I am trying to understand the paper "Option Pricing: A Simplified Approach" by Cox-Ross-Rubinstein (available online here).

To my frustration, I already don't understand the paper starting from just page 3. It is said there:

Consider forming the following levered hedge: (1) Write three calls at $C$ each, (2) buy two shares at $\\\$50$ each, and (3) borrow $\\\$ 40$ at $25\%$ to be paid back at the end of the period. Table 1 gives gives the return from this hedge for each possible level of the stock price at expiration. Regardless of the outcome, the hedge exactly breaks even on the expiration date. Therefore, to prevent profitable riskless arbitrage, its current cost must be zero; that is, $3C - 100 + 40=0$.

I take it from this that, apparently, the "cost" of this is equal to $3C - 100 + 40$.

I don't understand this. I always thought that the "cost" of something is what one would pay for it, that is, its value. In this case, it seems to me that the cost (the value) would therefore be equal to $-3C +100 -40$ (since e.g. we are buying two shares at $\\\$50$, so that would cost us $\\\$100$ for those two). This is exactly the negative of what is stated above. Of course, in this particular case it happens to be equal to zero anyway, so it doesn't matter, but the point is that the suggestion is made that the cost is equal to $3C - 100 + 40$. Thus, either this is indeed the cost, and what I thought to be the case is simply false and I'm insane; or, what is stated is wrong.

In the table right below the above text (see Table 1 in the paper), for the present date "$3C - 100 + 40$" is mentioned, whereas for expiration date the value of the portfolio (for instance in case that $S^* = 100$ it is $-150+200-50$) is mentioned.

What is meant by "cost" here? (It is, of course, undefined in the paper--who would need a definition of something so important?) It makes no sense to me at all. Could someone clarify this to me?

  • 1
    $\begingroup$ Hi: In the paragraph that starts with "Consider", it is explained that they want to create a hedge so that, no matter what happens to the stock price, they end up breaking even. So, when they refer to cost, they are referring to the cost of creating that hedge. In order to create the hedge, they need to write 3 calls ( which means they receive C 3 times ), spend 100 for the 2 shares and then borrow 40 ( so gain 40 ). So, that's where the 3C - 100 + 40 comes from. It's the value of what they have to do to create the hedging strategy. $\endgroup$
    – mark leeds
    Jun 3 at 5:48
  • $\begingroup$ I didn't check out the link you provided but if you want examples that might be more straightforward than the one you describe, the cox and rubenstein text has many. I haven't looked at it in probably 25 years but I remember that being the text that really opened my eyes as far as what equity options were. I don't know if it is still viewed as the bible on options pricing but I really liked it and it was the bible at one point in time. $\endgroup$
    – mark leeds
    Jun 3 at 5:52
  • $\begingroup$ It is just a sign convention, they assign a negative cost to an out of pocket expense, positive to a cash inflow. $\endgroup$
    – nbbo2
    Jun 4 at 2:12


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