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IN THIS CHAPTER

Using probability and statistics in finance

Finding alternatives for cash

Looking at efficient (and not-so-efficient) markets

Tackling options, futures and derivatives

Managing risk

Doing the maths (and the machines that can help)

Quantitative finance is the application of probability and statistics to finance. You can use it to work out the price of financial contracts. You can use it to manage the risk of trading and investing in these contracts. It helps you develop the skill to protect yourself against the turbulence of financial markets. Quantitative finance is important for all these reasons.

If you’ve ever looked at charts of exchange rates, stock prices or interest rates, you know that they can look a bit like the zigzag motion of a spider crossing the page. However, major decisions have to be made based on the information in these charts. If your bank account is in dollars but your business costs are in euros, you want to make sure that, despite fluctuations in the exchange rate, you can still pay your bills. If you’re managing a portfolio of stocks for investors and you want to achieve the best return for them at minimum risk, then you need to learn how to balance risk with reward. Quantitative finance is for banks, businesses and investors who want better control over their finances despite the random movement of the assets they trade or manage. It involves understanding the statistics of asset price movements and working out what the consequences of these fluctuations are.

However, finance, even quantitative finance, isn’t just about maths and statistics. Finance is about the behaviour of the participants and the financial instruments they use. You need to know what they’re up to and the techniques they use. This is heady stuff, but this book guides you through.

Defining Quantitative Finance

My guess is that if you’ve picked up a book with a title like this one, you want to know what you’re going to get for your money. Definitions can be a bit dry and rob a subject of its richness but I’m going to give it a go.

Quantitative finance is the application of mathematics – especially probability theory – to financial markets. It’s used most effectively to focus on the most frequently traded contracts. What this definition means is that quantitative finance is much more about stocks and bonds (both heavily traded) than real estate or life insurance policies. The basis of quantitative finance is an empirical observation of prices, exchange rates and interest rates rather than economic theory.

Quantitative finance gets straight to the point by answering key questions such as, ‘How much is a contract worth?’ It gets to the point by using many ideas from probability theory, which are laid out in Chapters 2 and 3. In addition, sometimes quantitative finance uses a lot of mathematics. Maths is really unavoidable because the subject is about answering questions about price and quantity. You need numbers for that. However, if you use too much mathematics, you can lose sight of the context of borrowing and lending money, the motivation of traders and making secure investments. Chapter 13 covers subjects such as attitudes to risk and prospect theory while Chapter 18 looks in more detail at the way markets function and dysfunction.

Just to avoid confusion, quantitative finance isn’t about quantitative easing. Quantitative easing is a process carried out by central banks in which they effectively print money and use it to buy assets such as government bonds or other more risky bonds. It was used following the credit crisis of 2008 to stimulate the economies of countries affected by the crisis.

Summarising the mathematics

I’m not going to pretend that quantitative finance is an easy subject. You may have to brush up on some maths. In fact, exploring quantitative finance inevitably involves some mathematics. Most of what you need is included in Chapter 2 on probability and statistics. In a few parts of the book, I assume that you remember some calculus – both integration and differentiation. If calculus is too much for you, just skip the section or check out Calculus For Dummies by Mark Ryan (Wiley). I’ve tried to keep the algebra to a minimum but in a few places you’ll find lots of it so that you know exactly where some really important results come from. If you don’t need to know this detail, just skip to the final equation.

Time and again in this book, I talk about the Gaussian (normal) distribution. Chapter 2 has a definition and explanation and a picture of the famous bell curve.

Please don’t get alarmed by the maths. I tried to follow the advice of the physicist Albert Einstein that ‘Everything should be made as simple as possible, but not simpler.’

Pricing, managing and trading

Quantitative finance is used by many professionals working in the financial industry. Investment banks use it to price and trade options and swaps. Their customers, such as the officers of retail banks and insurance companies, use it to manage their portfolios of these instruments. Brokers using electronic-trading algorithms use quantitative finance to develop their algorithms. Investment managers use ideas from modern portfolio theory to try to boost the returns of their portfolios and reduce the risks. Hedge fund managers use quantitative finance to develop new trading strategies but also to structure new products for their clients.

