# Arbitrage portfolio example

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.