为什么还要求一个派

问题如下：

Kleinert's analyst estimates a 50-50 chance that the price of SparCoin will either increase by 12% or decline by 10% at the put option's expiration date. Which of the following statements best describes the no-arbitrage option price implied by this assumption?

选项：

A.Since there is a 50% chance that the stock will fall to €94.73, there is a 50-50 chance of a €5.27 payout upon exercise and the no-arbitrage put is therefore worth €2.64 (= €5.27 / 2). B.Since there is a 50% chance that the stock will fall to €94.73, there is a 50-50 chance of a €5.27 payout upon exercise and given the risk-neutral probability of 0.47, the no-arbitrage put price is €2.48 (= €5.27 × 0.47). C.Since there is a 50% chance that the stock will fall to €94.73 and the risk-neutral probability is 0.47, the no-arbitrage put price is €2.78 (= €5.27 × {[1 – 0.47]/1.0037}).

解释：

Solution

C is correct.

A 12% increase in the stock price gives:

$S_1^u=R^u{S}_0=1.12\times €105.25=€\mathrm{117.88.}$

The put option will expire unexercised:

$p_1^u=\text{Max}\left(0,X-S_1^u\right)=\text{Max}\left(0,€100-€117.88\right)=€0.$

Alternatively, a 10% price decrease gives:

$S_1^d=R^d{S}_0=0.9\times €105.25=€\mathrm{94.73.}$

The put option will pay off:

$p_1^d=\text{Max}\left(1,X-S_1^d\right)=\text{Max}\left(0,€100-€94.73\right)=€\mathrm{5.27.}$

To price this option, the risk-neutral pricing formula gives the risk-neutral probability π as:

π = (1 + 0.0037 − 0.9)/(1.12 − 0.9) = 0.47.

The no-arbitrage price is:

p₀ = (0.47 × €0 + 0.53 × €5.27)/(1 + 0.0037) = €2.79/1.0037 = €2.78.

中文解析

本题考察的是使用二叉树对看跌期权进行定价。

题目比较常规，按照上面步骤计算即可。