Suppose you are a bank director. A customer enters your office and states that she would like to contract with you to
purchase five hundred shares of a certain telecom company ...

This is an article from the book "Five-minute mathematics" by Ehrhard Behrends which was published in 2008 by the American Mathematical Society (AMS). It is reproduced here with the kind permission of the AMS.

Suppose you are a bank director. A customer enters your office and states that she would like to contract with you to purchase five hundred shares of a certain telecom company next January 1. She says that she hopes and expects that the share price then will be at most 20 euros. If the price is greater, she would like you-the bank-to make up the difference.

These days, such arrangements are nothing unusual. They are called options. For the customer, the security that the proposed contact offers is not to be had for nothing. When the contract is signed, she will pay a certain premium. What should you do with this money in order to be able to fulfill your contractual obligations come January 1?

The magic word that allows this problem to be solved is hedging. In the mathematics of finance, hedging your bets amounts to the clever insuring against risk.

From the dictionary:

hedge: 1. A row of closely planted shrubs or low-growing trees forming a fence or boundary. 2. A line of people or objects forming a barrier: a hedge of spectators along the sidewalk. 3. a. A means of protection or defense, especially against financial loss: a hedge against inflation. b. A securities transaction that reduces the risk on an existing investment position. 4. An intentionally noncommittal or ambiguous statement.

hedgehog: 1. Any of several small insectivorous mammals of the family Erinaceidae of Europe, Africa, and Asia, having the back covered with dense, erectile spines and characteristically rolling into a ball for protection.

The underlying idea is simple and quite clever. In addition to the money received from your customer, you borrow money at market rates and use it-customer's money plus borrowed money-to purchase telecom shares.

And to what purpose? If the share price rises by January 1, they will be worth enough for you to be able to pay what you owe the options holder together with the money that you borrowed plus interest. If the price falls, that is a pity, but then the options holder has no demand against you, and the proceeds from the sale of the stock should suffice to pay back what you borrowed.

In sum, to insure against loss in a telecom stock transaction, some of the stock is purchased. No matter how things develop, you have placed a bet on both a rise and a fall in the stock price.

Mathematics comes into play in the determination of a fair price for the option, and how much stock the bank should purchase. Based on the "natural law of financial markets", namely, that there can be no profit without risk, the amount in question comes from solving a simple equation. What makes things complicated is that one must keep track of the markets throughout the option period. As stock prices and interest rates change, one must determine whether some of the stock should be sold or more money borrowed for an additional purchase.

## A Hedge for One Thousand Shares

Let us look at hedging through a concrete example. It is January. You would like to take out an option for the purchase of one thousand shares of Intergalactic Enterprises at the end of the year. If purchased today, the cost of the shares would be 10 000 euros. However, the price at the end of the year is uncertain. It could be 16 000 euros, but it could just as well be only 8000, depending on a host of factors. (For the sake of simplicity, let us assume that one of these prices will obtain at year's end and that we will do no trading during the year.) Come December, you will have 12 000 euros available, and if the price is 8000 euros, all will be well. However, if the price is 16 000, you would like the bank to make up the difference. How much should the bank ask you to pay for such an insurance policy, and what should it do with the money that you pay?

The bank officer from whom you are seeking to purchase the option calls down to the credit division and learns that the bank's internal interest rate is 6%: for a distribution of E/1.06 euros today, one has to pay back E euros at year's end. With this information, the options contract can be drawn up. The bank requires you to pay, in return for their guarantee,

 5000? 4000 1.06 ? 1226

euros. This number does not include the bank's fees, which are where its profit is realized. We are ignoring this issue.

You sign on the dotted line, and this is how things then develop. The credit division immediately turns over 4000/1.06 ? 3774 euros to the bank officer, who now has available 1226 + 3774 = 5000 euros. She uses this money to purchase 500 shares of Intergalactic, and then forgets about the whole matter until December.

Suppose that the shares have risen. The bank's portfolio for this transaction is now worth 8000 euros (recall our assumption that if the stock rises, one thousand shares would be worth 16 000 euros; the bank purchased only 500). The bank pays you the contracted 4000 euros, and with the 12 000 euros that you had budgeted, you can now purchase 1000 shares of Intergalactic for 16 000 euros. With the remaining 4000 euros, the bank officer repays the credit division.

If the share price has fallen, then the bank's portfolio is worth only 4000 euros, which is just enough to settle with the credit division. You, the option holder, get nothing, since your 12 000 euros is more than enough to purchase the shares.

The moral: with a hedging strategy, you were able to insure a risk of 4000 euros relatively cheaply (1226 euros), since 4000 is the amount you would have been short if the stock price had risen.