英文标题：
《Calibration and simulation of arbitrage effects in a non-equilibrium
  quantum Black-Scholes model by using semiclassical methods》
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作者：
Mauricio Contreras, Rely Pellicer, Daniel Santiagos and Marcelo
  Villena
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最新提交年份：
2015
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英文摘要：
  An interacting Black-Scholes model for option pricing, where the usual constant interest rate r is replaced by a stochastic time dependent rate r(t) of the form r(t)=r+f(t) dW/dt, accounting for market imperfections and prices non-alignment, was developed in [1]. The white noise amplitude f(t), called arbitrage bubble, generates a time dependent potential U(t) which changes the usual equilibrium dynamics of the traditional Black-Scholes model. The purpose of this article is to tackle the inverse problem, that is, is it possible to extract the time dependent potential U(t) and its associated bubble shape f(t) from the real empirical financial data? In order to give an answer to this question, the interacting Black-Scholes equation must be interpreted as a quantum Schrodinger equation with hamiltonian operator H=H0+U(t), where H0 is the equilibrium Black-Scholes hamiltonian and U(t) is the interaction term. If the U(t) term is small enough, the interaction potential can be thought as a perturbation, so one can compute the solution of the interacting Black-Scholes equation in an approximate form by perturbation theory. In [2] by applying the semi-classical considerations, an approximate solution of the non equilibrium Black-Scholes equation for an arbitrary bubble shape f(t) was developed. Using this semi-classical solution and the knowledge about the mispricing of the financial data, one can determinate an equation, which solutions permit obtain the functional form of the potential term U(t) and its associated bubble f(t). In all the studied cases, the non equilibrium model performs a better estimation of the real data than the usual equilibrium model. It is expected that this new and simple methodology for calibrating and simulating option pricing solutions in the presence of market imperfections, could help to improve option pricing estimations.
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中文摘要：
[1]中提出了一种期权定价的交互Black-Scholes模型，其中通常的恒定利率r被形式为r（t）=r+f（t）dW/dt的随机时间相关利率r（t）取代，考虑了市场缺陷和价格不一致性。白噪声振幅f（t）被称为套利泡沫，它产生了一个与时间相关的势U（t），改变了传统Black-Scholes模型通常的平衡动力学。本文的目的是解决反问题，也就是说，是否有可能从真实的经验金融数据中提取与时间相关的势U（t）及其相关的泡沫形状f（t）？为了回答这个问题，相互作用的Black-Scholes方程必须解释为具有哈密顿算符H=H0+U（t）的量子薛定谔方程，其中H0是平衡的Black-Scholes哈密顿量，U（t）是相互作用项。如果U（t）项足够小，相互作用势可以看作是微扰，因此可以用微扰理论近似地计算相互作用的Black-Scholes方程的解。在[2]中，通过应用半经典考虑，得到了任意气泡形状f（t）的非平衡Black-Scholes方程的近似解。利用这种半经典解和有关金融数据错误定价的知识，可以确定一个方程，该方程的解允许获得势项U（t）及其相关气泡f（t）的函数形式。在所有研究案例中，非均衡模型比通常的均衡模型对真实数据的估计效果更好。预计这种在存在市场缺陷的情况下校准和模拟期权定价解决方案的新的简单方法将有助于改进期权定价估计。
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Mathematical Finance        数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
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