A Mean–variance analysis of arbitrage portfolios

Based on the careful analysis of the definition of arbitrage portfolio and its return, the author presents a mean–variance analysis of the return of arbitrage portfolios, which implies that Korkie and Turtle's results ( B. Korkie, H.J. Turtle, A mean–variance analysis of self-financing portfolios, Manage. Sci. 48 (2002) 427–443) are misleading. A practical example is given to show the difference between the arbitrage portfolio frontier and the usual portfolio frontier.     

Faszination Finanzmathematik: Probleme, Methodik und Prinzipien

Zusammenfassung Es wird ein überblick über die Arbeitsgebiete und die wichtigsten Resultate der modernen Finanzmathematik gegeben. Hierbei soll durch ein klares Herausstellen der wichtigsten Prinzipien und eine Trennung zwischen ihrer Demonstration und der in der Praxis ben?tigten, theoretisch anspruchsvollen stochastischen Analysis einem gro?en Leserkreis ein verst?ndlicher Zugang zur Finanzmathematik erm?glicht werden.     

A quantum model of option pricing: When Black-Scholes meets Schrödinger and its semi-classical limit

The Black-Scholes equation can be interpreted from the point of view of quantum mechanics, as the imaginary time Schrödinger equation of a free particle. When deviations of this state of equilibrium are considered, as a product of some market imperfection, such as: Transaction cost, asymmetric information issues, short-term volatility, extreme discontinuities, or serial correlations; the classical non-arbitrage assumption of the Black-Scholes model is violated, implying a non-risk-free portfolio. From Haven (2002) [1] we know that an arbitrage environment is a necessary condition to embedding the Black-Scholes option pricing model in a more general quantum physics setting. The aim of this paper is to propose a new Black-Scholes-Schrödinger model based on the endogenous arbitrage option pricing formulation introduced by Contreras et al. (2010) [2]. Hence, we derive a more general quantum model of option pricing, that incorporates arbitrage as an external time dependent force, which has an associated potential related to the random dynamic of the underlying asset price. This new resultant model can be interpreted as a Schrödinger equation in imaginary time for a particle of mass 1/σ² with a wave function in an external field force generated by the arbitrage potential. As pointed out above, this new model can be seen as a more general formulation, where the perfect market equilibrium state postulated by the Black-Scholes model represent a particular case. Finally, since the Schrödinger equation is in place, we can apply semiclassical methods, of common use in theoretical physics, to find an approximate analytical solution of the Black-Scholes equation in the presence of market imperfections, as it is the case of an arbitrage bubble. Here, as a numerical illustration of the potential of this Schrödinger equation analogy, the semiclassical approximation is performed for different arbitrage bubble forms (step, linear and parabolic) and compare with the exact solution of our general quantum model of option pricing.     

No Arbitrage and the Growth Optimal Portfolio

Abstract

Recently, several papers have expressed an interest in applying the Growth Optimal Portfolio (GOP) for pricing derivatives. We show that the existence of a GOP is equivalent to the existence of a strictly positive martingale density. Our approach circumvents two assumptions usually set forth in the literature: 1) infinite expected growth rates are permitted and 2) the market does not need to admit an equivalent martingale measure. In particular, our approach shows that models featuring credit constrained arbitrage may still allow a GOP to exist because this type of arbitrage can be removed by a change of numéraire. However, if the GOP exists the market admits an equivalent martingale measure under some numéraire and hence derivatives can be priced. The structure of martingale densities is used to provide a new characterization of the GOP which emphasizes the relation to other methods of pricing in incomplete markets. The case where GOP denominated asset prices are strict supermartingales is analyzed in the case of pure jump driven uncertainty.     

Arbitrage and Equilibrium in Economies with Externalities

We introduce consumption externalities into a general equilibrium model with arbitrary consumption sets. To treat the problem of existence of equilibrium, a condition of no unbounded arbitrage, extending the condition of Page (1987) and Page and Wooders (1993, 1996) is defined. It is proven that this condition is sufficient for the existence of an equilibrium and both necessary and sufficient for compactness of the set of rational allocations.     

Likelihood Inferences for High-Dimensional Factor Analysis of Time Series With Applications in Finance

This article investigates likelihood inferences for high-dimensional factor analysis of time series data. We develop a matrix decomposition technique to obtain expressions of the likelihood functions and its derivatives. With such expressions, the traditional delta method that relies heavily on score function and Hessian matrix can be extended to high-dimensional cases. We establish asymptotic theories, including consistency and asymptotic normality. Moreover, fast computational algorithms are developed for estimation. Applications to high-dimensional stock price data and portfolio analysis are discussed. The technical proofs of the asymptotic results and the computer codes are available online.     

Arbitrage theory for non convex financial market models

We propose a unified approach where a security market is described by a liquidation value process. This allows to extend the frictionless models of the classical theory as well as the recent proportional transaction costs models to a larger class of financial markets with transaction costs including non proportional trading costs. The usual tools from convex analysis however become inadequate to characterize the absence of arbitrage opportunities in non-convex financial market models. The natural question is to which extent the results of the classical arbitrage theory are still valid. Our contribution is a first attempt to characterize the absence of arbitrage opportunities in non convex financial market models.     

Static arbitrage bounds on basket option prices

We consider the problem of computing upper and lower bounds on the price of an European basket call option, given prices on other similar options. Although this problem is hard to solve exactly in the general case, we show that in some instances the upper and lower bounds can be computed via simple closed-form expressions, or linear programs. We also introduce an efficient linear programming relaxation of the general problem based on an integral transform interpretation of the call price function. We show that this relaxation is tight in some of the special cases examined before.     

A General Benchmark Model for Stochastic Jump Sizes

Abstract

Under few technical assumptions and allowing for the absence of an equivalent martingale measure, we show how to price and hedge in a sequence of incomplete markets driven by Wiener noise and a marked point process. We investigate the structure of market prices of risk as markets become approximately complete and consider the limits of traded securities, characterizing explicitly the growth optimal portfolio and investigating arbitrage and diversification in such markets.     

Preservation of risk in capital markets

We model the capital market as a cooperative game. In this context, we formulate a property for solution concepts called preservation of risk. It may be viewed as a certain no-arbitrage-principle. In particular, we prove that the Shapley value is the only efficient solution concept that satisfies preservation of risk. Moreover, we derive an economic interpretation of the potential of the Shapley value. Finally, we relate our theoretical findings to the real world phenomenon called cornering the market.