Meeting the market participants

Who needs quantitative finance? The answer includes banks, hedge funds, insurance companies, property investors and investment managers. Any organisation that uses financial derivatives, such as options, or manages portfolios of equities or bonds uses quantitative finance. Analysts employed specifically to use quantitative finance are often called quants, which is a friendly term for quantitative analysts, the maths geeks employed by banks.

Perhaps the most reviled participants in the world of finance are speculators. (Bankers should thank me for writing that.) A speculator makes transactions in financial assets purely to buy or sell them at a future time for profit. In that way, speculators are intermediaries between other participants in the market. Their activity is often organised as a hedge fund, which is an investment fund based on speculative trading.

Speculators can make a profit due to

❯❯ Superior information

❯❯ Good management of the risk in a portfolio

❯❯ Understanding the products they trade

❯❯ Fast or efficient trading mechanisms

Speculators are sometimes criticised for destabilising markets, but more likely they do the opposite. To be consistently profitable, a speculator has to buy when prices are low and sell when prices are high. This practice tends to increase prices when they’re low and reduce them when they’re high. So speculation should stabilise prices (not everyone agrees with this reasoning, though).

Speculators also provide liquidity to markets. Liquidity is the extent to which a financial asset can be bought or sold without the price being affected significantly. (Chapter 18 has more on liquidity.) Because speculators are prepared to buy (or sell) when others are selling (or buying), they increase market liquidity. That’s beneficial to other market participants such as hedgers (see the next paragraph) and is another reason not to be too hard on speculators.

In contrast to speculators, hedgers like to play safe. They use financial instruments such as options and futures (which I cover in Chapter 4) to protect a financial or physical investment against an adverse movement in price. A hedger protects against price rises if she intends to buy a commodity in the future and protects against price falls if she intends to sell in the future. A natural hedger is, for example, a utility company that knows it will want to purchase natural gas throughout the winter so as to generate electricity. Utility companies typically have a high level of debt (power stations are expensive!) and fixed output prices because of regulation, so they often manage their risk using option and futures contracts which I discuss in Chapters 5 and 6, respectively.

Walking like a drunkard

The random walk, a path made up from a sequence of random steps, is an idea that comes up time and again in quantitative finance. In fact, the random walk is probably the most important idea in quantitative finance. Chapter 3 is devoted to it and elaborates how random walks are used.

Figure 1-1 shows the imagined path of a bug walking over a piece of paper and choosing a direction completely at random at each step. (It may look like your path home from the pub after you’ve had a few too many.) The bug doesn’t get far even after taking 20 steps.

© John Wiley & Sons, Ltd.

FIGURE 1-1: A random walk.

In finance, you’re interested in the steps taken by the stock market or any other financial market. You can simulate the track taken by the stock market just like the simulated track taken by a bug. Doing so is a fun metaphor but a serious one, too. Even if this activity doesn’t tell you where the price ends up, it tells you a range within which you can expect to find the price, which can prove to be useful.

Random walks come in different forms. In Figure 1-1, the steps are all the same length. In finance, though random walks are often used with very small step sizes, in which case you get a Brownian motion. In a slightly more complex form of Brownian motion, you get the geometric Brownian motion, or GBM, which is the most common model for the motion of stock markets. You can find out in detail about GBM in Chapter 3.

Knowing that almost nothing isn’t completely nothing

The orthodox view is that financial markets are efficient, meaning that prices reflect known information and follow a random walk pattern. It’s therefore impossible to beat the market and not worth paying anyone to manage an investment portfolio. This is the efficient market hypothesis, or EMH for short. This view is quite widely accepted and is the reason for the success of tracker funds, investments that seek to follow or track a stock index such as the Dow Jones Industrial Average. Because tracking an index takes little skill, investment managers can offer a diversified portfolio at low cost. Chapter 14 has much more about diversification and portfolios.

Academics often distinguish different versions of the efficient market hypothesis (EMH):

❯❯ Weak efficiency is when prices can’t be predicted from past prices.

