The solution of the Black-scholes equation is the price of a European call. And the option price assumes the underlying stock is a geometric Brownian motion with volatility $\sigma_{1}>0$.
Suppose, however, the underlying asset is really a geometric Brownian motion with volatility $\sigma_{2} > \sigma_{1}$, i.e. \begin{equation} dS(t) = \alpha S(t)dt + \sigma_{2}S(t)dW(t). \end{equation}
Consequently, the market price of the call is incorrect.
Can we set up a portfolio which has an arbitrage opportunity in the market? Furthermore, if there any methods to generate a portfolio arbitrage opportunity (how to consider this problem)?
Inspired by AFK, I try to answer this question by myself in mathematical way.
Firstly, the main idea of generating the portfolio with arbitrage opportunity is to buy a call option and at $\sigma_{1}$, and sell a call priced at $\sigma_{2}$. i.e. \begin{equation} X(t) = c(t,S(t)) - c^{\sigma_{2}}(t,S(t)) \end{equation} where $X(t)$ denote the value of portfolio, $c(t,S(t))$ is the value of the option at time $t$, and $c^{\sigma_{2}}$ is the value of the option priced in $\sigma_{2}$.
Actually, $c^{\sigma_{2}}$ was not exit in the market, but it doesn't matter since you can replicate it by hedging, which means, \begin{equation} X(t) = c(t,S(t)) - c_{x}(t,S(t))S(t) - \Gamma(t)M(t) \end{equation} Now, we want to show that X(t) has arbitrage opportunity.
It is trivial to see that X(0) = 0, then we want to show that dX(t) > 0 (Actually, we finally prove that de^{-rt}X(t) > 0). By Ito formula, we find that $$ dc(t,S(t)) = c_{t}dt + c_{x}dS(t) + 1/2c_{xx}d[S,S](t) $$ and $$ dX(t) = dc(t,S(t)) - c_{x}dS(t) - r(c - X(t) -c_{x}S(t))dt. $$ Then, substitute dc into this equation, we get $$ dX(t) = (c_{t}+1/2c_{xx}\sigma_{2}^{2}S(t)^{2}-rc+rc_{x}S(t))dt + rX(t) $$ Note that c(t,S(t)) follows the Black-scholes equation with $\sigma_{1}$, so we have $$ dX(t) - rX(t) = (c_{t}+1/2c_{xx}\sigma_{1}^{2}S(t)^{2}-rc+rc_{x}S(t))dt + 1/2c_{xx}(\sigma_{2}^{2} - \sigma_{1}^{2})S(t)^{2}dt $$ i.e. $$ de^{-rt}X(t) = 1/2c_{xx}(\sigma_{2}^{2} - \sigma_{1}^{2})S(t)^{2}dt $$ It is always positive ($\sigma_{2}>\sigma{1}$, and $c_{xx}>0$).
In summary, X(t) is a portfolio with X(0) = 0, and de^{-rt}X(t) is always positive, s.t. it has arbitrage opportunity.
If any problem in my idea and my proof, please let me know.
