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.
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.
Chapter 2
Understanding Probability and Statistics
IN THIS CHAPTER
Comprehending that events can be random
Gathering data to produce statistics of random variables
Defining some important distributions
If you’ve ever placed a bet on a horse or wondered whether your date for the evening is going to turn up, then you know a bit about probability and statistics. The concepts get more interesting if you have multiple events or events in succession.
For example, if you manage to pick both the first and second place horses in a race (an exacta) does that mean you have real skill? This common bet offered by bookies is almost as creative as some of the speculative products offered by bankers.
In this chapter, I start with a few ideas about probability and continue by showing you how they apply to statistical distributions. I examine applications of probability, starting with dice games.
I then look at what happens when you have many random events and a distribution of outcomes.
One distribution of special importance is the Gaussian distribution. It keeps on appearing, and I introduce you to its key properties. I also introduce you to the law of large numbers, which is a more mathematical way of looking at the outcome of a large number of random events.
Probability boils down to a number that refers to a specific situation or event. Statistics, on the other hand, is a way of reasoning from large amounts of data back to some general conclusion – a tool for dealing with data. The later sections of this chapter take you through some widely used results that help in understanding data sets.
The situations I present in this chapter come to look like financial markets, where day-by-day or even millisecond-by-millisecond prices are changing in a highly volatile fashion. So, this chapter gives you a taste of some of the key quantitative tools for understanding how modern financial markets work.
Humans have a deep fascination for outcomes that are not certain. That may be because humans learned early that outcomes in many situations are indeed uncertain. Dice games are the most common method used to examine probability, which is the chance of an event taking place or a statement being true. Dice games have engaged the interest of many famous mathematicians, and because the games are played for money, studying them can be considered the birth of quantitative finance.
Archaeological evidence shows that games of chance have been played for at least the past 34 centuries. Later (well, much later in fact, only several hundred years ago) mathematicians tried to understand the results of these games of chance and that is what led to what is now called probability theory, the mathematical study of randomness.
Probability is the mathematician’s way of analysing random events. To define random isn’t so easy and part of what makes the study of randomness important. The rising of the sun tomorrow isn’t a random event but what mathematicians (and almost everyone else) define as certain. Every certain event has a probability of one. An impossible event (such as having hot sunshine every day throughout the summer in England) has a probability of zero. However, whether it will be raining tomorrow or not is a random event with a probability somewhere between one and zero. That doesn’t mean you have no knowledge of whether it will rain, just that even if you have looked at the most reliable forecasts, you still cannot be certain one way or the other. Equally, the flip of a coin is a random event and so is the throw of a die. The outcomes cannot be predicted, at least if the coin or die isn’t loaded in some way.
Philosophers and mathematicians (for example, the French mathematician Laplace) have thought deeply about near-certain events such as the rising of the sun tomorrow. There’s no day on record when the sun didn’t rise, and the probability of the sun rising tomorrow is very, very close to 1, but that isn’t proof that it will continue to rise every day. I’m not trying to be apocalyptic; I’m just using facts to come to conclusions.
Mathematicians tend to evaluate the probability of a symmetrical coin turning up heads using their usual logic. It seems reasonable to assume that the likelihood of the coin turning up heads is the same as that of its turning up tails.
The probability
