A necessary and sufficient condition to rule out arbitrage profits in the single-period HJM binomial tree is that the following must hold:

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The pseudo-probabilities represent the probabilities of the stock going up and down in the binomial tree. If different bonds or money market accounts have different pseudo-probabilities, then there is an opportunity for arbitrage profits. By requiring the pseudo-probabilities to be equal for all zero-coupon bond/money market account pairs, we eliminate the possibility of arbitrage profits. Choices A and B are not necessary and sufficient conditions as they only consider the dollar return of one bond, whereas we need to consider all zero-coupon bond/money market account pairs. Choice D is incorrect as it only requires symmetric probabilities, which is not necessarily sufficient to rule out arbitrage profits.

A Necessary and Sufficient Condition to Rule Out Arbitrage Profits >\gt>

A Necessary and Sufficient Condition to Rule Out Arbitrage Profits $\gt$