1 Introduction
This course is concerned with a particular financial problem. One of the main economic roles of the financial sector is to facilitate the transfer of economic risks. This takes many forms, some of which you may have come across, others you probably haven’t. For example, most people choose to buy insurance to cover themselves against risks: for example, the risks of crashing a car or being burgled, and you can also purchase insurance to cover illness, unemployment, and many other occasional losses. Today, you can even ‘buy’ insurance on your insurance, in the form of ‘guaranteed’ no claims bonuses.
Other (financial) risks that many people are exposed to relate to borrowing costs — one day, you may choose to take out a mortgage, and if you do, you will be expected to choose whether the interest payments you make will be at a fixed rate (say 5% of the borrowed amount each year), or at a ‘variable’ rate, which will typically depend on the rate at which the bank can borrow money from customers or other financial institutions1. As someone who takes out a mortgage, it might be necessary to trade off the (possibly) riskier, unknown ‘variable’ rate against the fixed rate, to decide which the borrower might prefer. Different borrowers should have different preferences — typically, a variable rate may on average be lower, but for a borrower with very tight finances, knowing that the rate they pay will not change may allow them to budget with more certainty. Alternatively, there are products which have a variable rate, but which is ‘capped’ at a maximum rate that could be paid, which means that a holder of such a mortgage is exposed to only some of the risk of increased interest rates.
In a commercial setting, risk is pervasive. If you consider a company, say an airline, then they will be exposed to a number of risks that may or may not be directly related to their underlying business: for example, there are risks in activities that are directly related to the business, for example staff could go on strike due to the breakdown of pay discussions, or poor maintenance of aircraft could mean that the airline becomes unreliable, and customers choose competitors. Such ‘business risks’ are related to the specifics of how an individual company is managed. On the other hand, there are risks that the CEO of an airline may have less control over, and might not want the survival of the company to depend upon. Typically an airline may sell tickets well in advance of the actual date of the flight, however, on the day of the flight, the airline will have to fill their plane with sufficient fuel for the flight, and at the time of the ticket sale, the cost of this fuel will be unknown. It would be much better for the airline to fix the cost of the fuel needed at the time when they sell the tickets, so they know how much to charge the customer. One solution would be to go out and buy the airline fuel at the time of selling the ticket, but they would then have to have somewhere to store the fuel. Much better is to enter into a contract with a second party to provide the fuel at a fixed future date. The second party may be an oil company, who know that they can provide the fuel at the future date, but may also be a speculator, who believes that they will be able to buy the airline fuel in the future at a better price than they have agreed to sell it. In practice, the contract may not even involve physical delivery of the fuel — rather, the contract may just stipulate that the second party pays the airline the difference between the cost of purchasing airline fuel on the future date, and the cost agreed in the contract. The airline can then go and purchase the fuel normally, knowing that the difference in price between that expected, and the actual price paid, is covered by the agreement.
Consider instead an investment manager for a fund. The fund manager usually invests in an index of stocks, for example, the FTSE 100, but believes that the market is about to enter a period of high variability — there is a good chance that the index will increase substantially over the next month, for example, if a piece of economic news is good, but an outside chance that the index will fall substantially if the news is bad. What should the fund manager do? Imagine that she could enter a contract which gave her the option, at the end of the month, to buy the index of stocks at a fixed price \(K\), if she wished, but was not obligated to. Then if the index goes up, the fund manager can realise the gains by buying the index, however if the index falls, then there will be no loss. Of course, to persuade someone to enter into such a contract, the fund manager will have to offer an incentive, for example by paying a fee upfront to the other party in the contract.
The contracts in the last two examples are special examples of financial derivatives. In both cases, the value of the financial exchange made according to the contract derives its value from that of an underlying asset. In this case, the other assets are the price of airline fuel, or the value of the FTSE 100 index. In both cases, these correspond to the prices of quantities that can be bought and sold on an exchange or a financial market, and whose price today is easily determined (for example, by looking in a newspaper or online). A key question in this course will be: given some (probabilistic) model for the underlying asset, what should the price of a derivative contract be?
Example 1. A key connection will be to probabilities, and a key notion in derivatives pricing will be that we often need to use a form of modified probabilities. To see why, consider the following example:2 A bookmaker is taking bets on a two horse race. Based on a careful analysis, he correctly identifies that the first horse has a 25% chance of winning, and the second horse has a 75% chance of winning. As a result, the bookmaker determines that the odds on the horses should be 3-1 against, and 3-1 on. Note: in betting terms, odds of \(n\)-\(m\) against means that a gambler who places a bet of \(m\) will be rewarded with \(n\) and see their stake returned. If they lose, they receive nothing. A gambler who bets at \(n\)-\(m\) against will have a fair bet (average winnings zero) if the probability of winning is \(\frac{m}{m+n}\)). A bet of \(n\)-\(m\) on* is equivalent to a bet of \(m\)-\(n\) against. Usually, whichever form makes \(n\) larger is chosen, so we generally say 3-1 on, rather than 1-3 against. Note that these odds correspond to the bookmaker assigning ‘fair’ probabilities of a 25% chance of the first horse winning, and a 75% chance of the second horse winning.*
Suppose however that the bets taken by the bookmaker are not very sensitive to the odds the bookmaker sets (within reason), and the bookmaker expects to have 5,000 bet on the first horse, and 10,000 bet on the second horse. Then the bookmaker would make a loss of \(5,000 = 3 \times 5,000 - 10,000\) if the first horse wins, and a gain of \(1,667 = 5,000 - 10,000/3\) if the second horse wins. Note that the average profit of the bookmaker is \(0 = 0.25 \times -5,000 + 0.75 \times 1,667\).
On the other hand, if he were to set odds of 2-1 against for the first horse, and 2-1 on for the second horse, then the bookmaker ends up even which ever horse wins. For a bookmaker, they may prefer the second outcome (at least once they build in a bit of profit), since their profit or loss will not be heavily dependent on the outcome of a few races. Note that the implied probabilities given by the new odds are different: with the new odds, the new probabilities are 33% for the first horse, and 66% for the second horse.
This is an observation that will be crucial to much of what we discover in this course: in the presence of ‘market information’ (here, the amount bettors will place on each horse), the ‘correct’ probabilities may not be the true probabilities. Since these are the probabilities that expose the bookmaker to no risk, we will call these the risk-neutral probabilities. In what follows, we will mostly think of the market information as the current price of the underlying asset, and we want to use the risk-neutral probabilities to help us price derivative contracts.