A generalization of the Leibnitz rule of successive differentiation

This note generalizes the well-known Leibnitz rule of successive differentiation for the product of two functions to a similar result for the product of three functions.     

Pricing early exercise contracts in incomplete markets

We present a utility-based methodology for the valuation of early exercise contracts in incomplete markets. Incompleteness stems from nontraded assets on which the contracts are written. This methodology takes into account the individuals attitude towards risk and yields nonlinear pricing rules. The early exercise indifference prices solve a quasilinear variational inequality with an obstacle term. They are also shown to satisfy an optimal stopping problem with criterion given by their European indifference price counterpart. A class of numerical schemes are developed for the variational inequalities and a general approach for solving numerically nonlinear equations arising in incomplete markets is discussed.Accepted: May 2003, AMS Classification: 93E20, 60G40, 60J75The second author acknowledges partial support from NSF Grants DMS 0102909 and DMS 0091946.     

Local asymptotic admissibility of a generalization of Akaike's model selection rule

Summary A model selection rule of the form minimize [−2 log (maximized likelihood)+complexity] is considered, which is equivalent to Akaike's minimum AIC rule if the complexity of a model is defined to be twice the number of independently adjusted parameters of the model. Under reasonable assumptions, when applied to a locally asymptotically normal sequence of experiments, the model selection rule is shown to be locally asymptotically admissible with respect to a loss function of the form [inaccuracy+complexity], where the inaccuracy is defined as twice the Kullback-Leibler measure of the discrepancy between the true model and the fitted version of the selected model. This research was supported by NSF Grant No. MCS 80-02732.     

Pricing and hedging of American contingent claims in incomplete markets

0.IntroductionandSummaryThecelebratedpapersof[2]and[3],pavedthewayforpricingoptionsonstocks,onthebasisofthefollowingprinciple:inacompletemarket(suchastheoneinSection1.5),everycontingentclaimcanbeattainedexactlybyinvestinginthemarketandstartingwithala...     

Pricing and hedging derivative securities in markets with uncertain volatilities

We present a model for pricing and hedging derivative securities and option portfolios in an environment where the volatility is not known precisely, but is assumed instead to lie between two extreme values σ_minand σ_max. These bounds could be inferred from extreme values of the implied volatilities of liquid options, or from high-low peaks in historical stock- or option-implied volatilities. They can be viewed as defining a confidence interval for future volatility values. We show that the extremal non-arbitrageable prices for the derivative asset which arise as the volatility paths vary in such a band can be described by a non-linear PDE, which we call the Black-Scholes-Barenblatt equation. In this equation, the ‘pricing’ volatility is selected dynamically from the two extreme values, σ_min, σ_max, according to the convexity of the value-function. A simple algorithm for solving the equation by finite-differencing or a trinomial tree is presented. We show that this model captures the importance of diversification in managing derivatives positions. It can be used systematically to construct efficient hedges using other derivatives in conjunction with the underlying asset.     

Pricing and clearing combinatorial markets with singleton and swap orders

In this article we consider combinatorial markets with valuations only for singletons and pairs of buy/sell-orders for swapping two items in equal quantity. We provide an algorithm that permits polynomial time market-clearing and -pricing. The results are presented in the context of our main application: the futures opening auction problem. Futures contracts are an important tool to mitigate market risk and counterparty credit risk. In futures markets these contracts can be traded with varying expiration dates and underlyings. A common hedging strategy is to roll positions forward into the next expiration date, however this strategy comes with significant operational risk. To address this risk, exchanges started to offer so-called futures contract combinations, which allow the traders for swapping two futures contracts with different expiration dates or for swapping two futures contracts with different underlyings. In theory, the price is in both cases the difference of the two involved futures contracts. However, in particular in the opening auctions price inefficiencies often occur due to suboptimal clearing, leading to potential arbitrage opportunities. We present a minimum cost flow formulation of the futures opening auction problem that guarantees consistent prices. The core ideas are to model orders as arcs in a network, to enforce the equilibrium conditions with the help of two hierarchical objectives, and to combine these objectives into a single weighted objective while preserving the price information of dual optimal solutions. The resulting optimization problem can be solved in polynomial time and computational tests establish an empirical performance suitable for production environments.     

