*This article was shortlisted for the Plus New Writers Award 2006, an international competition pitting mathematical writers against each other in a fight to the death. The challenge was to write about a mathematical topic in an informative and entertaining way. Okay, so I didn't get the grand prize. But I did get the certificate.*

You may have heard stories that mathematicians are in demand in the financial industry. You may also have heard that they work with stuff called **derivatives** that apparently only have two purposes, one is to make people rich and the other is to destroy banks. Perhaps you've looked at a book about financial mathematics that was lying around, read a few pages and then thrown it in the corner because none of it seemed to make any sense.

Today, we're going to clear up some misconceptions and talk about the field of **quantitative finance**.

### What Makes a Price a Price

Bill makes a living selling tasty roasted peanuts from a street stall at 75p a bag. Ted, on the other side of the street, is buying peanuts at 90p a bag for his own little export business. Both of them are partially-sighted and can't see each other.

Your mother wouldn't be proud of you, but what the heck, you decide to take advantage. You buy peanuts from Bill for 75p and sell to Ted for 90p and make 15p profit. That's fabulous. Well done.

This is known as **arbitrage**, making a profit without risk. It's like free money. You're exploiting a pricing discrepancy in the marketplace. Bill and Ted are trading at prices so wide that it's a piece of cake to make a profit. But what's stopping you doing this again and again? Why can't you become rich on this scheme? The reason is that your transactions carry information between the two street vendors.

It's simple economics. If you buy lots of bags from Bill he'll realise that peanuts in demand and increase his price. If you sell a lot to Ted, he'll notice that peanuts are not as rare as he might have thought and reduce his price accordingly. With Bill raising his price and Ted dropping his... eventually, the prices will equalise. At that point, arbitrage vanishes in a puff of smoke. You go home and order a shiny new games console with your profits.

In the real world, there's not just Bill and Ted but a global army of arbitrage traders glued to their screens, looking for small price discrepancies where none should exist. Transactions are executed at breakneck speed, arbitrage opportunities appearing and disappearing within the blink of an eye. You and I sitting at home would never be able to catch such an opportunity, so I'm pleased you've got yourself the games console to pass the time instead.

This is related to the **Efficient Market Hypothesis (EMH)** that states that *market prices accurately represent all of the available information*. If any information is currently known about what might happen in the future, say, Bill's Peanut Emporium is susceptible to a hostile take-over by Napoleon's Praline Delights, the price on the market will already reflect this knowledge. The EMH is a separate topic in its own right, however, and we will say no more about it.

Importantly, though, the idea that **prices are arbitrage-free** is one of the keys to understanding quantitative finance.

### The Future is Peanuts

Ted gives up his export business and buys a local pub, because he discovers he's got a peanut allergy. However, he is distressed to learn that his punters love peanuts and Billy the Kid's bar down the road has peanuts aplenty. Ted makes the decision to offer peanuts to all of his customers free of charge to keep them sweet, or perhaps to keep them salty. Peanuts aren't free for Ted, of course, and he needs to be sure that the price of peanuts is not going to overwhelm his business.

He's got enough peanuts for this year but he'll need more for next year. He doesn't have the cash or space to buy next year’s supply, but he is worried that peanut prices will have gone through the roof by then. He finds out that there is a nice guy called Rufus who will sell him peanuts in a year’s time with a price agreed here and now. Ted’s bank manager is cool with this, because one uncertainty has been eliminated from Ted's business outlook. Rufus has taken the peanut price risk away.

The agreement Ted has made with Rufus is called a **futures contract**. It is one of many financial products called **derivatives**: financial products that derive their value from another financial quantity. Although our situation with peanuts is not particularly realistic, consider if Ted actually ran a plastics business – he would be concerned about the rising price of oil. It is important to appreciate that the primary use of derivatives is to **limit uncertainty**.

Now the real question is, how does Rufus know what price he should sell the peanut futures at? Does he travel into the future to find out? Does he just quote a ridiculously high price and walk away from it? It turns out there is a way to work out the correct price for the contract.

### Time is Money

The important thing is to remember arbitrage. If profit can be made without taking any risk, then the price is wrong. Rufus sits down in his office and observes that peanuts are currently selling at £1 a bag on the open market. He also notices that the current interest rate at the local bank is 10% a year. (As an aside, be aware that calculating interest is also something that requires some care as covered in an earlier *Plus* article [1].)

