From my textbook, I see that the theoretical lower bound for the price of a European call option on a non-dividend-paying stock is:

$S_0 - \mathrm{Ke}^\mathrm{-rT}$, where $S_0$ is the current stock price, $K$ is the option strike price, $r$ is the risk free rate, $T$ is the time.

According to the formula above, if the current stock price is \$20, strike price is \$18, risk free rate is 10%, time is 1 year, then the theoretical minimum call option price is $3.71. As the theory goes, if the option price is below \$3.71 (say, \$3), it will be possible for an arbitrageur to:

  1. Short the stock (cash inflow: \$20).
  2. Buy the call (cash outflow: \$3).
  3. Invest the remaining cash (\$17) at the risk free rate for 1 year, which will grow to \$18.79 in 1 year.
  4. If in 1 year, the stock price is above the strike price of \$18, the trader exercises the call option and uses that to close the short, and gains \$18.79 - \$18.00 = \$0.79.
  5. Otherwise, if the stock price is below the strike price of \$18, the trader buys stock at the market price to close the short. If the market price happens to be \$17, the trader gains \$18.79 - \$17.00 = \$1.79.

But all this makes no sense because shorting the stock (step 1) requires the trader to pay interest to whomever the trader is borrowing from. This interest is probably greater than the risk free rate. Given that all the steps above are impractical as a result of the interest that needs to be paid when shorting stocks, why is there a theoretical minimum in the first place? It seems to make the bad assumption that one can borrow stock at 0% interest.

What is wrong with my understanding of the issue?

  • $\begingroup$ Hi: The trader, who's shorting is borrowing a stock, not cash. So the rate is nowhere near the risk free rate because, in a way, who ever lent it to the trader, Mr. Lender, still owns it. Mr. Lender just signed a piece of paper saying that the trader can do what he wants with it temporarily. But the trader still has to give it back to Mr. Lender at some point in the future. The point is that Mr. Lender still experiences all the depreciation or appreciation that he would experience if he didn't lend it out. For this reason, he doesn't get the risk free rate from the trader. $\endgroup$ – mark leeds Mar 23 '20 at 6:25
  • $\begingroup$ @markleeds Why is rate for borrowing stock lower than the risk free rate? Isn't there a risk that Mr. Trader will not be able to return the stock to Mr. Lender? For example, Mr. Trader sold the borrowed stock, but is now not able to afford to re-buy the stock because the stock price has gone up. Mr. Trader then has to default. $\endgroup$ – Flux Mar 23 '20 at 6:49
  • $\begingroup$ If the trader buys it back at a higher price than it was when he lent it, then he lost $ in the transaction. If he buys back at a lower price than he profited. As far as breaking the law by not returning to owner, I'm not familiar with the rules regarding that. The key thing to keep in mind is that the LENDER is still profiting on the appreciation or losing on the depreciation so he still "owns" the stock. So he doesn't get the risk free rate. The risk free rate compensates for opportunity cost ( the money you lent out could have been put in bank ) which is non-existent in this case. $\endgroup$ – mark leeds Mar 24 '20 at 1:26
  • $\begingroup$ As you say, it is a "theoretical minimum". In practice, this is very close to reality if you are a large derivatives dealer and you consider blue chip stocks, if which case the stock borrow rate can be very close to $r$. But things get more messy for thinly traded or illiquid stocks. I disagree that the rate to borrow a stock is below the risk-free rate, this is dependent on many factors, among which whether it is an easy to borrow stock. I've seen stock borrow rates close to 50% per year. $\endgroup$ – Daneel Olivaw Apr 8 '20 at 11:57

For shorting a stock what you would do is to have a margin account with your broker. As an example, in the American jurisdiction, according to Regulation T from the Federal Reserve, you would provide a 150% of the value of your position as initial margin (50% of additional marging). And the daily margining would be done against your margin account (both incremental and excess margin). You would also earn the margin interest rate associated with your margin account, the one that the broker agrees with you (close to the overnight interest rate).

So, in a practical view and thinking about the margining in the short positions, there is not a specific interest rate to be applied to the short position. And, after all, if there was a way in which the shorting wasn't done by a margin account and it was produced through a interest rate to be paid to the broker, for me it would be taken as a transaction cost rather than a rate to compare to the risk free rate. The lower bound you are analysing as well as most of the academic theoretical formulas is ignoring all the transaction costs.

In practice, the short position doesn't entail cost by itself (you can have general costs associated with the agreement you have with the broker you are operating with, on an annual basis for example). You could think that if you need to provide 30 dollars (150 per cent of 20) you would have a negative cost of opportunity in having those 30 dollars blocked from the time the short position is opened, but you are also getting the brokers margin interest (close to overnight interest rate) for that as a compensation.

  • $\begingroup$ Mr. Broker pays a margin interest to Mr. Lender to compensate Mr. Lender for tying up his stock for the duration that the short position is opened. Mr. Broker also pays an interest to Mr. Trader for the funds in his margin account. So what does Mr. Broker gain from all this? It looks like Mr. Broker always loses. What are the incentives for Mr. Broker to offer short selling services in the first place? $\endgroup$ – Flux Mar 24 '20 at 3:00
  • $\begingroup$ Mr. Lender doesn't even know that his Stock Y is in a short position. Mr. Broker does this management by his own risk, as a broker's mission is to provide liquidity. If Mr. Lender sometime decides to sell the stock, Mr. Broker uses Stock Y from other clients, and if he cannot do this, he can then force Mr. Trader to close the short position. Mr. Lender doesn't need to receive an interest, he is not even aware of the short position. Mr. Broker is proving him the agreed service and Mr. Lender is being able to buy Stock Y thanks to that agreed service. $\endgroup$ – ramoncelma Mar 24 '20 at 16:25
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    $\begingroup$ Now, if you wonder what does Mr. Broker gain from all of this... Mr. Broker is receiving broker fees, it can be at transaction level (each time Mr. Trader and Mr. Lender perform a transactions through Mr. Broker, they are charged for this). It could also be implicit in the bid-ask spread, that in the theoretical lectures like the one you are reading is ignored. Also brokers charge service fees to their clients with a specified frequency (monthy, annual,...). Also brokers offer financial analysis services to their clients. $\endgroup$ – ramoncelma Mar 24 '20 at 17:18
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    $\begingroup$ And lastly, when I said that Mr. Broker is paying interest on the margin accounts that their clients have with them, it is not interest that Mr. Broker is paying directly from his pocked, this is interest that Mr. Broker is getting from going with the margin account money to the Money Market and invest it in overnight transactions. And the rate earned by the margin account owners is less than what Mr. Broker is earning from that. Lastly, Mr. Broker is able to operate in market exchanges because they are members of the exchange, and sometimes the exchange also pays them. Hope it helped :) $\endgroup$ – ramoncelma Mar 24 '20 at 17:26

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