Risky asset allocation and consumption rule in the presence of background risk and insurance markets

This study examines joint decisions regarding risky asset allocation and consumption rate for a representative agent in the presence of background risk and insurance markets. Contrary to the conclusion of the “mutual fund separation theorem”, we show that the optimal risky asset mix will reflect an agent’s risk attitude as long as background risk is not independent of investment risk. This result can, however, be used to solve the “riskyasset allocation puzzle”. We also unveil that optimal insurance to shift background risk is determined through establishing a hedging portfolio against investment risk and is an arrangement maintaining the balance between growth and volatility of expected consumption. Because the optimal insurance we obtain generally leads to a smoother consumption path, it may plausibly explain the “equity premium puzzle” in the financial literature.     

Generalized Ramsey Theory for Graphs V. the Ramsey Number of a Digraph

Continuing this series of papers on generalized Ramsey theoryfor graphs, we define the Ramsey number r(D_l, D₂) of two digraphsD₁ and D₂ as the minimum p such that every 2-colouring of thearcs (directed lines) of DK_P (the complete symmetric digraphof order p) contains a monochromatic D₁ or D₂. It is shown(Theorem1) that this number exists if and only if D₁ or D₂ is acyclic.Then r(D), the diagonal Ramsey number of a given acyclic digraphD, is defined as r(D, D). Notation: D' is the converse of D,GD is the underlying graph of D, DG is the symmetric digraphof G, and T_p is the transitive tournament of order p. Let r(m,n) be the traditional Ramsey number of the two complete graphsK_m and K_n. Finally, let S_n be the star with n arcs from onepoint u to n points v_i. Assuming the Ramsey numbers under discussionexist, we prove the following results: THEOREM 2. r(D₁, D₂) = r(D₁' D₂'). THEOREM 3. r(D₁, D₂) r(GD₁, GD₂). THEOREM 4. r(D₁, D₂) r(T_P1, T_P2) if both D₁ and D₂ (with p₁and p₂ points respectively) are acyclic. THEOREM 5. r(T_m, T_n) = r(m, n). THEOREM 6. r(m, ri) r(T_m, DK_n) r(2^m–1, n). THEOREM 7. r(S_m, S_n) = r(S_m, S_n') = m+n. Finally, we establish all Ramsey numbers r(D₁, D₂) for digraphswithout isolates and with less than four points, and all diagonalRamsey numbers r(D) of acyclic digraphs without isolates withless than five points.     

A generalization of Descartes’ rule of signs and fundamental theorem of algebra

Descartes’ rule of signs yields an upper bound for the number of positive and negative real roots of a given polynomial. The fundamental theorem of algebra implies a similar property; every real polynomial of degree n ? 1 has at most n real zeroes. In this paper, we describe axiomatically function families possessing one or another of these properties. The resulting families include, at least, all polynomial functions and sums of exponential functions. As an application of our approach, we consider, among other things, a method for identifying certain type of bases for the Euclidean space.     

随机市场中均值-方差模型最优投资策略的时间不相容性及其修正

由于方差算子在动态规划意义下不可分,导致随机市场中多期均值一方差模型的最优投资策略不满足时间相容性,即Bellman最优性原理．为此,首先提出了随机市场中比Bellman最优性原理更弱的时间相容性,并证明在投资区间的任意中间时刻,当投资者的财富不超过某一给定的财富阈值时,最优投资策略满足弱时间相容性;当投资者的财富超过该阈值时,最优投资策略将不再是弱时间相容的,且导致投资者变为非理性,即他会同时极小化终期财富的均值和方差．在这种情形下,通过放松自融资约束,对最优投资策略进行了修正,使得其满足：修正策略可使投资者回归理性;相对于终期财富,修正策略可以获得与最优投资策略相同的均值和方差．在策略修正过程中,投资者可以从市场中获得一个严格正的现金流．这些结果表明修正策略要优于原最优投资策略,拓展了现有关于确定市场下多期均值．方差模型的求解以及策略时间相容性的结论．     

Optimal material ordering policy and allocation rule for a manufacturer making multiple products

