Why must a replicating portfolio be self-financing?

If I have a trading strategy such that at each time $t$ I own $\Delta_t$ units of stock $S_t$ and $\psi_t$ units of bond $B_t$, it is a replicating strategy for some claim with time $T \geq t$ payoff $X$ if the value of this portfolio $V_T = X$.

Written out, the time $t$ value of the portfolio is $V_t = \Delta_t S_t + \psi_t B_t$, and is self-financing if $dV_t = \Delta_t dS_t + \psi_t dB_t$; that is, changes in value are brought on only by changes in asset prices, not the strategy.

Now, why is a self-financing strategy so important for option pricing? If $V_T = X$ but this strategy is NOT self-financing, doesn't the time $t$ price of the option still have to be $V_t$, else arbitrage? If it's NOT self-financing, there may be cash injections/consumption along the way to the payoff, but so be it - it still replicates the payoff, so at each $t$ the price of the option must still be $V_t$.

Please let me know how I've confused myself so badly!

The idea of a replicating portfolio is that at time t all you need to provide is an amount $V_t$ so that in all outcomes of the world you end up with a portfolio worth $X$ without providing any additional money.
I suppose you could extend the portfolios to include deterministic influx/outfluxes of money and then you could present value all those changes. However, if an influx/outflux at any time depends on the outcome of your process, then that means at time $t$ you cannot say exactly how much cash is needed to end up at $X$. It's not that it provides an arbitrage but instead it just doesn't provide you with what you are looking for.
• I think this is a good counterexample, but technically shouldn't the strategy be previsible, by definition of a trading strategy? E.g. in discrete time, at time $T-1$ set up a replicating portfolio (must assume complete market?) for $V_{T-1}$. Then it would still be incorrect to say the value of the portfolio today (and hence the option) is $e^{-r(T-1)}V_{T-1}$ since we don't know $V_{T-1}$ today. Would this be more correct? – bcf May 28 '15 at 18:08