# Binomial Model - completeness in presence of arbitrage

Consider a uniperiodal binomial model where I buy one bond of value $$B_0$$ and rate $$r=0.1$$, and $$h$$ stocks with price $$S_0=5$$. The value of the portfolio at time $$t=0$$ is

$$V_0 = B_0 + hS_0,$$

that should be equal to the price $$p$$ of a call option with strike price $$K$$, given that the binomial model is complete. If I select the price of the stock at time $$t=1$$ to be $$S_1=6$$ or $$8$$, the market is not arbitrage-free, but I could determine that for $$0 there exists a price $$p>0$$, corresponding to $$-0.1, and $$B_0=6$$. In other words, the call option is replicable (meaning that $$p>0$$, right?) if $$K<5.5$$ in a market where arbitrage is possible.

Physically it seems acceptable for me. In a market where I always gain money, I just need to buy a call option always in-the-money. Is it reasonable?