical observation of the options implied volatility with the same expiration date, across different exercise prices. However, its shape has been under discussion seeming to be dependent on the option underlying security. In this paper, and filling up a scarce empirical research on the topic, we used liquid equity options on 9 stocks traded on the London International Financial Futures and Options Exchange (LIFFE) between August 1990 and December 1991. We tested two different hypothesis for testing two different phenomena: (1) the increase of the smile as maturity approaches; (2) and the association between the smile and the volatility of the underlying stock. In order to estimate implied volatilities for unavailable exercise prices, we modelled the smile using cubic B-spline curves. We found empirical support for the smile intensification (the U-shape is more pronounced) as maturity approaches as well as when volatility rises. However, we found two major sources of disagreement with the literature on stochastic volatility models. First, as maturity approaches, out-of-the-money options implied volatility tends to be higher than the implied volatility of in-the-money options. Second, as the volatility of the underlying asset increases, the implied volatility of in-the-money options tends to be higher than implied volatility of out-of-the-money options.Received: September 2001, Accepted: September 2003, JEL Classification: G13Correspondence to: João L. C. DuqueWe thank Professor Dean A. Paxson (University of Manchester), António Sousa Câmara (University of Strathclyde), Ser-Huang Poon (University of Lancaster) and the attendees of the 26th EFA Annual Conference for helpful comments on previous versions of this paper. We also want to thank to two anonymous referees for their relevant comments and suggestions. Financial support granted by the Fundação para a Ciência e a Tecnologia (FCT) and the Programa Praxis XXI is gratefully acknowledged.     

On the Non-Existence of Redundant Options

In a seminal paper, Ross (Q J Econ 90:75–89, 1976) shows that if security markets are resolving, then there exist (non-redundant) options that generate complete security markets. Complementing his work, Aliprantis and Tourky (2002) show that if security markets are strongly resolving and the number of primitive securities is less than half the number of states, then every option is non-redundant. Our paper extends Aliprantis and Tourky’s result to the case when their condition on the number of primitive securities is not imposed. Specifically, we show that if there exists no binary payoff vector in the asset span, then for each portfolio there exists a set of exercise prices of full measure such that any option on the portfolio with an exercise price in this set is non-redundant. Since the condition that there exists no binary payoff vector in the asset span holds generically, redundant options are thus rare. I am grateful to an anonymous referee for very helpful comments. Research support from the School of Business at The George Washington University is gratefully acknowledged     

Asset markets and the information revealed by prices

Summary When nominal assets serve to transfer revenues across states of the world, noninformative rational expectations equilibria exist. At noninformative prices, the restricted information under which individuals optimize can be modelled as restricted participation of the individuals in asset markets. When assets are nominal, the indeterminacy of equilibrium prices, and, generically, allocations as well, which characterizes economies with restricted participation guarantees that noninformative equilibrium prices exist.This work was supported in part by grant No 26 from the program Pole d' Attraction Interuniversitaire — Deuxieme phase of the Belgian government of CORE, Universite' Catholique De Louvain. The second author acknowledges the generous support of the Graduate School of Business of Columbia University. We wish to thank David Cass for very helpful comments.     

A life cycle analysis of social security

Summary We develop an applied general equilibrium model to examine the optimal social security replacement rate and the welfare benefits associated with it. Our setup consists of overlapping generations of 65-period lived individuals facing mortality risk and individual income risk. Private credit markets, including markets for private annuities, are closed by assumption. Unlike previous analyses, we find that an unfunded social security system may well enhance economic welfare. In our benchmark economy, the optimal social security replacement rate is 30%, and an empirically more plausible replacement rate of 60% raises welfare compared with an economy with no social security system.We would like to thank Andy Atkeson, V. V. Chari, Steve Davis, Paul Evans, Lars Hansen, Tim Kehoe, Nobu Kiyotaki, Ed Prescott, José-Victor Ríos-Rull, Richard Rogerson, Tom Sargent, Nancy Stokey, Dick Sweeney, Robert Townsend, and the participants of the NBER Economic Fluctuations Small Group Workshop on Micro and Macro Perspectives on the Aggregate Labor Market in Palo Alto, the NBER General Equilibrium Theory Conference in Minneapolis, the Money and Banking Workshop at the University of Chicago, and the NBER Summer Institute. An earlier version of this paper was titled A Dynamic Stochastic General Equilibrium Analysis of Social Security. This material is based upon work supported by the National Science Foundation under Grant No. SES-9210291. We also thank the Minnesota and San Diego Supercomputer Centers for their support.     

