A combinatorial approach for pricing Parisian options

This paper provides a discrete time algorithm, in the framework of the Cox–Ross–Rubinstein analysis (1979), to evaluate both Parisian options with a flat barrier and Parisian options with an exponential boundary. The algorithm is based on a combinatorial tool for counting the number of paths of a particle performing a random walk, that remains beyond a barrier constantly for a period strictly smaller than a pre-specified time interval. As a result, a binomial evaluation model is derived that is very easy to implement and that produces highly accurate prices. Received: 19 March 2001 / Accepted: 17 March 2002 The author thanks Prof. Ivar Massabó for helpful comments and discussions. This research has been partially supported by MIUR (research on “Modelli per la Finanza Matematica”)     

A note on preference for flexibility

A result of Kreps (1979) on preference for flexibility is extended from two to three periods (formally from preferences over sets to preferences over sets of sets). An intuitively easier route to Kreps' original result is also presented, making the proof essentially ready for use in a decision theory class. Received: 23 April 2001 / Accepted: 1 September 2001 First version October 1997.     

An efficient binomial method for pricing¶American options

We present a new method for obtaining fast and accurate estimates of the price of an American put option by binomial trees. The method is based on the interpolation of suitable values obtained by modifying the contractual strike. A time-saving procedure allows us to derive all the interpolating data from a unique standard backward procedure. Received: 16 July 2001 / Accepted: 19 April 2002 {The authors would like to thank an anonymous referee for helpful comments. We also thank Antonino Zanette for his help in the refinements of the numerical procedures.     

Knightian decision theory. Part I

A theory of choice under uncertainty is proposed which removes the completeness assumption from the Anscombe–Aumann formulation of Savage's theory and introduces an inertia assumption. The inertia assumption is that there is such a thing as the status quo and an alternative is accepted only if it is preferred to the status quo. This theory is one way of giving rigorous expression to Frank Knight's distinction between risk and uncertainty. Received: 8 October 2001 / Accepted: 8 November 2001 The author is grateful to Martin Shubik for encouragement and long discussions. He is also grateful to Donald Brown, Edward Leamer, David Schmeidler, Martin Shubik, Sidney Winter, Asad Zaman and Arnold Zellner for calling his attention to relevant literature. Note of the Editor: Except for a few minor editorial changes, this article publishes content formerly circulated as Discussion Paper no.~807 of the Cowles Foundation at Yale University, November 1986.     

An elementary core equivalence theorem¶in a countable economy

We consider an infinite exchange economy with countably many traders, which can be regarded as a natural extension of finite exchange economies to an infinite one. In our countable economy the core defined in the traditional manner would be empty. To avoid this unwanted situation we have to strengthen the notion of “improves upon”. We will achieve this based on the idea that forming coalitions involve costs. Received: 23 August 2000 / Accepted: 12 April 2002 I am very grateful to István Dancs, Péter Medvegyev, Miklós Pintér, Csaba Sándor, András Simonovits, László Szili, Péter Tallos and the anonymous referees for comments and suggestions. Parts of this research were done during the author's Bolyai János Research Fellowship provided by the Hungarian Academy of Sciences (MTA). Of course, all errors remain the author's responsibility.     

Single factor models with Markovian spot interest rate: an analytical treatment

In the spirit of the Heath–Jarrow–Morton methodology, we provide an analytical characterization of bond prices within the context of single factor term structure models in which the spot rate follows a Markov process and the volatility structure of zero coupon bond returns is stochastic. Also, a perturbative analysis of the extended Cox–Ingersoll–Ross model is proposed. Received: 7 February 2001 / Accepted: 8 June 2002     

Bargaining over an uncertain outcome:¶the role of beliefs

We study the Nash bargaining solution of a problem in which two agents bargain over an uncertain outcome. Under the assumptions of risk neutrality and of constant absolute risk aversion, we study the way that the solution varies, ex ante, when we vary the beliefs of one agent. Changing an agent's beliefs in a way that makes them “more distant” from the other agent's beliefs makes the second agent better off. Received: 10 May 2001 / Accepted: 22 August 2001     

Discrete representations of a continuum economy

We consider an atomless economy in which the continuum of agents is represented by a real interval. By dividing the interval and associating to every agent in each subinterval the same initial endowments and preferences, we define sequences of discrete economies as approximations to the initial continuum economy. We obtain convergence results for the core (or, alternatively, for the set of Walrasian allocations) of the continuum economy in terms of the cores of the approximating discrete economies. Finally, we state some counterexamples which provide a boundary for more general results in this framework. Received: 29 December 2000 / Accepted: 28 April 2002 C. Hervés and E. Moreno acknowledge support by Research Grant BEC2000-1388-C04-01 from the Dirección General de Investigación Científica y Técnica (DGICYT), Spanish Ministry of Education.?E. Moreno acknowledges support from the Spanish Ministry of Education through a post-doctoral fellowship in 1997 and from a Training and Mobility of Researchers (E.C.) fellowship in 1998 while visiting Universidade Nova de Lisboa.     

