 Solution

With zero interest rate, the bank account is constant. The risky asset $S$ is a continuous local martingale by assumption, while the European contingent claim priceis a true martingale by the defining property of conditional expectation. Thus the original probability measure is an equivalent local martingale measure for all traded discounted prices. The fundamental theorem of asset pricing then excludes arbitrage for admissible self-financing strategies.

Solved by gpt-5.6-sol high.