Past exam of the martingales and financial mathematics course of the University of Oxford / 2010 / B / 4 / b / Solution

The risk-neutral probability must make the expected stock multiplier equal the bank multiplier:Both up and down paths have positive probability under this measure. A self-financing portfolio's discounted value is then a Q-martingale, since the discounted bank asset is constant and the discounted stock is a martingale. An arbitrage with zero initial wealth would have nonnegative terminal value and positive value in some possible state, giving a strictly positive Q-expectation and contradicting its zero discounted initial value. Thus the market with $e^r=1.1$ is arbitrage free.