There are two separate branches of finance that require advanced quantitative techniques: derivatives pricing, and risk and portfolio management. One of the main differences is that they use different probabilities such as the risk-neutral probability (or arbitrage-pricing probability), denoted by "Q", and the actual (or actuarial) probability, denoted by "P".
Derivatives pricing: the Q world The goal of derivatives pricing is to determine the fair price of a given security in terms of more
liquid securities whose price is determined by the law of
supply and demand. The meaning of "fair" depends, of course, on whether one considers buying or selling the security. Examples of securities being priced are
plain vanilla and
exotic options,
convertible bonds, etc. Once a fair price has been determined, the sell-side trader can make a market on the security. Therefore, derivatives pricing is a complex "extrapolation" exercise to define the current market value of a security, which is then used by the sell-side community. Quantitative derivatives pricing was initiated by
Louis Bachelier in
The Theory of Speculation ("Théorie de la spéculation", published 1900), with the introduction of the most basic and most influential of processes,
Brownian motion, and its applications to the pricing of options. Brownian motion is derived using the
Langevin equation and the discrete
random walk. Bachelier modeled the
time series of changes in the
logarithm of stock prices as a
random walk in which the short-term changes had a finite
variance. This causes longer-term changes to follow a
Gaussian distribution. The theory remained dormant until
Fischer Black and
Myron Scholes, along with fundamental contributions by
Robert C. Merton, applied the second most influential process, the
geometric Brownian motion, to
option pricing. For this M. Scholes and R. Merton were awarded the 1997
Nobel Memorial Prize in Economic Sciences. Black was ineligible for the prize because he died in 1995. The next important step was the
fundamental theorem of asset pricing by Harrison and Pliska (1981), according to which the suitably normalized current price
P0 of security is arbitrage-free, and thus truly fair only if there exists a
stochastic process Pt with constant
expected value which describes its future evolution: {{NumBlk|:|P_{0} = \mathbf{E}_{0} (P_{t}) | }} A process satisfying () is called a "
martingale". A martingale does not reward risk. Thus the probability of the normalized security price process is called "risk-neutral" and is typically denoted by the
blackboard font letter "\mathbb{Q}". The relationship () must hold for all times t: therefore the processes used for derivatives pricing are naturally set in continuous time. Quantitative analysts (
quants) in the "Q-world" focus on the pricing of derivative products. Their role requires a detailed understanding of the mathematical models used for specific financial instruments. Securities are priced individually, and thus the problems in the Q world are low-dimensional in nature. Calibration is one of the main challenges of the Q world: once a continuous-time parametric process has been calibrated to a set of traded securities through a relationship such as (), a similar relationship is used to define the price of new derivatives. The main quantitative tools necessary to handle continuous-time Q-processes are
Itô's stochastic calculus,
simulation and
partial differential equations (PDEs).
Risk and portfolio management: the P world Risk and portfolio management aims to model the statistically derived probability distribution of the market prices of all the securities at a given future investment horizon. This "real" probability distribution of the market prices is typically denoted by the blackboard font letter "\mathbb{P}", as opposed to the "risk-neutral" probability "\mathbb{Q}" used in derivatives pricing. Based on the P distribution, the buy-side community takes decisions on which securities to purchase in order to improve the prospective profit-and-loss profile of their positions considered as a portfolio. Increasingly, elements of this process are automated; see for a listing of relevant articles. For their pioneering work,
Markowitz and
Sharpe, along with
Merton Miller, shared the 1990
Nobel Memorial Prize in Economic Sciences, for the first time ever awarded for a work in finance. The portfolio-selection work of Markowitz and Sharpe introduced mathematics to
investment management. With time, the mathematics has become more sophisticated. Thanks to Robert Merton and
Paul Samuelson, one-period models were replaced by continuous time,
Brownian-motion models, and the quadratic
utility function implicit in
mean–variance optimization was replaced by more general increasing, concave utility functions. Furthermore, in recent years the focus shifted toward estimation risk, i.e., the dangers of incorrectly assuming that advanced time series analysis alone can provide completely accurate estimates of the market parameters. See . Much effort has gone into the study of financial markets and how prices vary with time.
Charles Dow, one of the founders of
Dow Jones & Company and
The Wall Street Journal, enunciated a set of ideas on the subject which are now called
Dow Theory. This is the basis of the so-called
technical analysis method of attempting to predict future changes. One of the tenets of "technical analysis" is that
market trends give an indication of the future, at least in the short term. The claims of the technical analysts are disputed by many academics. While numerous empirical studies have examined the effectiveness of technical analysis, there remains no definitive consensus on its usefulness in forecasting financial markets. == Criticism ==