❯❯ Semi-strong efficiency is when prices can’t be predicted with all available public information.

❯❯ Strong efficiency goes a step further than semi-strong efficiency and says that prices can’t be predicted using both public and private information.

Anomalies are systematically found in historical stock prices that violate even weak efficiency. For example, you find momentum in most stock prices: If the price has risen in the past few months, it will tend to rise further in the next few months. Likewise, if the price has fallen in the past few months, it will tend to continue falling in the next few months. This anomaly is quite persistent and is the basis for the trend following strategy of many hedge funds.

Somehow, though, the EMH smells wrong. Even though you can find many vendors of market information, EMH has a cost. It’s no coincidence that some of these vendors are very wealthy indeed. Also, if you examine publicly available information, you soon find that such information is not perfect. Often the information is delayed, with the numbers published days or even weeks following the time period they apply to. Some exceptions exist and you can read about one of them in the sidebar, ‘The impact of US employment numbers’.

THE IMPACT OF US EMPLOYMENT NUMBERS

One of the most widely anticipated numbers in finance is the so-called nonfarm payroll issued by the US Bureau of Labour Statistics. In fact, the nonfarm payroll isn’t just a number but a report with almost 40 pages. You can find the November 2015 report at www.bls.gov/news.release/pdf/empsit.pdf. Formally, this report is called the employment situation. Its headline figure is the nonfarm payroll employment and its companion figure is the unemployment rate, so it gives a picture of the employment situation in the United States.

This number is hugely impactful globally and can move the value of currencies, stock markets and bond markets across the world within seconds of its release. In the US, though, the number is released one hour before the opening of the New York Stock Exchange so that traders get a chance to absorb the information before trading begins. Aside from the data being for the largest economy in the world, other factors make it influential:

• The nonfarm payroll is timely. It’s issued on the first Friday in the month following the one it relates to. For example, the September 2015 report was issued on Friday 2 October 2015 at exactly 8:30 a.m. Eastern Daylight Time. This is no mean feat given the amount of information contained in it.

• The nonfarm payroll is comprehensive. It has surveys including small business and the self-employed so the information is credible.

• Although estimates and statistical models are used in some of the numbers, revisions are made to these numbers in subsequent months as more information becomes available. The existence of timely revisions based on a well-defined process supports market confidence in the numbers.

Be warned: If you’re trading any instruments when the nonfarm payroll figures come out, you may be in for some significant turbulence!

It’s far more likely that markets are not informationally efficient and that many participants for reasons of cost or availability are not perfectly informed. It’s also highly likely that most participants are not able to instantly work out in detail the consequences of the information presented to them. This working out may take some time.

Indeed, if markets were informationally efficient, there would be no incentive to seek out information. The cost wouldn’t justify it. On the other hand, if everyone else is uninformed, it would be rewarding to become informed as you can trade successfully with those who know less than you.

The point that in an efficient market there’s no incentive to seek out information and so therefore no mechanism for it to become efficient is the Grossman-Stiglitz paradox, named after the American economists Sanford Grossman and Joseph Stiglitz. The implication is that markets will be efficient but certainly not perfectly efficient.

Only with deep research into market data do markets have a chance of becoming efficient. That’s the norm in financial markets, but pockets of inefficiency are always left that market traders and savvy investors can attempt to exploit. Also, attempts to use the results of deep research drive the intense trading found in many markets. In Chapter 8, I talk about techniques for analysing historical price data for patterns.

Recognising irrational exuberance

Most markets are responding constantly to a flow of news on companies, economies, interest rates and commodities. They also react to changes in the supply and demand for the financial asset in question. If more fund managers decide to buy a stock than sell it, its price tends to rise. The greater the demand for loans from companies, the higher the interest rate lenders demand.

