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I'm trying to solve the following problem from "Probability and Statistics" book by Morris H. DeGroot and Mark J. Schervish.

Suppose that common stock in the up-and-coming company A is currently priced at \$200 per share. As an incentive to get you to work for company A, you might be offered an option to buy a certain number of shares of the stock, one year from now, at a price of \$200. For simplicity, suppose that the price $X$ of the stock one year from now is a discrete random variable that can take only two values (in dollars): $260$ and $180$. Let $p$ be the probability that $X = 260$. You want to calculate the value of these stock options. (For simplicity, we shall ignore dividends and the transaction costs of buying and selling stocks.) Assume that an investor could earn 4% risk-free on any money invested for this same year. (Assume that the 4% includes any compounding.)

I solve it in the following way. Let's say that $Y$ is an amount of money you make if you buy an option.

$$ Y = \begin{cases} 60 - c, & \mbox{with probability } p \\ -c, & \mbox{with probability } 1 - p \end{cases} $$

where c $-$ option price. Then the amount of money you make in 1 year with this stock would be $E(Y) = p(60 - c) - c(1-p)$. On the other hand you can make $1.04c$ risk-free. Therefore a fair stock price would be:

$$ p(60 - c) - c(1-p) = 1.04c. \\ c = \frac{60p}{2.04} $$

In the book authors for some reason ignore an option price and say that

$$ Y = \begin{cases} 60, & \mbox{with probability } p \\ 0, & \mbox{with probability } 1 - p \end{cases} $$

I am quite puzzled as to what I miss here.

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    $\begingroup$ Well, their $Y$ is just the future payout of the option. To price the option, you do need to consider the entire portfolio. Or, equivalently, discount the option payout back to start time. $\endgroup$
    – lulu
    Commented Jun 14 at 16:51
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    $\begingroup$ For the author version, usually we call that to be the payoff of the option. In your version, we call $c$ to be the price of the option, and $Y$ is the profit/loss for longing that call option. I am wondering if some terms are just abused or there are some misunderstandings $\endgroup$
    – BGM
    Commented Jun 14 at 16:51
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    $\begingroup$ The equation should be $1.04c = 60p$ (and then you get $p$ from $1.04\cdot 200 = 260p+180(1-p)$) $\endgroup$
    – spaceisdarkgreen
    Commented Jun 14 at 17:07
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    $\begingroup$ @Claptar I invest $c$ (the price of the option now). My expected value in the future is $60p$ (if the stock increases I end up with 60, otherwise nothing). A risk neutral return on that same amount $c$ would leave me with $1.04c$ in the future. (And same logic for investing in the stock to solve for $p$). You can also solve this by replicating the option by buying some quantity of stock and borrowing some amount of money risk-free. $\endgroup$
    – spaceisdarkgreen
    Commented Jun 14 at 17:12
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    $\begingroup$ @spaceisdarkgreen Oh, I see where I felt for the confusion, thanks. If I want to say that $Y$ is the amount of money I make is $Y$ then I should say that $E(Y) = 0.04c$. $\endgroup$
    – Claptar
    Commented Jun 14 at 17:21

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If I say that $Y$ is an amount of money you make if you buy an option.

$$ Y = \begin{cases} 60 - c, & \mbox{with probability } p \\ -c, & \mbox{with probability } 1 - p \end{cases} $$

Then $E(Y) = 0.04c$, as $0.04c$ is the amount of money you make if you invest it risk-free. Therefore a fair stock price would be:

$$ E(y) = p(60 - c) - c(1-p) = 0.04c. \\ c = \frac{60p}{1.04} $$

To find $p$ we assume that the price of the stock doesn't change in 1 year, so $E(X) = 1.04 \cdot 200$. Therefore $p = 0.35$ and $c = \text{\$}20.19$.

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