Interpreting Portfolio Cost in a Replicating Option Hedge
Summary
This document raises a sign-convention question about the current cost of a levered hedge in the Cox–Ross–Rubinstein option-pricing framework. The portfolio consists of written calls, purchased shares, and borrowed funds. Its payoff at expiration is described as breaking even across possible stock prices, leading the paper to use the present-value expression for the hedge when applying the no-arbitrage condition.
The central distinction is between the cash flows from entering positions and the value or cost assigned to the combined portfolio. Buying shares and borrowing cash contribute with opposite signs from writing calls, depending on whether the expression represents the portfolio’s net cash received or the amount paid to establish it. The question highlights why a consistent sign convention matters when interpreting a zero-cost replication argument. The document contains the question and quoted setup, but no answer or broader derivation, so it does not fully resolve the convention or discuss transaction costs and other market frictions.
Key ideas
- A hedge can combine written options, stock holdings, and borrowing.
- A payoff that breaks even at expiration is used in a no-arbitrage argument about its current value.
- The signs in a portfolio expression depend on whether it records cash received or cash paid.
- The document poses the sign-convention issue without providing a full derivation.
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# Confusion about "cost" in option pricing paper by Cox-Ross-Rubinstein paper # Confusion about "cost" in option pricing paper by Cox-Ross-Rubinstein paper 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?
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.