You're on the trading desk and need to price a European derivative that pays exactly $S_T^2$ at maturity $T=1$. Assume the non-dividend-paying underlying follows GBM with volatility $\sigma = 20\%$ and risk-free rate $r = 5\%$. The current stock price is $100$. What is the expected payoff in 1 year, and what is the fair value today?

你在交易台上，需要给一个收益为 $S_T^2$ 的欧式衍生品报价。假设标的遵循无红利的几何布朗运动，波动率20%，无风险利率5%，当前股价100。1年后的期望支付是多少？今天这个期权的公允价值是多少？