Suppose Rufus sells a peanut futures contract for 1 bag in a year's time at £1.05. Bill, who owns a bag of peanuts, could do the following. He sells a bag immediately for £1 and then puts the money into the bank for a year. He also buys a peanut futures contract from Rufus to be settled in one year's time for £1.05. After the year is up, the bank gives him £1.10 that he uses to pay for the peanut futures contract. So Bill is left, after one year, with a bag of peanuts and 5p. For no risk whatsoever, he has made 5p profit. There is clearly arbitrage possible here, so Rufus should not sell at this price.

What if Rufus sells peanut futures at £1.15? Well, Rufus could make a tidy profit himself out of the deal. He could borrow £1 from the bank, agreeing to pay it back with 10% interest at the end of the year. With this cash, he buys a bag of peanuts and sells a peanut futures contract to Ted to settle in a year at £1.15. In a year's time, he receives £1.15 for the bag of peanuts and pays back £1.10 back to the bank. Oh my gosh! He's made 5p with no risk to himself. Arbitrage is in evidence again.

In both schemes, profit was made from the difference between the futures price F and the value of the current price S after being invested in the bank for one year. If the annual interest rate is r then the arbitrage profit can be written

Arbitrage Profit in One Year = | F - S(1 + r) |.

To eliminate arbitrage, this profit must be zero. So the correct futures price is

F = S(1 + r). (1)

In our example, Rufus should price the futures contract at £1.10 per bag. This brings us to another important concept in finance.

Rufus knows all about the **time value of money**. Rufus knows that £1 today is not £1 in a year’s time, it’s actually £1.10. It should, at the very least, have earned interest in the bank. Letting cash rest in the bank is considered to be a risk-free course of action and it means that we should always consider cash to be rising in value at the **risk-free interest rate**.

The **future value** (FV) of some cash in a year’s time, therefore, is equal to its **present value** (PV) increased by the risk-free rate. This is summarised by the formula

FV = PV(1 + r). (2)

Compare the formula for the futures price (1) to the definition of FV (2). We can see that the peanut futures price is simply the *future value of the current price*. This price is *not* a prediction of the price of a bag of peanuts in the future.

### And After This, It Goes Nuts

These are the basics and we've ignored some complications. Is there a cost to storing peanuts that we should take into account? Are there costs involved by making futures transactions? What if the borrow and lend interest rates are different? What happens when interest rates change over time? What if it's not peanuts but a stock that pays dividends?

The meat of quantitative finance, though, is the ability to price more complicated products. Suppose, instead of a futures contract, Rufus offers what is called an **option**. In this contract, Ted has the *option* to buy peanuts at a set price, called the **strike price**, in a year. Now, if the peanut price on the market happens to be lower than the strike price Ted can discard his agreement with Rufus and buy direct from the market.

That sounds great for Ted, but Rufus needs to come up with a price for the option. The same arguments for the futures contract do not apply here. If the market price of a bag of peanuts is lower than the strike price at the end of the contract then, as Ted doesn't actually buy from Rufus, the future value of the contract is *zero*.

One approach is to look at the possible price movements over a year and work out the future value of the option in each case. From this, you can find the average future value and calculate the corresponding present value using (2).

However, there is a known formula for the option price. Robert Merton, Fischer Black and Myron Scholes came up with the hideous-looking **Black-Scholes formula** [2]. If you want to see it, take a look at the relevant page on Wikipedia [3]. It may be hideous, but Merton and Scholes received the 1997 Nobel Prize in Economics in part for this discovery.

It should be noted that the Black-Scholes formula is limited in scope. In general, there is no convenient formula to find the price of a financial product and numerical algorithms must be used instead.

### So Long and Thanks for all the Peanuts

I’m afraid that’s all we have time for.

It should now be clear that the primary function of quantitative finance is not to predict the markets nor determine why a butterfly flapping its wings in South Kensington could drive an investor in the Outer Hebrides to buy peanuts. It is to accomplish something far more humble and deceptively simple: **determine a fair price**.

If you're interested in pursuing this subject you’ll need to have a good grasp of statistics and calculus. Statistics are important because derivative pricing is all about probabilities. Calculus is vital because partial differential equations are what you can end up with after the smoke clears. (You can read more on derivatives in an earlier *Plus* article [4].)

Good luck. Bill and Ted are in need of your help.

### Bibliography

[1] Webb, J (2000). “Have we caught your interest?” Plus 11, June 2000. (http://plus.maths.org/issue11/features/compound/)

[3] Wikipedia. “Black-Scholes”. (http://en.wikipedia.org/wiki/Black-Scholes)

[4] Dickson, J (2001). “Rogue trading?“ Plus 16, Sep 2001. (http://plus.maths.org/issue16/features/derivatives/index.html)

### The Author

*...also writes fiction on the site Hammerport where he pretends to be psuedononymous. These stories have nothing to do with mathematics. Usually.*