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.
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.
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.
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.
❯❯ 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.
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.
❯❯ 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
