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For the European call option, risk-neutral valuation givesDefine the stock-numeraire measure in a binomial market byIt is a probability measure because the risk-neutral martingale property gives ${\mathbb{E}}_Q{S}_N=S_0(1+r{)}^N$. Consequently

Tilting one up move by its stock factor changes its probability toSimilarly $1-\hat{q}=(1-q)(1+a)/(1+r)$. Thus under $\hat{Q}$ the number of up moves is binomial with parameter $\hat{q}$, and