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I'm reading an article trying to derive option pricing with a simple approach, but I got stuck. In the second paragraph of this article (Name – Options Pricing: A Simplified Approach), which takes just about 2 minutes to read, I got stuck trying to understand why you would want to take a loan or lend money while trying to replicate this portfolio with a hedge.

When looking at this I understand that you need the call obviously since it serves as the upside for a downfall in price and on the other hand the investment (buying to shares) serves as the upside towards an increased price.

I thought the point would be to show that these parts cancel out and thus one could set an option price. But that's not the whole story!

No, instead it seems like you always end up with +10 dollars, whatever the case. Which can't be. Thus, they've included a loan of 40 bucks with an interest rate of 25% (10$) which gives the total result of 0 in both cases.

But why would you just take a purposeless loan of 40 bucks? Anybody understands that you could just exclude it and thus save 10 bucks, whatever the outcome; just by investing the money needed from your own account.

So what am I missing? Why the loan and how come you end up with a profit whatever the case if you exclude the unnecessary loan since that contradicts the no-arbitrage?

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    $\begingroup$ When you do the replication of the option, you need a cash position to fund your buys and sells of the stock position. $\endgroup$ Commented May 25 at 2:34

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More generally, in finance almost all replication arguments always assume that you have no cash to begin with (usually also there are simplifications such as assuming that there is no credit risk and you can borrow/lend cash frictionlessly).

In this case assume that the price of the Call is 20USD, you sell 3 of these, and you buy 2 shares at 50USD each. Your day one cash position is 60 - 100 = -40. You need to borrow that 40USD (to get your cash position back to flat) to fund yourself (and that typically happens at a non zero rate). Therefore, you need to factor in the cost for borrowing this amount to term in the valuation of the option.

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  • $\begingroup$ I see! But doesn't that put people with cash to begin with at a competitive advantage? Or is that 25% just supposed to represent the fact that there's a riskless rate of return, implying that for this hedge to come out at zero? it needs to come out a plus by the riskless rate or return (e^r) since it's always somewhat of a plus-side to have the cash during the period. @user68819 $\endgroup$ Commented May 25 at 20:41
  • $\begingroup$ Goes back to my first statement. You should always start with no cash, if you have a surplus of cash it needs to be invested. Take a simple example, you short sell a share..you would borrow the share, sell it for $cash, use this cash to collateralise the borrow of the share,borrow the share, deliver the share, net you end up with no cash on hand, but you do have a short position in the share. $\endgroup$
    – user68819
    Commented May 25 at 21:02
  • $\begingroup$ The arbitrage arguments made, don't depend on how much cash any particular person has. They are general and can be executed with someone with no cash on hand (in an ideal world) $\endgroup$
    – user68819
    Commented May 25 at 21:05
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I don't really have much to add to the other comments, but I just want to emphasize the general principle of opportunity cost and the intuition behind it.

You should only invest in something if you can get a (risk-adjusted) rate of return equal or better to what you can get elsewhere. In a world of no-arbitrage and efficient markets, we don't expect to find opportunities with "better" return, but we need to make sure we're at least as well of as the status quo.

It therefore doesn't matter whether you have the money yourself initially, or whether you take a loan. Having the money yourself should not change the price you are willing to provide your services for. Can I just add this principle holds in every single aspect of running a business in general and not just for derivatives pricing.

Using the example and as user68819 pointed out, there is a cash shortage of 40 USD that needs to be plugged from day one. Think of this as an investment opportunity.

Imagine you have this money already. It is probably sitting in an account and is earning the risk free rate of return, so you expect this to earn 10$ a year with the given example.

So in order for you to want to "invest" in providing the option, you have to make sure the 40 dollar cash you provide is being remunerated at at least the same rate as it already does sitting in the bank account, ie. $10 anually.

Otherwise you are effectively giving someone else $10 and you are running a charity for your own money and would be better off just leaving it in the bank account.

Therefore derivatives pricing (and literally the pricing of anything, even bubble gum in the convenience store) should be priced pretending you're borrowing the money to begin with.

That way you make sure that 1) your asset earns enough to cover the funding expense and 2) if you provide the money yourself, that you effectively are paying yourself interest on your own money commensurate with what it would earn doing nothing in a bank account.

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  • $\begingroup$ Good point: in practice the arbitrageur would use his own money, but in evaluating the P&L it is important to take into account the opportunity cost and a good way to do this is to pretend that you are borrowing from someone else and have to pay interest. It is an assumption made for accounting purposes if you will. $\endgroup$
    – nbbo2
    Commented May 29 at 9:00

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