# Why is there a theoretical lower bound on the price of call options?

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?

• 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. – mark leeds Mar 23 '20 at 6:25
• @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. – Flux Mar 23 '20 at 6:49
• If the trader buys it back at a higher price than it was when he lent it, then he lost $in the transaction. If he buys back at a lower price than he profited. As far as breaking the law by not returning to owner, I'm not familiar with the rules regarding that. The key thing to keep in mind is that the LENDER is still profiting on the appreciation or losing on the depreciation so he still "owns" the stock. So he doesn't get the risk free rate. The risk free rate compensates for opportunity cost ( the money you lent out could have been put in bank ) which is non-existent in this case. – mark leeds Mar 24 '20 at 1:26 • As you say, it is a "theoretical minimum". In practice, this is very close to reality if you are a large derivatives dealer and you consider blue chip stocks, if which case the stock borrow rate can be very close to$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. – Daneel Olivaw Apr 8 '20 at 11:57