I'm trying to solve the following problem:
Analyst report says that stock will be taken over with probability $50$%. If it’s not taken over, it’ll move to $5$ dollars with $40$% chance, and to 0 dollars with $60$% chance. If it’s taken over, it’ll move to $15$ dollars with $80$% chance, and $x$ with $20$% chance. The stock is trading at $10$ dollars. How much would you pay, assuming no time value of money, for the option to buy the stock at $22$ dollars after everything settles? This means you get to see if it’s taken over or not, and then get to see where it
Supposedly the correct answer is $\$0.80$ but I can't seem to arrive at that answer. Here is my thinking:
The value of the call ($C$) should be such that the expected payout for the contract is $\$0$.
$\frac34$ possible outcomes result in stock prices below the strike price, therefore the call is out of the money and we lose $\$C$. The probabilities for each of these outcomes sum to $0.9$.
In the last case, where the stock is worth $\$x$ with probability $0.1$, our payout is $x-22-C$
Therefore the price of the call C should be $-0.9C + 0.1(x-22-C) = 0$ solved for $C$. But I cannot figure how to possibly derive a value for the option without knowing $x$ (or at the very least the distribution of x) because if x was hypothetically infinitely large, the option would be worth a very large amount, whereas if $x = 0$, the option would be worthless (it is guaranteed to expire out of the money)
I also cannot figure out if I need to incorporate the spot price ($\$10$) into a calculation like this, although my work seems intuitively correct to me.