call 价格的三张图

Draw the graph

专题: Finance / 金融
难度: L4
来源: QuantQuestion

题目详情

This question concerns the standard European call option on a non- dividend- paying stock. You are asked to draw three closely related graphs as follows:

Please draw the graph of call price at maturity (time $T$ ) versus terminal stock price, $S(T)$ .
Please draw the graph of call price at time $t$ versus the futures price $F(t, T)$ . The futures price $F(t, T)$ is observed at time $t$ , prior to maturity. The futures contract and the option both mature at the same date.<sup>57</sup>
Now draw the graph of call price versus stock price at time $t$ , prior to maturity.

Explain carefully the relationships between the three graphs.

解析

到期 payoff： $\max(S_T-K,0)$ ，在 $K$ 处折点，右侧斜率 1。
$t<T$ 时，call 价格对期货 $F(t,T)$ ：平滑、凸； $F$ 很小时趋近 0； $F$ 很大时近似

C\approx e^{-r\tau}(F-K),\quad \tau=T-t,

因此渐近斜率为 $e^{-r\tau}$ （小于 1）。

$t<T$ 时对现货 $S$ ：同样平滑凸；与对 $F$ 的图通过 $F=Se^{r\tau}$ （无分红）互相映射：

C(S,t)=C(F,t),\quad \frac{\partial C}{\partial F}=e^{-r\tau}\frac{\partial C}{\partial S}.