  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.
