非均衡偏度、市场危机与期权定价：超对称市场的非线性朗之万模型
Non-Equilibrium Skewness, Market Crises, and Option Pricing: Non-Linear Langevin Model of Markets with Supersymmetry
摘要：
本文提出了一个易于处理的模型的非线性动态的市场收益率使用朗之万方法。由于相互作用势的非线性，该模型承认小和大的回报波动的制度。朗之万动力学映射到一个等效的量子力学（QM）系统。借用超对称量子力学（SUSY QM）的思想，该QM系统的参数化基态波函数（WF）被用作模型的直接输入，该模型还固定了非线性朗之万势。使用一个双组分高斯混合物作为一个基态WF与非对称双阱势产生一个易于处理的低参数模型与解释的参数，被称为NES（非平衡偏斜）模型。超对称性（SUSY），然后用一种分析易处理的方式找到模型的时间依赖的解决方案。额外的近似值产生了NES模型的最终实用版本，其中真实测量和风险中性收益分布由三个分量高斯混合物给出。这产生了一个封闭形式的近似期权定价在NES模型的混合物的三个布莱克-斯科尔斯价格，提供准确的校准，无论是良性或不良市场环境的期权价格，而只使用一个单一的波动率参数。这些结果与大多数其他期权定价模型形成鲜明对比，例如局部、随机或粗糙波动率模型，这些模型需要更复杂的噪声规范来拟合市场数据。
英文摘要：
This paper presents a tractable model of non-linear dynamics of market returns using a Langevin approach. Due to non-linearity of an interaction potential, the model admits regimes of both small and large return fluctuations. Langevin dynamics are mapped onto an equivalent quantum mechanical (QM) system. Borrowing ideas from supersymmetric quantum mechanics (SUSY QM), a parameterized ground state wave function (WF) of this QM system is used as a direct input to the model, which also fixes a non-linear Langevin potential. Using a two-component Gaussian mixture as a ground state WF with an asymmetric double well potential produces a tractable low-parametric model with interpretable parameters, referred to as the NES (Non-Equilibrium Skew) model. Supersymmetry (SUSY) is then used to find time-dependent solutions of the model in an analytically tractable way. Additional approximations give rise to a final practical version of the NES model, where real-measure and risk-neutral return distributions are given by three component Gaussian mixtures. This produces a closed-form approximation for option pricing in the NES model by a mixture of three Black-Scholes prices, providing accurate calibration to option prices for either benign or distressed market environments, while using only a single volatility parameter. These results stand in stark contrast to the most of other option pricing models such as local, stochastic, or rough volatility models that need more complex specifications of noise to fit the market data.
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