Computing arbitrage upper bounds on basket options in the presence of bid–ask spreads

We study the problem of computing the sharpest static-arbitrage upper bound on the price of a European basket option, given the bid–ask prices of vanilla call options in the underlying securities. We show that this semi-infinite problem can be recast as a linear program whose size is linear in the input data size. These developments advance previous related results, and enhance the practical value of static-arbitrage bounds as a pricing technique by taking into account the presence of bid–ask spreads. We illustrate our results by computing upper bounds on the price of a DJX basket option. The MATLAB code used to compute these bounds is available online at www.andrew.cmu.edu/user/jfp/arbitragebounds.html.     

Risk averse asymptotics in a Black–Scholes market on a finite time horizon

We consider the optimal investment and consumption problem in a Black–Scholes market, if the target functional is given by expected discounted utility of consumption plus expected discounted utility of terminal wealth. We investigate the behaviour of the optimal strategies, if the relative risk aversion tends to infinity. It turns out that the limiting strategies are: do not invest at all in the stock market and keep the rate of consumption constant!     

The uncertainties about the relationships risk–return–volatility in the Spanish stock market

The relationships between the market risk premium, its conditional variance and the risk-free rate in the Spanish stock market are studied in this paper. Using daily data, the above mentioned relations are analyzed by quasi maximum likelihood for an EGARCH-M(1,1) model with normal innovations and by nonparametric maximum likelihood for a semiparametric EGARCH-M(1,1) model with arbitrarily distributed innovations. It is worth mentioning that the conclusions differ from one model to the other.     

On the results of N. P. Korneichuk obtained in 1990–1999

We present a brief survey of Korneichuk’s works published in 1990–1999.     

Chapter 15 DEA/AR analysis of the 1988–1989 performance of the Nanjing textiles corporation

This article employs new data envelopment analysis/assurance region (DEA/AR) methods to evaluate the efficiency of the 35 textile factories of the Nanjing Textiles Corporation (NTC), Nanjing, China. The returns to scale (RTS) of these factories were studied without assuming that the optimal DEA solutions were unique. All DMUs are identified with pointsE (Extreme Efficient),E (Efficient but not an extreme point) andF (Frontier but not efficient). We then further identify the nonfrontier DMUs with pointsNE, NE andNF according to whether they are projected onto a point inE, E, orF en route to evaluating their performances. All of the inefficient factories were in classNF and had unique optimal primal-dual solution pairs. Consequently, the solution pairs satisfy the strong complementary slackness condition (SCSC). Application of cone-ratio (CR) ARs reduced significantly the number of factories in classE, and showed that some AR-efficient factories were more flexible in adopting the mixture of central planning and market economies that China currently is trying to use. Also, linked-cone (LC) ARs were applied to measure maximum and minimum profit ratios. The SCSC multiplier space approach was utilized to analyze the sensitivity of the efficiency results to potential errors in the data with and without ARs. The results in this article suggest that collective units had a better performance than state-owned units in the two consecutive years analyzed.This paper was written while the author was at the School of Economics and Management, Southeast University, Nanjing 210018, P.R. China.     

Mixed Brownian–fractional Brownian model: absence of arbitrage and related topics

We consider a mixed Brownian–fractional-Brownian model of a financial market. The class of self-financing strategies is restricted to Markov-type smooth functions. It is proved that such strategies satisfy a parabolic equation that can be reduced to heat equation. Then it is proved that the mixed model is arbitrage-free. Finally, the capital of the model is presented as the limit of a sequence of semimartingales.     

Equilibrium in an ambiguity-averse mean–variance investors market

In a financial market composed of n risky assets and a riskless asset, where short sales are allowed and mean–variance investors can be ambiguity averse, i.e., diffident about mean return estimates where confidence is represented using ellipsoidal uncertainty sets, we derive a closed form portfolio rule based on a worst case max–min criterion. Then, in a market where all investors are ambiguity-averse mean–variance investors with access to given mean return and variance–covariance estimates, we investigate conditions regarding the existence of an equilibrium price system and give an explicit formula for the equilibrium prices. In addition to the usual equilibrium properties that continue to hold in our case, we show that the diffidence of investors in a homogeneously diffident (with bounded diffidence) mean–variance investors’ market has a deflationary effect on equilibrium prices with respect to a pure mean–variance investors’ market in equilibrium. Deflationary pressure on prices may also occur if one of the investors (in an ambiguity-neutral market) with no initial short position decides to adopt an ambiguity-averse attitude. We also establish a CAPM-like property that reduces to the classical CAPM in case all investors are ambiguity-neutral.     

A note on arbitrage‐free pricing of forward contracts in energy markets

Arbitrage theory is used to price forward (futures) contracts in energy markets, where the underlying assets are non‐tradeable. The method is based on the so‐called ‘fitting of the yield curve’ technique from interest rate theory. The spot price dynamics of Schwartz is generalized to multidimensional correlated stochastic processes with Wiener and Lévy noise. Findings are illustrated with examples from oil and electricity markets.     

