Arbitrage portfolio example

Question

Can you give me a concrete example of a self financing portfolio which gives arbitrage opportunity in the two-dimensional Black-Scholes model?

By the two-dimensional Black-Scholes model I mean

$$dS_{1}\left(t\right)=S_{1}\left(t\right)\left[\mu_{1}dt+\sigma_{1}dW\left(t\right)\right]$$ $$dS_{2}\left(t\right)=S_{2}\left(t\right)\left[\mu_{2}dt+\rho\sigma_{2}dW_{1}\left(t\right)+\sqrt{1-\rho^{2}}\sigma_{2}dW_{2}\left(t\right)\right]$$

where $S_{1}\left(t\right)$ and $S_{2}\left(t\right)$ are the underlyings; $W_{1}\left(t\right)$ and $W_{2}\left(t\right)$ are independent Wiener processes; $\mu_{1}$, $\mu_{2}$, $\sigma_{1}$, $\sigma_{2}$ and $\rho$ are constants; and the risk-free interest rate is also constant in the dynamic of the risk-free product: $$dB\left(t\right)=rB\left(t\right)dt.$$

In the previous underlying dynamic: $\rho\sigma_{2}dW_{1}\left(t\right)+\sqrt{1-\rho^{2}}\sigma_{2}dW_{2}\left(t\right)$ represents a kind of Cholesky decomposition.

Answer 1

You can choose an arbitrage for the classical 1-dimensional Black-Scholes model, and not use $S_2$ at all.

Such an arbitrage is e.g. presented in Example 3.5 in "Fractional Processes As Models In Stochastic Finance" by Bender et al (2011).