麻烦解释一下三个选项。

问题如下：

If risk-free investments yielded a higher return over the next year, which of the following statements best describes how this would affect the no-arbitrage value of the put option on SparCoin shares?

选项：

A.An increase in the risk-free rate will have no effect on SparCoin's put option price, as it is solely a function of the probability and degree of share price increase or decrease upon option expiration.

B.An increase in the risk-free rate will increase the value of the put option, as it will increase the risk-neutral probability of a price decline.

C.An increase in the risk-free rate will decrease the value of the put option, as it will both increase the risk-neutral probability of a price increase π and decrease the present value of the expected option payoff.

解释：

C is correct.

Rising interest rates reduce the value of a put option. Increasing the risk-free rate will increase the risk-neutral probability of a price increase π and decrease the present value of the expected option payoff. Since the value of a put option is inversely related to the price of the underlying asset, an increased probability of an upward price move will reduce the expected payoff from the put. Consequently, both of these effects will reduce the put option value as the return on risk-free investments increases.

中文解析

增加无风险利率将增加价格上涨的风险中性概率π，并减少预期期权支付的现值。

但是由于看跌期权的价值与标的资产的价格成反比，价格上涨概率的增加将最终减少看跌期权的预期收益。

因此，随着无风险投资回报的增加，看跌期权的价值在下降。