Question: Consider a market with one risk-free asset and one risky asset that follows the binomial tree model. Suppose that whenever stock goes up, you can predict that it will go down at the next step. Find a self-financing (but not necessarily predictable) strategy with V (0) = 0, V (1) ≥ 0 and 0 ≠ V (2) ≥ 0.
This exercise indicates that predictability is an essential assumption in the No-Arbitrage Principle. An investor who could foresee the future behaviour of stock prices (here, if stock goes down at one step, you can predict what it will do at the next step) would always be able to find a suitable investment strategy to ensure a risk-free profit.