Markets don’t always behave in this sensible way, however. Sometimes, they defy gravity and keep on rising, which is called a bubble. Figure 1-2 shows an example of this in a chart for the share price of British Telecom, a fixed-line telecom operator. In September 1996, the Chairman of the US Federal Reserve Bank warned of irrational exuberance in markets. Unusual circumstances, especially low interest rates, were making markets overly excited. He was dead right. The Internet had just been invented so even traditional companies such as British Telecom saw their share price rocket upward. The market ignored Chairman Alan Greenspan when he made his warning, although the Japanese stock market respectfully dipped several per cent on the day of his speech. In a way, the market was right and farsighted: The Internet was going to be big, it was just that British Telecom wasn’t Google. After rising to a very sharp peak in early 2000, British Telecom shares crashed back down to earth and continued on in their usual way.

© John Wiley & Sons, Ltd.

FIGURE 1-2: Share price chart for British Telecom plc.

One thing for sure is that with crazy behaviour like this, the statistics of the price movements for shares don’t obey Gaussian statistics. In Chapter 2, I explain quantities such as kurtosis, a measure of how much statistical distributions deviate from the Gaussian distribution. A large positive value for the kurtosis means that the probability of extreme events is far more likely than you’d expect from a Gaussian distribution. This situation has come to be called a fat-tailed distribution. Statistics is the way of measuring and analysing the market price data used in quantitative finance, and I try to emphasise this throughout the book.

Another possibility, of course, is that prices crash rapidly downwards far more often than you’d expect. The fear of prices crashing downwards is palpable. Market participants want to protect themselves against nasty events like that. To do that, you need financial instruments such as options and futures, which I explain in detail in Chapters 5 and 6, respectively. Options are a form of financial insurance. For example, if you think that the stock market is going to crash, then you buy an option that compensates you if that happens. If the market doesn’t crash, you’ve lost just the premium you paid for the option, just like an insurance contract.

George Soros, a billionaire hedge fund manager, attempted to explain these irrational market events with a concept he called reflexivity. He replaced the efficient market hypothesis view that the market is always right with something else:

❯❯ Markets are always biased in one direction or another. An example of this bias is the British Telecom shares illustrated in Figure 1-2. The market thought that all things telecom would be highly profitable.

❯❯ Markets can influence the events that they anticipate. Financial markets can be stabilising. If a recession is anticipated and the currency declines, this situation should boost exports and help prevent a recession.

George Soros’s ideas are controversial, but they help to explain some major market distortions. He’s been proven correct on enough occasions to have been successful using his insights.

Wielding Financial Weapons of Mass Destruction

Cash is the most fundamental of all financial assets. Economists write that money has three functions. It serves as a:

❯❯ Store of value

❯❯ Means of exchange

❯❯ Unit of account

These three functions are familiar to anyone with a savings account (store of value) who has done some shopping (means of exchange) and carefully compared prices (unit of account). Whether in the form of nickel, plastic or paper, cash is the key.

Two alternatives to cash – one ancient, one modern – are good to know about:

❯❯ Gold has been used for thousands of years as a store of value and also as a means of exchange. Most central banks in the world hold substantial quantities in vaults. This practice is partly a relic of the time when paper money could be exchanged for gold at the central bank. Although this ended in the United States in 1971, many investors still hold gold as part of their investment portfolios.

❯❯ Like gold, the bitcoin is a currency not under the control of any government. However, bitcoin isn’t physical. It’s been described as a cryptocurrency because bitcoin is completely digital and relies heavily on encryption techniques for security. It can be used for payments just like other forms of cash, but at the moment these transactions are small compared with, say, the volume of credit card transactions.

One of the appeals of both gold and bitcoin is that they’re not under government control. In the past, governments have used their power to print money, which undermined the value of the currency. The currencies then no longer function well as a store of value. By investing in gold, which is limited in supply, this undermining can’t happen.

Cash exists in the form of many currencies such as the US dollar, the Japanese Yen and the Chinese renminbi. These countries all have their own central banks, and one of the key functions of these banks is to set the interest rate for the currency. This interest is money that you earn by depositing cash at the central bank. Normally, only other banks are permitted to use central banks in this way, but these interests rates are one of the key parameters in quantitative finance. The interest rate at a central bank is often called the risk-free rate because the assumption is that a central bank can’t go bankrupt. Chapter 4 has some of the maths involved with interest rates that’s the basis behind lots of quantitative finance calculations.