On rational belief equilibria

Summary We study equilibria in which agent's belief are rational in the sense of Kurz [1994]. The market is formulated by specifying a stochastic demand function and a continuum of producers, each with a quadratic cost function who must select their output before knowing prices. Holding Rational Beliefs about future prices, producers maximize expected profits. In a Rational Belief Equilibrium (RBE) agents select diverse forecast functions but each one is rational in the sense that it is based on a theory which cannot be rejected by the data. It is shown that there exists a continuum of RBE's and they could entail very different patterns of time series for the economy and consequently different aggregate levels of longterm volatility. Since the model contains exogenously specified random variables, the difference in the level of long-term volatility of prices among the different RBE's arises endogenously as an amplification of the volatility of exogenous variables. The paper derives exact bounds on the possible levels of such amplification.This research was supported by NSF Grant IRI-8814954 to Stanford University. The author is thankful to Steven N. Durlauf and Carsten K. Nielsen for useful comments on an earlier draft.     

Spanning,valuation and options

Summary We model the space of marketed assets as a Riesz space of commoditics. In this setting two altenative characterizations are given of the space of continuous options on a bounded asset,s, with limited liability. The first characterization represents every continuous option ons as the uniform limit of portfolios of calls ons. The second characterization represents an option as a continuous sum (or integral) of Arrow-Debreu securities, with respect tos. The pricing implications of these representations are explored. In particular, the Breeden-Littzenberger pricing formula is shown to be a direct consequence of the integral representation theorem.Research supported in part by NSF Grant SES83-19611     

The limitation of monotonicity property of option prices: an empirical evidence

Many option pricing models are based on the assumption that the underlying asset price follows one-dimensional diffusion process. An alternative approach is to test the properties that should hold for all models based on a given stochastic process for the underlying asset. Following Pérignon (2006 Pérignon, C. 2006. Testing the monotonicity property of option prices. Journal of Derivatives, 14: 61–76. [Crossref] , [Google Scholar]), we test the empirical validity of the monotonicity property for option prices by collecting all transaction data from 1 July 2006 to 31 December 2006 for option contracts traded on the Taiwan Futures Exchange (TAIFEX). We find that sampled intraday option prices violate the monotonicity property between 29.97% and 57% of the time, and that call and put prices often increase, or decrease, together. We also find evidence to show that the frequent violations of the monotonicity property are to a large extent attributable to microstructure effects and that they arise from rational trading tactics.     

Non-existence and inefficiency of equilibria with American options

Summary We analyze two examples of economies with incomplete financial markets. In the first model we consider a stock and an American put option on the stock. Although there is only one commodity and asset payoffs therefore do not depend on spot prices, we derive robust non-existence of equilibria. In the second model we consider an economy where a stock is the only asset available for trade. We show that it is impossible to complete the market by introducing American put options and that equilibria are inefficient. This example is also robust.We wish to thank Jonathan Berk, Virginia France, Flavio Menezes and a referee of this journal for useful comments. This research was partially funded by the National Science Foundation (SES-8821723).     