On the order of the operators in the Douglas–Rachford algorithm

The Douglas–Rachford algorithm is a popular method for finding zeros of sums of monotone operators. By its definition, the Douglas–Rachford operator is not symmetric with respect to the order of the two operators. In this paper we provide a systematic study of the two possible Douglas–Rachford operators. We show that the reflectors of the underlying operators act as bijections between the fixed points sets of the two Douglas–Rachford operators. Some elegant formulae arise under additional assumptions. Various examples illustrate our results.     

On the Price of Risk of the Underlying Markov Chain in a Regime-Switching Exponential Lévy Model

Regime-switching models (RSM) have been recently used in the literature as alternatives to the Black-Scholes model. Several authors favor RSM as being more realistic since, by construction, they model those exogenous macroeconomic cycles against which asset prices evolve. In the context of derivatives pricing, these models lead to incomplete markets and therefore there exist multiple Equivalent Martingale Measures (EMM) yielding different pricing rules. A fair amount of literature (Buffington and Elliott, Int J Theor Appl Finance 40:267–282, 2002; Elliott et al., Ann Finance 1(4):23–432, 2005) focuses on conveniently choosing a family of EMM leading to closed-form formulas for option prices. These studies often make the assumption that the risk associated with the Markov chain is not priced. Recently, Siu and Yang (Acta Math Appl Sin Engl Ser 25(3):339–388, 2009), proposed an EMM kernel that takes into account all risk components of a regime-switching Black-Scholes model. In this paper, we extend the results and observations made in Siu and Yang (Acta Math Appl Sin Engl Ser 25(3):339–388, 2009) in order to include more general Lévy regime-switching models that allow us to assess the influence of jumps on the price of risk. In particular, numerical results are given for Regime-switching Jump-Diffusion and Variance-Gamma models. Also, we carry out a comparative analysis of the resulting option price formulas with existing regime-switching models such as Naik (J Financ 48:1969–1984, 1993) and Boyle and Draviam (Insur Math Econ 40:267–282, 2007).     

On the solutions of the Dullin–Gottwald–Holm equation in Besov spaces

This paper is concerned with the Cauchy problem for the Dullin–Gottwald–Holm equation. First, the local well-posedness for this system in Besov spaces is established. Second, the blow-up criterion for solutions to the equation is derived. Then, the existence and uniqueness of global solutions to the equation are investigated. Finally, the sharp estimate from below and lower semicontinuity for the existence time of solutions to this equation are presented.     

On the Dimension of the Space of Dark States in the Tavis–Cummings Model

Mathematical Notes - The space of minimal energy of a qubit system is the dark subspace of quantum states of a system of two-level atoms in the finite-dimensional Tavis–Cummings (TC) model of...     

Is there chaos in the Spanish labour market?

One could argue that there is a resurgence of the non-linear modelling in economics. Some instruments have been developed to measure the complexity or instability of the analysed systems. At the present work some of these developed techniques are applied to verify the non-linearity present in the time series of Spanish unemployment, as well as to quantify the degree of complexity of the system that has generated the series. Using these techniques we find evidence of chaos in Spanish unemployment time series.     

A path to the Arrow–Debreu competitive market equilibrium

We present polynomial-time interior-point algorithms for solving the Fisher and Arrow–Debreu competitive market equilibrium problems with linear utilities and n players. Both of them have the arithmetic operation complexity bound of )) for computing an -equilibrium solution. If the problem data are rational numbers and their bit-length is L, then the bound to generate an exact solution is O(n ⁴ L) which is in line with the best complexity bound for linear programming of the same dimension and size. This is a significant improvement over the previously best bound )) for approximating the two problems using other methods. The key ingredient to derive these results is to show that these problems admit convex optimization formulations, efficient barrier functions and fast rounding techniques. We also present a continuous path leading to the set of the Arrow–Debreu equilibrium, similar to the central path developed for linear programming interior-point methods. This path is derived from the weighted logarithmic utility and barrier functions and the Brouwer fixed-point theorem. The defining equations are bilinear and possess some primal-dual structure for the application of the Newton-based path-following method. Dedicated to Clovis Gonzaga on the occassion of his 60th birthday. This author was supported in part by NSF Grants DMS-0306611 and DMS-0604513. The author would like to thank Curtis Eaves, Osman Güler, Kamal Jain and Mike Todd for insightful discussions on this subject, especially on their mathematical references and economic interpretations of the fixed-point model presented in this paper.     

Proof of Stability in the Brower–Paul Problem

Doklady Mathematics - We study the stability of equilibrium in the problem known as “a ball on a rotating saddle,” which was first considered by the famous Dutch mathematician Brauer in...     

Global indeterminacy of the equilibrium in the Chamley model of endogenous growth in the vicinity of a Bogdanov–Takens bifurcation

This paper studies the dynamics implied by the Chamley (1993) model, a variant of the two-sector model with an implicit characterization of the learning function. We first show that under some “regularity” conditions regarding the learning function, the model has (a) one steady state, (b) no steady states or (c) two steady states (one saddle and one non-saddle). Moreover, via the Bogdanov–Takens theorem, we prove that for critical regions of the parameters space, the dynamics undergoes a particular global phenomenon, namely the homoclinic bifurcation. Because these findings imply the existence of a continuum of equilibrium trajectories, all departing from the same initial value of the predetermined variable, the model exhibits global indeterminacy.