If you take out a loan to buy a house or expand your business, the loan is said to be a floating-rate loan if the interest rate changes when the central bank in your country changes its interest rate. The load is fixed-rate if it stays the same when the central bank changes the interest rate. However, given that the period over which loans are repaid can be long, locking into one type of loan gives you no flexibility. If you have a floating-rate loan, you may decide that you want to keep the interest payments fixed in future. That may help you sleep at night. The solution to this fixing is called an interest-rate swap. This instrument allows you to swap from a fixed-rate loan to a floating-rate loan or vice versa. Chapter 4 has a section which gives you the maths behind this.

Interest-rate swaps are one of the most important instruments used by banks to manage risk. They also use more sophisticated tools as well and Chapter 12 provides an introduction to some of the most common interest-rate derivatives. These derivatives have proved very popular with real-estate investors who typically borrow large sums of money and want to put limits on interest payments.

Cash in one currency can be exchanged for cash in another currency. This transaction is called foreign exchange, often abbreviated as FX. The FX market isn’t organised on an exchange and normally consists of dealers working in banks. This market is the largest financial market in the world with huge volumes of transactions per day.

Because different currencies have different interest rates, you can potentially make money by

❯❯ Selling a currency with a low interest rate

❯❯ Buying currency with a high interest rate

❯❯ Earning a high interest rate

Such transactions are called the carry trade and are a big factor in influencing foreign exchange rates.

Going beyond cash

Borrowing money from a bank to expand a business is fine, but other ways are possible too:

❯❯ Bonds are a form of loan to a business. The borrower (or business owner) receives the principal from the lender and in return promises to pay a regular interest payment called a coupon. On the bond’s maturity date, the lender gets her principal back. The clever bit, though, is that this bond is a financial instrument. This means that the lender can sell it to someone else. Then the buyer is entitled to the coupon payments and the principal repayment on maturity.

❯❯ Owning stocks or shares in a business means you’re a part owner of the business, are entitled to dividend payments and can vote at the annual general meeting in support (or otherwise) of the managers.

Businesses issue shares in exchange for cash from investors but they have no fixed repayment date as a bond does. Dividend payments are at the discretion of the management and can vary and, in fact, be non-existent. Because of this, shares are often considered riskier than bonds.

Bonds and shares are the building blocks for most investment portfolios. Bonds are risky because the borrower can default and fail to pay her coupons. Shares are risky because the company may be unable to pay a dividend. Shareholders have no right to any repayment of capital so are more likely to lose everything. Chapter 4 gives you the lowdown on the bond and stock markets.

If you’re thinking that you’re never going to invest in shares or bonds because you may never get your money back, then you’re not alone. However, the financial markets have created a solution to this, using two instruments, options and futures that can be used to control and manage the risk of investing in the stock and bond markets. They’re both flexible contracts that I cover in great detail in Chapters 5 and 6, respectively. Quantitative finance developed rapidly in the 1980s after people figured out a mathematical way to price options. You can find out about pricing in Chapters 10 and 11.

SETTING CONTRACTS IN STONE

Is anything ever written in tablets of stone? Apparently so. Some of the oldest examples of documents written in stone are Babylonian futures contracts. These were agricultural futures – contracts agreeing to sell or buy grain at a time in the future at a price agreed now. The point of these contracts is to reduce the impact of price fluctuations on farmers or buyers of grain such as bakers. Knowing a price in advance makes business easier. Exactly the same sort of contracts are used today, although they’re mainly traded electronically on the CME (Chicago Mercantile Exchange).

Inventing new contracts

Every business likes to show off shiny new products so as to boost sales, but the financial industry has been better than most at creating new products; some would say too successful. After a long career at the heights of the financial world, the former chairman of the US Federal Reserve Bank Paul Volcker said that he’d encountered only one financial innovation in his career, and that was the automatic teller machine (ATM).

