Quantitative finance

Quantitative finance is about applying mathematics and statistics to finance. Quantitative finance helps you to price contracts such as options, manage the risk of investment portfolios and improve trade management. 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 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.

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.

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.

I tried to follow the advice of the physicist Albert Einstein that ‘Everything should be made as simple as possible, but not simpler.’

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. Because speculators are prepared to buy (or sell) when others are selling (or buying), they increase market liquidity.

In contrast to speculators, hedgers like to play safe. They use financial instruments such as options and futures 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.

Random Behaviour of Assets

Random Walk

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.

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.  The bug doesn’t get far even after taking 20 steps.

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.

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.

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.

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.