Black-Scholes期权定价模型的自适应波动选择

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摘要翻译：
给出了标准Black-Scholes期权定价模型的非线性波动方案。用自适应非线性Schr\'Odinger(NLS)方程形式化地定义了代表金融市场受控布朗行为的自适应波模型，将期权定价的波函数定义为股票价格和时间。该模型包括两个参数：波动率（发挥离散频率系数的作用），可以是固定的，也可以是随机的，以及依赖于利率的自适应市场潜力。波函数表示量子概率振幅，其绝对平方为概率密度函数。从与自由量子力学粒子相关的德布罗意平面波包出发，用Jacobi椭圆函数给出了NLS方程的四种解析解。与Black-Scholes模型最吻合的是自适应激波NLS解，它可以与自适应孤立波NLS解有效地结合。使用无监督的Hebbian学习或有监督的Levenberg-Marquardt算法来估计自适应市场热潜力的可调“权重”。在随机波动的情况下，它本身是由波函数表示的，因此我们得到了两个耦合的NLS方程（允许闭式解）的所谓Manakov系统，它具有共同的自适应市场势，定义了一个双向时空联想记忆。关键词：Black-Scholes期权定价，自适应非线性Schr\'Odinger方程，市场热势，受控随机波动率，自适应Manakov系统，受控布朗行为
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英文标题：
《Adaptive-Wave Alternative for the Black-Scholes Option Pricing Model》
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作者：
Vladimir G. Ivancevic
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最新提交年份：
2009
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Pricing of Securities        证券定价
分类描述：Valuation and hedging of financial securities, their derivatives, and structured products
金融证券及其衍生产品和结构化产品的估值和套期保值
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一级分类：Quantitative Finance        数量金融学
二级分类：Computational Finance        计算金融学
分类描述：Computational methods, including Monte Carlo, PDE, lattice and other numerical methods with applications to financial modeling
计算方法，包括蒙特卡罗，偏微分方程，格子和其他数值方法，并应用于金融建模
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英文摘要：
  A nonlinear wave alternative for the standard Black-Scholes option-pricing model is presented. The adaptive-wave model, representing 'controlled Brownian behavior' of financial markets, is formally defined by adaptive nonlinear Schr\"odinger (NLS) equations, defining the option-pricing wave function in terms of the stock price and time. The model includes two parameters: volatility (playing the role of dispersion frequency coefficient), which can be either fixed or stochastic, and adaptive market potential that depends on the interest rate. The wave function represents quantum probability amplitude, whose absolute square is probability density function. Four types of analytical solutions of the NLS equation are provided in terms of Jacobi elliptic functions, all starting from de Broglie's plane-wave packet associated with the free quantum-mechanical particle. The best agreement with the Black-Scholes model shows the adaptive shock-wave NLS-solution, which can be efficiently combined with adaptive solitary-wave NLS-solution. Adjustable 'weights' of the adaptive market-heat potential are estimated using either unsupervised Hebbian learning, or supervised Levenberg-Marquardt algorithm. In the case of stochastic volatility, it is itself represented by the wave function, so we come to the so-called Manakov system of two coupled NLS equations (that admits closed-form solutions), with the common adaptive market potential, which defines a bidirectional spatio-temporal associative memory.   Keywords: Black-Scholes option pricing, adaptive nonlinear Schr\"odinger equation, market heat potential, controlled stochastic volatility, adaptive Manakov system, controlled Brownian behavior
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PDF链接：
https://arxiv.org/pdf/0911.1834

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关键词：SCHOLES choles 期权定价模型 Black Holes adaptive nonlinear stochastic 频率 波动