A two-stage supply chain with a risk-averse retailer and a risk-neutral supplier is considered, among which the demand decreases sharply and market price fluctuates randomly with emergencies. A "modified profit-CVaR" criteria is put forward to measure the impact of risk aversion. Then a quantity flexibility contract model under random price is constructed based on the "modified profit-CVaR" when retailer is risk-neutral, together with the optimal ordering and wholesale price strategy for supply chain to respond to emergencies. Meanwhile, a numerical simulation is set up to analyze the influence of the aversion degree and the demand function volatility degree on the optimal decision and the supply chain members' profits. The results show that:in the case of risk aversion and stochastic price, the quantity flexibility contract can make the supply chain achieve coordination, and can make the supply chain optimal decisions have strong robustness to the degree of risk aversion and demand fluctuation.