Material ordering and allocation are important decisions for manufactures making multiple products, because those firms usually possess flexible production systems which can produce different products based on the same raw material. In this paper, we investigate the ordering policy (OP) and allocation rule (AR) of the raw materials for a manufacturer selling multiple products. The manufacturer’s decision-making problem is analyzed under three scenarios: (1) joint decisions on OP and AR, (2) fixed AR, and (3) predetermined OP. We show that the latter two are not special cases of the first scenario, and they require different solution methods. Our objective is to derive the optimal solutions analytically. For the first scenario, we obtain the closed-form solution that is indeed optimal for the nonconcave profit function. For the fixed AR scenario, the products with twice-differentiable demands are studied, and the exact optimal OP for the raw material is achieved. Finally, if the OP is predetermined, we prove that the profit function is concave in AR and provide the associated optimality conditions, for which the optimal AR can be reached numerically. Different from the pervious heuristic approaches, these mathematically tractable solutions are easy to be applied by the practitioners.     

Geometry in Grassmannians and a generalization of the dilogarithm

In this paper we introduce some polyhedra in Grassman manifolds which we call Grassmannian simplices. We study two aspects of these polyhedra: their combinatorial structure (Section 2) and their relation to harmonic differential forms on the Grassmannian (Section 3). Using this we obtain results about some new differential forms, one of which is the classical dilogarithm (Section 1). The results here unite two threads of mathematics that were much studied in the 19th century. The analytic one, concerning the dilogarithm, goes back to Leibnitz (1696) and Euler (1779) and the geometric one, concerning Grassmannian simplices, can be traced to Binet (1811). In Section 4, we give some of this history along with some recent related results and open problems. In Section 0, we give as an introduction an account in geometric terms of the simplest cases.     

On the generalization of a retrocirculant

We study the properties of matrices of the form P(σ)A where σ is induced by an automorphism of an abelian group G and A is a group matrix. P(σ)A is a generalization of a retrocirculant. We also determine the eigenvalues of P(σ)A.     

A dynamic analysis of protection and environmental policy in a small trading developing country

We analyze a dynamic model of protection and environmental policy in a small trading developing country (DC). The DC government protects the import competing (and the polluting) sector of the economy with a tariff. The employment and output effects of three different pollution taxes are analyzed. These taxes incorporate different assumptions about the DC government's ability to commit to its announced policy. First, we describe the taxes, we study the dependence of these taxes on the tariff, and we show that in general an activist environmental policy is called for, irrespective of the length of time to which the government can commit to its announced policy. Second, we identify a situation in which the conduct of environmental policy raises welfare unambiguously, and the situations in which it does not do so. Finally, we show that the time inconsistency of certain optimal programs can prevent the DC government from achieving its environmental and employment objectives.     

Competitive analysis of a dispatch policy for a dynamic multi-period routing problem

We analyze an on-line algorithm (dispatch policy) for a dynamic multi-period routing problem. The objective is to minimize the total cost over all periods. We show that the competitive ratio of this policy for instances with customers located on the non-negative real line is .     

Cyclic behavior in dynamic investment decisions for deregulated energy markets

In the first years after the deregulation of the electricity industry, investment into new generation capacity has not taken place on a large scale in any central european country. Recent increases in prices indicate that investment could be very profitable. However, the fear is that the need for new capacity can be overestimated and that could lead to a decrease in prices and profits and consequently to a reduction/stop of new investments. The aim of this paper is to model and analyze factors that influence the stability of electricity prices. The electricity market is modeled using a Cournot game and the stability of electricity prices is analyzed by simulations. The research was supported by the grant 1/3001/06 of the Grant Agency of Slovak Republic (VEGA) and grant VVGS 36/2006.     

Effects of pollution restrictions on dynamic investment policy of a firm

The purpose of this paper is to determine the effects of different pollution standards on the firm's resource allocation decisions. To do so, a dynamic model of the firm is developed in which it is assumed that production causes pollution as an inevitable byproduct. Concerning its investment policy, we suppose that the firm can choose between investing in productive capital goods and investing in abatement efforts.It is shown that, in some cases, future abatement expenses have a negative impact on the present level of productive investment, even if the pollution standard is not binding at the moment. This implies a really dynamic optimal investment policy for the firm, which cannot be obtained within a comparative static analysis.This research has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences. Comments by Frank van der Duyn Schouten and Piet Verheyen (Tilburg University) and by Raymond Gradus (Dutch Ministry of Finance, The Hague) are gratefully acknowledged.