论文标题
内源性随机套利气泡和黑色 - choles模型
Endogenous Stochastic Arbitrage Bubbles and the Black--Scholes model
论文作者
论文摘要
本文开发了一个模型，该模型在黑色 - choles方程中明确地包含了随机套利的存在。在这里，套利是由随机气泡产生的，它概括了文献中获得的确定性套利模型。它被认为是套利气泡的通用随机动态，然后得出了广义的黑色 - choles方程。所得的方程式与随机波动率模型相似，但是没有不确定的参数作为风险市场价格。提出的理论具有与弱和强套利泡沫极限相关的渐近行为。对于套利气泡的波动率为零（确定性气泡）的情况，弱极限对应于通常的黑色 - choles模型。强极限情况也给出了黑色 - choles模型，但是基本资产的平均值取代了利率。当气泡随机时，该理论还具有弱且强大的渐近限制，这会引起与黑色 - choles模型相似的期权价格动力学。显式公式是针对高斯和对数正常随机气泡的。因此，黑色 - choles模型可以被认为是更通用随机模型的“低能量”极限。
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.
