内生随机套利泡沫与布莱克-斯科尔斯模型
Endogenous Stochastic Arbitrage Bubbles and the Black--Scholes model
摘要：
本文提出了一个模型，将随机套利的存在明确的Black-Scholes方程。在这里，套利是由一个随机泡沫产生的，它推广了文献中得到的确定性套利模型。我们认为套利泡沫是一个一般的随机动力学过程，并由此导出了一个广义的Black-Scholes方程。所得方程与随机波动率模型的方程相似，但不存在风险市场价格的待定参数。所提出的理论具有与弱套利泡沫极限和强套利泡沫极限相关的渐近行为。对于套利泡沫的波动率为零的情况（确定性泡沫），弱极限对应于通常的布莱克-斯科尔斯模型。强极限情形也给出了一个布莱克-斯科尔斯模型，但标的资产的平均值代替了利率。当泡沫是随机的，该理论也有弱和强的渐近极限，引起类似于布莱克-斯科尔斯模型的期权价格动态。导出了高斯和对数正态随机气泡的显式公式。因此，布莱克-斯科尔斯模型可以被认为是一个更一般的随机模型的“低能量”极限。
英文摘要：
This paper develops a model that incorporates the presence of stochastic arbitrage explicitly in the Black--Scholes equation. Here, the arbitrage is generated by a stochastic bubble, which generalizes the deterministic arbitrage model obtained in the literature. It is considered to be a generic stochastic dynamic for the arbitrage bubble, and a generalized Black--Scholes equation is then derived. The resulting equation is similar to that of the stochastic volatility models, but there are no undetermined parameters as the market price of risk. The proposed theory has asymptotic behaviors that are associated with the weak and strong arbitrage bubble limits. For the case where the arbitrage bubble's volatility is zero (deterministic bubble), the weak limit corresponds to the usual Black-Scholes model. The strong limit case also give a Black--Scholes model, but the underlying asset's mean value replaces the interest rate. When the bubble is stochastic, the theory also has weak and strong asymptotic limits that give rise to option price dynamics that are similar to the Black--Scholes model. Explicit formulas are derived for Gaussian and lognormal stochastic bubbles. Consequently, the Black--Scholes model can be considered to be a "low energy" limit of a more general stochastic model.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
