This question already has an answer here:
Suppose we follow the assumptions of the Black-Scholes Model, including unlimited borrowing, continuous prices, and frictionless markets. For simplicity assume the risk-free rate is 0.
In this world, why can we not construct a call option at strike K (assume K>S) by doing the following (assume K=100 for this case):
1) If stock price (S) hits 100, borrow money and buy stock.
2) If stock price then falls below 100, sell stock at 99.999999 (=100) and return money (continuous price assumption).
3) If stock price never hits 100, do nothing.
In theory, this generates the same payoff as a call option, but for free. Assuming we are in the world of the Black-Scholes assumptions, where does this argument break down?