Competitive equilibria in overlapping generations Experiments

Summary The paper presents the results of four overlapping generations experiments performed at the California Institute of Technology. Overlapping generations markets were created in which each agent had a two period life span. With the exception of the first period, there were eight agents trading in each period; four buyers (two young and two old) and four sellers (two young and two old). Parameters were selected so that a small set of equilibria existed. The markets were open for twenty-nine periods with a demand shift occurring at the fifteenth and sixteenth periods.This work provides a method of computing all competitive equilibria for a class of environments—called the opposing shift environments. The main conclusion of the experiments is that the experimental price data converge to near the stationary portions of the competitive equilibria. Demographic dynamics are also explored as part of the price adjustment process. Dynamic features found in non-overlapping environments are also observed in the overlapping generations case.We thank E. J. Nanale, a Caltech undergraduate student, who helped design and conduct the experiments. Special thanks are due to Hsing Yang Lee for his help in adapting the Multiple Unit Double Auction Program to accomodate the overlapping generations environment and for his help with the data processing. We also thank David Grether who found a serious problem with the statistical analysis in an early draft. The research assistance of Kay-Yut Chen was also very valuable to the completion of this work. The comments and suggestions of P. Rangazas were very helpful. The research of the second author was supported in part by a National Science Foundation Grant and the California Institute of Technology Laboratory for Experimental Economics and Political Science.     

When sunspots don't matter

Summary We show that a finite, competitive economy isimmune to sunspots if (i) preferences are strictly convex, (ii) the set of feasible allocations is convex, and (iii) the contingent-claims market is perfect. The conditions (i)–(ii) cover some, but not all, economies with nonconvex technologies. Based on an indivisible-good example, we show that even economies with strictly convex preferences and full insurance arenot in general immune from sunspots. We also show that (1) the sufficient conditions (i)–(iii) are not necessary for sunspot immunity and (2)ex-ante efficiency is not necessary for immunity from sunspots.This paper is based on an earlier paper, Indivisibilities in Production, and Sunspot Equilibrium, presented at the 1990 S.E.D.C. Meetings, Minneapolis-St. Paul, June 1990. The research support of NSF Grant SES-9012780, the Center for Analytic Economics, and the Thorne Fund is gratefully acknowledged.     

Accounting for the impact of higher order moments in foreign equity option pricing model

This paper investigates the pricing of foreign equity option whose value depends on foreign equity prices and exchange rate. We assume that the underlying asset returns of foreign equity option is not a Brownian motion, and use the Gram-Charlier series expansion to augment a normal density with two additional terms to capture the effects of skewness and kurtosis. The empirical study shows that the higher order moments (skewness and kurtosis) clearly affect the estimated prices of foreign equity options. This approach enables us to capture more accurately the foreign equity option prices.     

An improved framework for approximating option prices with application to option portfolio hedging

As the price of the underlying asset changes over time, delta of the option changes and a gamma hedge is required along with delta hedge to reduce risk. This paper develops an improved framework to compute delta and gamma values with the average of a range of underlying prices rather than at the conventional fixed ‘one point’. We find that models with time-varying volatility price options satisfactorily, and perform remarkably well in combination with the delta and delta-gamma approximations. Significant improvements are achieved for the GARCH model followed by stochastic volatility models. The new approach can ensure significant improvement in modelling option prices leading to better risk-management decision-making.     

On uniquely implementing cooperation in the prisoners' dilemma

Summary We consider the problem of a principle who wishes to induce two agents playing a one shot prisoner's dilemma to behave cooperatively. We assume that the principal cannot observe the actions of the agents, and is not able to change the strategy sets or payoff functions in the underlying game. The only power the principle has is to randomly delay the arrival of payoffs. Specifically, agents choose their one shot strategies, and then the principle randomly determines whether these are cheap talk, or if payoffs should be distributed. If the round is cheap talk, then each agent observes the strategy choice of the other and play moves to a new round. This continues until payoffs are distributed. We establish conditions under which the probability of cheap talk can be chosen at the beginning of the induced game in such a way that full cooperation is the only equilibrium outcome. The sufficiency condition is met by a wide class of economic interpretations of the prisoners' dilemma, including those involving strategic complementarities among players.The authors wish to thank Dilip Abreu, Robert Aumann, Michael Baye, James Friedman, Richard McLean, Herve Moulin, Ariel Rubinstein, Rajiv Vohra and Simon Wilkie for their comments. Also, participants in seminars and conferences at Arizona, Bellcore, Brown, Illinois, Northwestern, Princeton, Rochester, Vanderbilt and West Virginia have provided stimulating comments. We also thank the referee for many detailed and useful suggestions. The third author's research was supported in part by NSF grant SES-9213145.     

