If there is an inconsistent pricing strategy then by defintion we have strong arbitrage

Question

Background Information:

An Inconsistent pricing strategy is a self financing strategy $\phi$ with $V_T(\phi)= 0$ and $V_0(\phi) \neq 0$

A strong arbitrage is a self-financing strategy $\phi$ with $V_0(\phi) = 0$ and $V_T(\phi) > 0$

Question:

Suppose there exists an Inconsistent pricing strategy. Prove from the definition that there must exist a strong arbitrage.

Attempted proof - Let $\phi$ be a self-financing strategy such that $V_0(\phi)\neq 0$ and $V_T(\phi) = 0$.

I am confused how this is possible to prove seems like we have a direct contradiction. Any suggestions are greatly appreciated.

Answer 1

We assume that $V_0(\phi)<0$; otherwise, we can consider the strategy $-\phi$. Then, we buy extra $-V_0(\phi)/S_0^0$ share of the risk-free asset $S^0$, from the $k+1$ assets $S^0, S^1,\ldots, S^k$, which is the deposit or money-market account, and hold until maturity $T$, that is, we consider the trading strategy $\psi$, where \begin{align*} \psi_i = \begin{cases} -V_0(\phi)/S_0^0, & \text{ if } i=0,\\ 0, & \text{ otherwise}, \end{cases} \end{align*} without any intermediate adjustment. It is then clear that $V_0(\psi+\phi)=0$, and $V_T(\psi+\phi)>0$. In other words, there exists a strong arbitrage strategy (e.g., $\psi+\phi$).