# The Concepts and Practice of Mathematical Finance Training Course

## 1.1 What is risk?

It is arguable that risk is the key concept in modern finance. Every transaction can be viewed as the buying or selling of risk. The success of an organization is determined by how much return it can achieve for a given level of risk. Before we can justify these statements, we need to achieve some understanding of what risk is.

In a typical pure mathematical ploy, let us start by trying to understand the absence of risk. A riskless asset is an asset which has a precisely determined future value. Do such assets exist? The fundamental example is that of a government bond. We can buy a government bond for say a £100 today and know that we will receive say £5 a year, (called a coupon payment), until a pre-determined date, when we receive our £100 back.

Is this asset truly riskless? There is of course a possibility that the government will renege on its promise to pay. (This is known as defaulting.) But if we pick the right government this possibility is sufficiently remote that we can for practical purposes neglect it. If this seems unreasonable, consider that if the British, American or German government reached such straits, the world’s financial system would be in such a mess that there would be precious few banks left to employ financial mathematicians.

In fact, the existence of such riskless assets is so fundamental both to financial mathematics and to the modern finance industry, that the fiscal policy of the American and British governments of running budget surpluses, and therefore reducing the number of bonds they have issued, caused great consternation.

The reader who is tempted to chuckle at the predicament of the finance industry should consider that financial institutions fund pensions by buying long-maturity government bonds and using the interest coupons to pay the pension. The shortage of long-maturity bonds therefore makes pensions harder to fund and ultimately results in smaller pensions.

We can now define a risky asset to be an asset which is not riskless. That is it is an asset of uncertain future value; risk can be regarded as a synonym for uncertainty. The most basic example of such an asset is a share of a public limited company and we shall return to this example again and again. However, it is important to realize that almost anything except a riskless government bond is such an asset. For exam- ple, we could hold foreign currency and be exposed to the risk that the exchange rate will change against us, or we could buy a flat in London and be exposed to the possibility that there is a property crash, as occurred in the early 1990s.
The sharp reader will have noted that the definition in the paragraph is not quite right, in that an investor would not actually care about the riskiness of an asset if the worst possible future value of the asset was greater than today’s value.
However, we have to be slightly careful about what we mean by value here. Unless there is no inflation, £1 a year from now will buy less than £1 today. This means that £1 a year from now is effectively worth less than £1 today. In addition, even in a non-inflationary world, most people prefer jam today to jam tomorrow and so would not be happy to receive the same amount of money back in a year with no compensation.

A better view of riskiness is that the asset can return less than the same amount invested in a riskless government bond for the same period. A good example of such an asset is the premium bond. In the United Kingdom, one can buy a government bond, called a premium bond, redeemable at any time which pays no coupon but instead the holder gets a free entry in a prize draw paying up to a million pounds a month.

This seems too good to be true at first, but the issue is, of course, that the bond is not very different from investing some money and using the interest to buy lottery tickets. That said, the expected winnings for the amount of interest foregone is much better for premium bonds than for lottery tickets. The investor is effectively buying risk.

## 1.2 Market efficiency

Before we can understand why risk is so important we first have to understand the concept of market efficiency, which underlies most of financial mathematics and modern economics. This concept roughly states that in a free market, all available information about an asset is already included in its price. Therefore there is no such thing as a good buy – the only value an asset has is its market value and it is meaningless to attempt to think otherwise.

Is this hypothesis correct? To see that it cannot be wholly so, consider the apocryphal story of the two economists who see a ten dollar bill lying in the gutter. The first one goes to pick it up but the second one tells him not to be silly as if it were real, someone else would have already picked it up.

However, the true moral of this story is that market efficiency only works if someone does not believe in it – the first person who does not believe in market efficiency picks up the ten dollar bill and thereafter it is no longer lying in the gutter. Warren Buffett is the most famous example of a non-believer who has very effectively made a lot of money through his disbelief.

He has largely done so by buying shares in companies he believes are undervalued by the market. Indeed, until Bill Gates overtook him he was the richest man in the world, and he made his money by beating the market. So although market efficiency is not wholly correct, there are enough people attempting to be the next Buffett, for it to be sufficiently correct that we can work under the assumption that it is true.

What does this mean for us? Well, the first thing it means is that it is pointless to for us to try to predict the future price of a share by looking at graphs of its past prices. All this information is already encoded in the share price. This is sometimes called the Markov property, and is also called the weak efficiency of markets as it’s a consequence of the strong form mentioned above.

It is interesting to note that the modem white-collar crime of insider trading is really based on the principle of market efficiency. Insider trading is trading based on knowledge which is not publicly available and therefore not included in the share price.

For example an employee of a company might know that the company was about to announce unexpectedly large losses or profits which would move the share price in an obvious direction, and take advantage in advance. The perception of this as a crime rather than a natural action is fairly recent, and is based on the ubiquity of the concept of market efficiency.

Given that all assets are correctly priced by the market, how can we distinguish one from another? Part of the information the market has about an asset is its riskiness. Thus the riskiness is already included in the price, and since it will reduce the price, the value of the asset without taking into account its riskiness must be higher than that of a less risky asset.

This means that in a year from now we can expect the risky asset to be worth more than the less risky one. So increased riskiness means greater returns, but only on average – it also means a greater chance of losing money. From this point of view, an asset’s price reflects the value it is likely to have in the future reduced by a factor depending upon its riskiness.

To illustrate these ideas, let us consider a simple game. Suppose we toss a coin, if it comes up heads I give you £3, if it comes up tails you give me £1. Unless beset by moral qualms, you would consider this game a very good deal and play it – your expected winnings would be:

and your maximum losses would only be 1. Suppose we play a slightly different game, I pay you £13 on heads, you pay me £1 ]. on tails. Your expected winnings are still £l but are you still so keen to play? If not why not? If you are still keen, let’s take the payment on heads to be £103, and on tails to be £1.01.

At some point, when the stakes become high enough you will stop regarding the game as a good deal. The point where you stop depends upon personal risk preferences; the stopping point is where the expected gains stop outweighing your aversion to the possibility of losing money.

Now suppose the game is changed a little again. The sum you lose is paid to me today and we toss a coin a year from now. If the coin comes up heads, I return your money to you and pay you my losses, otherwise I keep your money. What has changed? During the year in between, I have put the money on deposit with a bank and earned some interest.

If you were not playing the game you could have done so also. The amount of return you would want from the risky game would increase to express the interest foregone. And since you could have made money from the interest payment without taking any risks, you will demand that the expected winnings be greater than the amount of interest you could have earned.

The moral is that there is no such thing as a guaranteed high return. The reader would be well-advised to remember this the next time he sees a guaranteed high return in a newspaper or Internet advertisement.
Let’s return to the concept of weak market efficiency. This says that all the past movements of an asset’s market price is already expressed by today’s price.

At this point, the prospects of a financial mathematician could be regarded as being pretty bleak. Why? This tells us that trying to predict the future price from past data is a waste of time – there is no periodicity nor trends to be read. The only mathematical information is today’s price which tells us very little. In that case, why is financial mathematics a burgeoning field?

The job of a financial mathematician is not to predict prices but instead to relate the movements of price in one asset to that of another. These price movements are viewed as being driven by information arriving in the market and since that information is by definition unknown until it arrives, we can view it as being random.

The key point in mathematical finance is to use market instruments which are affected by the same information in such a way as to cancel out randomness. This process is called hedging. The objective of mathematical finance is to understand how to do this and to understand the consequences.