Volcker’s sceptical remark points out that the nature of the contracts that people enter into are not fundamentally different from ancient contracts. Energy futures were first created in the 1970s but they’re similar to agricultural futures, which have been around for thousands of years. Indeed, they’re now traded on exactly the same exchanges. Trading is now electronic and greatly accelerated, but the function of these contracts is exactly the same. The success of energy futures led to the introduction of financial futures contracts on interest rates and bonds. They were, and are still, a big success.

Just as in the futures market, the variety of option contracts available has proliferated. Initially, most options were share options, but they soon found use in the foreign exchange and bond markets. You can also buy commodity options such as for crude oil, which have proved very popular too.

New option styles have also been introduced. In this book, I stick to what are known as plain vanilla contracts which give the holder the right, but not the obligation, to buy or sell an underlying asset at a predetermined price (the strike price) at a specified time in the future. In the plain vanilla contract, the option payoff (the amount that you may get paid when the contract expires) depends only on a single strike price (the price that has to be reached for there to be any payoff to the option) whereas for barrier options, and other more complicated options, other prices are involved too.

THE 2008 BANKING CRISIS IN A NUTSHELL

In September 2008, the US investment bank Lehman Brothers filed for bankruptcy. This event was the first time in decades that a major US bank had collapsed. In the UK, major retail banks had to be bailed out by the government, and in Germany the second largest bank, Commerzbank, was partly nationalised.

These banks were deemed too big to fail, meaning that the government felt compelled to intervene fearing that allowing the banks to fail would create a crisis across the entire banking system.

This financial crisis was a complicated event (you can find whole books on it – not just a paragraph) but it boils down to the fact that the banks lent way too much money and lent some of it to people who were unlikely ever to pay it back. You can be forgiven for thinking they just weren’t doing their job properly.

A lot of this lending was done using mortgage-backed securities. These securities are a bit like bonds where the coupon payments and final principal repayments come from a portfolio of residential mortgages. By ingenious methods, the banks made these securities appear less risky than they really were. These methods allowed the bank to earn yet more fees from the lending but at the expense of building a financial time bomb.

Finally, credit derivatives give protection against defaulting loans. The most common of these derivatives are credit-default swaps in which the buyer of the swap makes a regular series of payments to the seller; in exchange, the seller promises to compensate the buyer if the loan defaults.

Derivatives are useful because market participants who can’t bear certain risks can shift them (at a price) to someone who can. As a whole though, trading in derivatives can lead to risk being concentrated in a small number of dealers with fatal consequences for the likes of Lehman Brothers. As the investor Warren Buffett presciently observed years before the 2008 crisis, ‘derivatives are financial weapons of mass destruction’.

Despite the explosive possibilities inherent in the derivatives market, the use of derivatives continues because of the constant need to mitigate financial risks. Better regulation will hopefully reduce the nasty accidents that have happened.

Analysing and Describing Market Behaviour

Quantitative finance is primarily about prices, but because markets are almost efficient, price changes are almost random. Also, you may be interested in not one price but many prices – all the prices in an investment portfolio, for example. I explain some of the statistical tools that you can use to deal with this problem in the next sections.

Measuring jumpy prices

The measure of the jumpiness of prices is called volatility. Chapter 7 is all about volatility and the different ways that you can calculate it. Normally price changes are called returns even if they’re negative, and the volatility is the standard deviation of these returns. The higher the volatility, the jumpier the prices.

Because of the instability of financial markets, volatility is constantly changing. Prices can go through quiet spells but then become very jumpy indeed. This means that calculating volatility isn’t as simple as calculating a normal standard deviation, but Chapter 7 shows you how.

Keeping your head while using lots of data

Most financial institutions are trading, selling or investing many different financial assets, so understanding the relationships between the prices of these assets is useful. In Chapter 9, I show you a special technique for gaining this understanding called principal components analysis (PCA). This technique helps because it can point out patterns and relationships between assets and even help you build predictive models. This is no mean feat given the almost random changes in asset prices, but PCA can do it.

