Quantitative finance is the branch of finance that uses mathematical models, statistical methods and computation to describe markets and to make decisions in them. Its main subjects are the pricing of derivatives, the measurement and control of risk, the construction of portfolios and the design of systematic trading strategies. It is also known as mathematical finance or financial engineering, although those terms are sometimes used for narrower parts of the field.
The people who practise it are called quantitative analysts, or quants. Quantitative finance is used by investment banks, asset managers, hedge funds, insurers, central banks and regulators, and it draws on probability theory, statistics, numerical analysis, economics and computer science.
Origins and development
The field is usually traced to 1900, when the French mathematician Louis Bachelier submitted a thesis that modelled share prices as a random process, anticipating the mathematics of Brownian motion. His work was little noticed for half a century.
Modern quantitative finance was built on a small number of papers. In 1952 Harry Markowitz showed how to weigh expected return against risk when choosing a portfolio, the basis of modern portfolio theory. In 1964 William Sharpe set out the capital asset pricing model, which relates an asset's expected return to its exposure to the market. In 1973 Fischer Black, Myron Scholes and Robert Merton published the option pricing theory now called the Black-Scholes-Merton model. Scholes and Merton received the Nobel Memorial Prize in Economic Sciences in 1997; Black had died in 1995. Around 1980 Michael Harrison, David Kreps and Stanley Pliska gave the theory its general mathematical form, linking the absence of arbitrage to pricing under a risk-neutral probability measure.
The two main branches
Practitioners often divide the field by the kind of probability it works with.
- Derivatives pricing, the risk-neutral world. The aim is to find the fair price of a contract today and the trades needed to hedge it. Prices are computed as discounted expected payoffs under a risk-neutral measure calibrated to current market prices, not to forecasts. This branch serves the sell side: banks and dealers that make markets in options, swaps and structured products.
- Risk and portfolio management, the real-world side. The aim is to estimate how prices are actually likely to behave, in order to forecast returns, measure risk and allocate capital. Models are fitted to historical data. This branch serves the buy side, including fund managers engaged in quantitative trading and factor investing.
The two branches use different tools. The first leans on stochastic calculus and partial differential equations. The second leans on statistics, econometrics and optimization.
Core methods
Asset prices are modelled as a stochastic process, most simply a geometric Brownian motion in which returns over short intervals are random with a constant volatility. Extensions allow volatility itself to vary, or prices to jump. When a pricing formula cannot be solved exactly, the value is found numerically, by lattice methods such as the binomial option pricing model, by finite-difference solution of the pricing equation or by Monte Carlo simulation.
On the statistical side, the standard toolkit includes regression, time series analysis, volatility models such as GARCH, portfolio optimization and risk measures such as value at risk. Since the 2010s machine learning has been added, for tasks ranging from return forecasting to the analysis of text and other unstructured data.
Applications
Quantitative finance underlies much of the daily work of markets. Dealers quote and hedge options using model-derived sensitivities. Banks compute regulatory capital for their trading books with approved risk models. Clearing houses set margin with statistical models of price moves. Asset managers use optimizers to build portfolios and attribution models to explain performance. Execution algorithms split large orders using models of market impact. Insurers and pension funds use related techniques to value long-dated guarantees.
Criticism and limits
The models rest on assumptions that markets regularly violate. Returns are not normally distributed, volatility and correlations change, and liquidity can disappear when it is most needed. The near-failure of the hedge fund Long-Term Capital Management in 1998, whose partners included Scholes and Merton, showed how a leveraged strategy built on historical relationships can break down. In the 2008 financial crisis, models used to price and rate mortgage-linked credit products were widely blamed for understating the chance that many borrowers would default together.
A further criticism is that models change the markets they describe. When many firms use similar models, their trades become crowded and their hedging can amplify price moves. Practitioners therefore treat models as tools for consistent comparison and risk control, to be combined with stress tests, limits and judgement, and not as exact descriptions of how prices behave.