Consider the one period binomial model with $S_0=4,S_1(H)=8,S_1(T)=2$

and the risk$-$free interest rate is $r=\frac{1}{4}.$ Recall in class we calculated the

time $t=0$ price of a call option that expires at time $t=1$ with strike price

$K=5$ is $V_0=1\mathrm{.}2.$ Consider an agent who begins with wealth $x_0=0$

and at time $t=0$ buys ${}_0$ shares of stock and ${}_0$ options $($these can be

positive or negative or zero$).$ This leaves the agent with a cash position of

$-{}_{0}4-{}_{0}1\mathrm{.}2.$ If this is positive, it is invested in the bond; if it is negative,

it represents a loan. What is the value of the agents portfolio at time $t=1?$

Assume that the probability of a head $0.$ Show that there

$isno$ arbitrage opportunity.

Solution JUST FIGURE OUT WHAT THE VALUE OF THE AGENTS PORTFOLIO IS AT T$=1.$