Valuing your options

Black-Scholes is the equation that launched a thousand models. Technically, it’s a partial differential equation for the price of an option. The reason you need such a complicated equation to model the price of an option is because of the random nature of price movements. Chapter 10 is the go-to place to find out more about Black-Scholes.

If you’re a physicist or chemist, you may recognise part of the Black-Scholes equation as being similar to the diffusion equation that describes how heat moves in solids. The way you solve it is similar, too.

An option gives you the right, but not the obligation, to buy or sell a financial asset, such as a bond or share, at a time in the future at a price agreed now. The problem is that because prices move in random fashion you have no idea what the price will be in the future. But you do know how volatile the price is, and so from that you have an idea what range the future price is in. If the asset price is highly volatile, the range of possible future prices is large. So, the price of an option depends on the following factors:

❯❯ The risk-free rate of interest

❯❯ The volatility of the asset

❯❯ The time to expiry

❯❯ The strike price

The Black-Scholes equation makes assumptions about the statistical distribution of the asset returns. You can find the details of this geometric Brownian motion model in Chapter 3. Chapter 10, gives you an alternative way of calculating option prices using probability theory. You don’t need the complicated partial differential equation to do this, but you still need the maths that you can find in Chapter 2.

You even have a third way to calculate option prices using simulation. With a simulation, you use the idea that asset prices follow a random walk and use your computer to generate lots of paths that the price may take in the future. From this, you can calculate the probability of the price hitting the strike price. You use this probability to work out today’s price for the option.

Managing Risk

Quantitative finance and the associated futures and option contracts provide the tools for managing financial risk. With futures, you can fix the price now of purchases or sales that you know you need to make in the future. Options can give you more flexibility in protecting yourself against adverse price movements, but the drawback is that you have to pay a premium up front.

To quantify the overall riskiness of a portfolio of risky financial assets, you can use the Value at Risk (VaR) number. VaR is widely used by fund managers, banks and companies using derivatives. It gives senior managers an indication of how much risk they’re taking on. Regulators use VaR to figure out how much capital a bank must hold. Chapter 15 explains this measure.

Hedging and speculating

You can use options either for speculation or hedging. Options have some leverage built in, in other words, the returns can be similar to using borrowed money to buy shares. This similarity makes them attractive to some market participants. You can quickly earn many times more than your original premium, but you can easily end up with nought. This game is for professionals.

Options are, however, great tools for hedging. If you have a large investment portfolio, but you think that the stock market may go down, you can buy a put option which pays you compensation if the market goes down before the option expires.

The price of options is very much influenced by how much time is left before they expire. The sensitivity of the option price to the time to expiry is called theta, after the Greek letter. Chapter 11 shows you how to calculate theta and some of the other Greeks, which are useful if you’re trading options.

Generating income

Most options written are worthless when they expire. That makes the business of writing them attractive – your customer pays you a premium to buy an option from you and, highly likely, it expires worthless. You can see why bankers like to sell options to their clients and why some become rich from it. Of course, a downside also exists to selling options. The option may not expire worthless. Your client may have had a great insight when buying a call option and that share price shoots up, and you have to pay your client a large payoff. Ouch!

To mitigate the risk of selling options, you can and should delta hedge, which means to buy or sell the underlying asset associated with your option. Chapter 11 shows you how to calculate the value of delta for a plain vanilla equity option. If you don’t delta hedge and take a naked position, then you run the risk of large losses.

Building portfolios and reducing risk

Investment managers build large portfolios of shares, bonds and other financial assets. These portfolios are often part of pension funds or made available to private investors as mutual funds. How much of each asset should the manager buy for the portfolio? This decision depends on the manger’s objective but if, like many others, she wishes to maximise returns and reduce risk, she can use a framework called modern portfolio theory (MPT for short). MPT is not so modern now as it was first worked out by the economist Markowitz in 1952, but the framework and concepts are still applicable today. You can read about it in Chapter 14.

For your portfolio, you need to know the following:

❯❯ The expected return of your assets

❯❯ The volatility of your assets

❯❯ The correlations (statistical relationships calculated from price returns) between your assets

From this, you can calculate the portfolio that meets your objectives. That may mean minimising the risk but it may also mean achieving some minimum level of return.

