= 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> price
$$
C_t=\mathbb E(\sqrt{S_T}\mid\mathcal F_t)
$$
is 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.
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