So the question asks: L et $S(0) = 120$ dollars, $u = 0.2$, $d = −0.1$ and $r = 0.1$. Consider a call option with strike price $X = 120$ dollars and exercise time $T = 2$. Find the option price and the replicating strategy.
So the solution is:
The option price at time $0$ is $22.92$ dollars. (Yes, I got the same answer)
In addition to this amount, the option writer should borrow $74.05$(?) dollars and buy $0.8081$ (?) of a share.
At time 1, if $S(1) = 144$, then the amount of stock held should be increased to 1 share, the purchase being financed by borrowing a further $27.64$ dollars, increasing the total amount of money owed to $109.09$ dollars. (Understood. $144*(1-0.8081) = 27.64$, $27.64+74.05+7.4=109.09$)
If, on the other hand, $S(1) = 108$ dollars at time $1$, then some stock should be sold to reduce the number of shares held to $0.2963$(why?), and $55.27$ (?) dollars should be repaid, reducing the amount owed to $26.18$ (what?) dollars. (In either case the amount owed at time 1 includes interest of 7.40 dollars on the amount borrowed at time 0.)
So basically, I really can't see why it decides to buy $0.8081$ share at time 0?
Also, where does the number $0.2963$ come from? Even though the share should be reduced to $29.63%$, how does it come out with the repaid amount, $55.27$ and the amount owned, $26.18$?