In practice, using MPT has proved difficult because both correlations and expected returns are hard to estimate reliably. But some timeless ideas do exist that were usefully highlighted by MPT. The main one is diversification, which has been described as the only free lunch in finance because of its almost universal benefits. By placing investments over a wide number of assets, you can significantly reduce the risk for the same level of return. Equivalently, you can boost your return for the same level of risk. By paying special attention to the correlation between the assets in your portfolio you gain maximum benefit from diversification. If the correlation between your assets is small or even negative, the benefit is large. Sadly that’s not easy to achieve because, for example, many stocks and shares are correlated, but at least you know what to look for. Chapter 14 talks more about tools to manage portfolios, including correlation and diversification.

Computing, Algorithms and Markets

Data can be gathered directly by monitoring activity on the Internet – especially trade data: the price, time and quantity of financial instruments bought and sold. The large amounts of data now captured means that more specialised databases are used to store it and more sophisticated machine learning techniques are used to model it. The better your models are, the more successfully you can trade, and the more data you generate for further analysis. A poet once wisely wrote that you can’t feed the hungry on statistics. You can’t eat data, but data is now a big industry employing – and feeding – many people. You may be one of them.

Seeing the signal in the noise

The problem with large amounts of data is what to do with it. The first thing is to plot it. Plotting allows you to spot any obvious relationships in the data. You can also see whether any data is missing or bad, which is an all-too-frequent occurrence.

Several kinds of plot are especially useful in finance:

❯❯ Line plot: A line plot or chart shows how a value Y (normally shown on the vertical axis) varies with a value indicated on the horizontal axis. The Y values are shown as a continuous line. A line plot is good for showing how a price or interest rate or other variable (Y) changes with time. You can overlay several line plots to compare the movement of several assets.

❯❯ Scatter plot: A plot of two variables, X and Y, against each other where each pair of values (X,Y) is drawn as a point. Scatter plots can look like a swarm of bees but are good for revealing relationships you may otherwise not spot. For example, you may want to plot the daily returns of a stock against the daily returns of a stock index to see how correlated they are.

❯❯ Histogram: Also known as a bar chart, a histogram is great for showing the distribution of the returns of a financial asset.

In Chapter 8 I show you how to investigate a bit deeper into histograms and discover a better representation of the returns distribution.

The Gaussian distribution is so frequently encountered in quantitative finance that you can easily forget that there are often more complex distributions behind your data. To investigate this, you can use the expectation maximisation algorithm, which is a powerful iterative way for fitting data to models. Go to Chapter 8 to find out more about this.

Keeping it simple

If you build models for the expected returns of an asset you’re trading or investing in, you need to take great care. If you apply a volatility adjustment to the returns of your asset, the returns look much like Gaussian random noise. Normally, Gaussian noise is what’s left after you build a model. So, because markets are nearly efficient, you have little to go on to build a model for returns. Also, you certainly can’t expect anything that has much predictive power.

The temptation in building a model is to introduce many parameters so as to fit the data. But given the lack of information in the almost random data you encounter in finance, you won’t have enough data to accurately determine the parameters of the model.

Always choose the simplest model possible that describes your data. Chapter 17 shows you in more depth how to fit models in these situations and statistics you can use to determine whether you have a good model or not.

Looking at the finer details of markets

In Chapter 18, you can find out more about markets in real life. Some of this information isn’t pretty, but it is important. One important mechanism is market impact, the amount by which prices move when you buy or sell an asset. In a way, this impact is the reason markets are important – prices change with supply and demand. The example using Bayes’ theorem shows how markets can take on new information and reflect it in changed prices. Doing so is the way that markets can become almost efficient.

Trading at higher frequency

More and more financial trading is completely automated. Computers running powerful algorithms buy and sell stocks and futures contracts often with holding periods of less than a second – sometimes less than a millisecond. This high frequency trading (HFT) must use maths and algorithms. It is part of quantitative finance and many quants are involved with the development of trading algorithms.