Optimality conditions¶and bubbles in sequential economies¶and bounded relative risk-aversion

We provide specific qualifications in order that Kuhn–Tucker type Euler equations and transversality conditions at infinity hold in stochastic equilibrium models with heterogeneous agents and where assets are traded in sequential markets. It is not assumed that uncertainty is modeled as an event-tree structure or that preferences are necessarily bounded. We also describe an important class of preferences based on bounded relative risk aversion which yields relevant simplifications. Our results are used to establish conditions that rule out asset pricing bubbles. Specific examples of economies with bubbles are also discussed. Received: 28 January 2002 / Accepted: 19 July 2002 We are grateful to the editor and an anonymous referee for their valuable comments. This research was partially supported by MURST (Italy), National Group on “Nonlinear Dynamics and Stochastic Models in Economics and Finance”.     

Mean-variance hedging for interest rate models¶with stochastic volatility

The mean-variance hedging approach for pricing and hedging claims in incomplete markets was originally introduced for risky assets. The aim of this paper is to apply this approach to interest rate models in the presence of stochastic volatility, seen as a consequence of incomplete information. We fix a finite number of bonds such that the volatility matrix is invertible and provide an explicit formula for the density of the variance-optimal measure which is independent of the chosen times of maturity. Finally, we compute the mean-variance hedging strategy for a caplet and compare it with the optimal stategy according to the local risk minimizing approach. Received: 14 July 2000 / Accepted: 10 April 2001     

Optimal impulse control for cash management¶with quadratic holding-penalty costs

This paper studies optimal control for an infinite horizon cash management problem where the cash fund fluctuates as a Brownian motion. Holding-penalty costs are assumed to be a quadratic function of the cash level and there are fixed and proportional transaction costs. Using the “impulse technique”, we prove that optimal control exists and takes the form of a control band policy. Received: 4 January 2001 / Accepted: 5 June 2001     

The discrepancy distribution of stationary multiplier rules for rounding probabilities

The problem of rounding finitely many (continuous) probabilities to (discrete) proportions N _i/n is considered, for some fixed rounding accuracy n. It is well known that the rounded proportions need not sum to unity, and instead may leave a nonzero discrepancy D=(∑N _i) −n. We determine the distribution of D, assuming that the rounding function used is stationary and that the original probabilities follow a uniform distribution. Received: April 1999     

DS-optimal designs in one way ANOVA

Characterization and construction of optimal designs using the familiar optimality criteria, for example A-, D- and E-optimality are well studied in the literature. However the study of the Distance Optimality (DS-) criterion introduced by Sinha (1970) has very recently drawn attention of researchers. In the present article, we consider the singularly estimable full rank problem of estimating the full set of elementary treatment contrasts using the DS optimality criterion in the set up of a one way ANOVA model. Using a limit argument it turns out that a CRD in which difference between any two allocation numbers is at the most unity is uniquely DS-optimal. Acknowledgement. We are thankful to Prof. B. K. Sinha for suggesting the problem to us and many helpful discussions with him. We are also thankful to the referees for drawing our attention to the reference of Bischoff (1995) and many helpful comments.     

Two-level search design for main-effect plus two plan

One important class of screening designs is the search design first proposed by Srivastava (1975). A new class of two-level factorial search designs which are capable of estimating all main-effect plus two interactions is provided. We first give a necessary and sufficient condition for the main-effect plus two plan and then show that the proposed search design always satisfies such a condition. Received October 2000     

D-optimal weighing designs for six objects

For all integers m≥6, we determine the maximum value of det X ^T X, where X is an m×6 (0, 1)-matrix, and exhibit (D-optimal) matrices X for which the maximum occurs. For D-optimal matrices X, the uniqueness of the Gram matrix X ^T X is discussed. Received: May 2000