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.     

Continuous GRASP with a local active-set method for bound-constrained global optimization

Global optimization seeks a minimum or maximum of a multimodal function over a discrete or continuous domain. In this paper, we propose a hybrid heuristic—based on the CGRASP and GENCAN methods—for finding approximate solutions for continuous global optimization problems subject to box constraints. Experimental results illustrate the relative effectiveness of CGRASP–GENCAN on a set of benchmark multimodal test functions.     

A biased random-key genetic algorithm for road congestion minimization

One of the main goals in transportation planning is to achieve solutions for two classical problems, the traffic assignment and toll pricing problems. The traffic assignment problem aims to minimize total travel delay among all travelers. Based on data derived from the first problem, the toll pricing problem determines the set of tolls and corresponding tariffs that would collectively benefit all travelers and would lead to a user equilibrium solution. Obtaining high-quality solutions for this framework is a challenge for large networks. In this paper, we propose an approach to solve the two problems jointly, making use of a biased random-key genetic algorithm for the optimization of transportation network performance by strategically allocating tolls on some of the links of the road network. Since a transportation network may have thousands of intersections and hundreds of road segments, our algorithm takes advantage of mechanisms for speeding up shortest-path algorithms.     

Objective evaluation of acute adverse events and image quality of gadolinium-based contrast agents (gadobutrol and gadobenate dimeglumine) by blinded evaluation. Pilot study

Purpose

The purpose was to objectively evaluate a recently FDA-approved gadolinium-based contrast agent (GBCA) in comparison to our standard GBCA for acute adverse events and image quality by blinded evaluation.

Methods

Evaluation was made of a recently FDA-approved GBCA, gadobutrol (Gadavist; Bayer), in comparison to our standard GBCA, gadobenate dimeglumine (MultiHance; Bracco), in an IRB- and HIPAA-compliant study. Both the imaging technologist and patient were not aware of the brand of the GBCA used. A total of 59 magnetic resonance studies were evaluated (59 patients, 31 men, 28 women, age range of 5–85 years, mean age of 52 years). Twenty-nine studies were performed with gadobutrol (22 abdominal and 7 brain studies), and 30 studies were performed with gadobenate dimeglumine (22 abdominal and 8 brain studies). Assessment was made of acute adverse events focusing on objective observations of vomiting, hives, and moderate and severe reactions. Adequacy of enhancement was rated as poor, fair and good by one of two experienced radiologists who were blinded to the type of agent evaluated.

Results

No patient experienced acute adverse events with either agent. The target minor adverse events of vomiting or hives, and moderate and severe reactions were not observed in any patient. Adequacy of enhancement was rated as good for both agents in all patients.

Conclusions

Objective, blinded evaluation is feasible and readily performable for the evaluation of GBCAs. This proof-of-concept study showed that both GBCAs evaluated exhibited consistent good image quality and no noteworthy adverse events.     

Prescribed diagonal Schouten tensor in locally conformally flat manifolds

We consider the pseudo-euclidean space ${(\mathbb{R}^n, g)}$ , with n ≥  3 and ${g_{ij} = \delta_{ij} \varepsilon_i, \varepsilon_i = \pm 1}$ and tensors of the form ${T = \sum \nolimits_i \varepsilon_i f_i (x) dx_i^2}$ . In this paper, we obtain necessary and sufficient conditions for a diagonal tensor to admit a metric ${\bar{g}}$ , conformal to g, so that ${A_{\bar g}=T}$ , where ${A_{\bar g}}$ is the Schouten Tensor of the metric ${\bar g}$ . The solution to this problem is given explicitly for special cases for the tensor T, including a case where the metric ${\bar g}$ is complete on ${\mathbb{R}^n}$ . Similar problems are considered for locally conformally flat manifolds. As an application of these results we consider the problem of finding metrics ${\bar g}$ , conformal to g, such that ${\sigma_2 ({\bar g })}$ or ${\frac{\sigma_2 ({\bar g })}{\sigma_1 ({\bar g })}}$ is equal to a given function. We prove that for some functions, f ₁ and f ₂, there exist complete metrics ${\bar{g} = g/{\varphi^2}}$ , such that ${\sigma_2 ({\bar g }) = f_1}$ or ${\frac{\sigma_2 ({\bar g })}{\sigma_1 ({\bar g })} = f_2}$ .     

Rayleigh-schrödinger series for defective spectral elements of compact operators in banach spaces

We prove a posteriori error bounds and convergence of Rayleigh-Schrödinger Series for bases of maximal invariant subspaces of compact operators in Banach spaces. These results generalize those for eigenvectors associated with simple eigenvalues. Recursive formulae are given for the numerical computation of the series' coefficients.

This work has been partially supported by Grant 89–0879 Fondecyt Chile.     

Depth-profile analysis of nanostructures by SIMS: Depth resolution function

A new model of the depth-resolution function for secondary-ion mass spectrometry, which takes into account recoil implantation, ion mixing, and surface roughness formation under ion irradiation, is considered. A simple three-parameter equation is proposed to describe the depth-resolution function. Analytical expressions are obtained for two parameters.