The Call Option Lower Bound and Stock Shorting Costs

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


# 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:

- Short the stock (cash inflow: \$20).

- Buy the call (cash outflow: \$3).

- Invest the remaining cash (\$17) at the risk free rate for 1 year, which will grow to \$18.79 in 1 year.

- 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.

- 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?

## Answer by ramoncelma (score 1)

https://quant.stackexchange.com/a/51782

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.