Market depth,domestic investors and price monotonicity violations

We re-examine the monotonicity violations of option price dynamics considering the roles of market depth and domestic investors. Violations caused by option price movements in conflict with underlying price movements tend to occur less frequently as the market depth increases, especially in the case of out-of-the-money options. In contrast, violations caused by option prices that remain sticky despite underlying price changes occur more frequently as the market depth increases. Both of these relationships are amplified by domestic investors.     

Stock options and managerial optimal contracts

Summary. In this paper we are concerned with the performance of stock option contracts in the provision of managerial incentives. In our simple framework, we restrict the space of contracts available to the principal to those conformed by a fixed payment and a call option on the firms stock. As compared to the fixed payment and the option grant, we find that the strike price plays an intermediate role in the provision of insurance and incentives. We also develop some methods for the calibration of a standard principal-agent model based upon observed CEO earnings schedules and the volatility of the firms value in the stock market. These methods are useful to address some important issues such as the performance of stock option contracts, the degree of risk aversion compatible with current earnings profiles and the sensitivity of compensation to changes in firms characteristics.Received: 9 April 2003, Revised: 13 September 2004, JEL Classification Numbers: C6, D83. Correspondence to: Manuel S. SantosWe have benefitted from helpful discussions with Marco Celentani, Hector Chade, Alejandro Manelli and Ed Schlee. We are especially grateful to an anonymous referee for very detailed comments.     

Non-marketed options, non-existence of equilibria, and non-linear prices

This paper presents a surprising example that shows that the lattice theoretic properties in Mas-Colell's (1986) seminal work are relevant to the existence of equilibrium even when the commodity space is finite dimensional.The example is a two-period securities model with a three-dimensional portfolio space and two traders. The paper identifies a non-marketed call option that fails to have a minimum cost super-replicating portfolio. Using this option, we construct an economy that satisfies all of Mas-Colell's assumptions, except that the three-dimensional commodity space is not a vector lattice. In this economy, there is no Walrasian equilibrium and the second theorem of welfare economics fails.Our example has important finite- as well as infinite-dimensional implications. It is also an example of a “well behaved” economy in which optimal allocations that are not supported by linear Walrasian prices are decentralized by the non-linear prices studied in Aliprantis-Tourky-Yannelis (2001).     

Bubbles as payoffs at infinity

Summary We 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.This paper was circulated under the title Stochastic bubbles in Markov economies. We acknowledge with gratitude numerous conversations with Mark Fisher, the editorial advice of David Levine and the useful comments of anonymous referees. This paper should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or its staff.     

Risk,variety and volatility: growth,innovation and stock prices in early industry evolution

The paper studies the patterns of volatility in firm growth rates and stock prices during the early phase of the life-cycle of an old economy industry, the US automobile industry from 1900-1930, and a new economy industry, the US PC industry from 1974-2000. In both industries, firm growth rates are more volatile in the period in which innovation is the most radical. This is also the period in which stock prices are more volatile. The comparison sheds light on the co-evolution of industrial and financial volatility and the relationship between this co-evolution and mechanisms of Schumpetarian creative destruction. Results provide insight into the debate on whether the statistical behavior of firm growth rates is well represented by Gibrats Law.JEL Classification: L11, 030, G12I thank Massimiliano Tancioni for his excellent research assistance. Support from the following grants is much appreciated: European Commission Key Action Improving the socio-economic knowledge basecontract HPSE-CT-2002-00146, and the Open University RDF Grant contract no. 793.