Question $3[8$ points$]$

Consider an economy with two dates $(t=0,1)$ and three states at $t=1.$ The following three

stocks are traded:

$x_1=(10,0,30),x_2=(0,20,40),x_3=(20,20,0)$

The $t=0$ prices of these stocks are given as follows

$(p_1,p_2,p_3)=(12,14,8)$

$($a$)$ Is there an arbitrage? $[1$p$]$

Suppose an investment firm sells options.

$($b$)$ What is the $t=0$ price $($premium$)$ of a call option on stock $3$ with exercise price $E=12?$

$2p$

$($c$)$ What is the $t=0$ price $($premium$)$ of a put option on stock $2$ with exercise price $E=34?$

$[2$p$]$

Suppose a start$-$up company wants to go public. The firm has total costs of $$100,000$ at date

$t=1$ and sales of $$130,000$ in state $1,$$$160,000$ in state $2,$ and $$240,000$ in state $3.$ The firm

wants to issue $1,000$ IPO shares. $($A share is endowed with a cash flow right of $0\mathrm{.}1\%$ of the

total profits of the firm.$)$

$($d$)$ The underwriter suggests an IPO price of $$62$ per share. Will this IPO be successful,

i$.$e$.$ will there be a positive demand for the shares? $[3$p$]$