Application of mathematical and statistical methods in finance
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling in the financial field.
In general, there exist two separate branches of finance that require advanced quantitative techniques: derivatives pricing on the one hand, and risk and portfolio management on the other.[1] Mathematical finance overlaps heavily with the fields of computational finance and financial engineering. The latter focuses on applications and modeling, often with the help of stochastic asset models, while the former focuses, in addition to analysis, on building tools of implementation for the models. Also related is quantitative investing, which relies on statistical and numerical models (and lately machine learning) as opposed to traditional fundamental analysis when managing portfolios.
French mathematician Louis Bachelier's doctoral thesis, defended in 1900, is considered the first scholarly work on mathematical finance. But mathematical finance emerged as a discipline in the 1970s, following the work of Fischer Black, Myron Scholes and Robert Merton on option pricing theory. Mathematical investing originated from the research of mathematician Edward Thorp who used statistical methods to first invent card counting in blackjack and then applied its principles to modern systematic investing.[2]
The subject has a close relationship with the discipline of financial economics, which is concerned with much of the underlying theory that is involved in financial mathematics. While trained economists use complex economic models that are built on observed empirical relationships, in contrast, mathematical finance analysis will derive and extend the mathematical or numerical models without necessarily establishing a link to financial theory, taking observed market prices as input. See: Valuation of options; Financial modeling; Asset pricing. The fundamental theorem of arbitrage-free pricing is one of the key theorems in mathematical finance, while the Black–Scholes equation and formula are amongst the key results.[3]
Today many universities offer degree and research programs in mathematical finance.
History: Q versus P
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".
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.[4][5] Brownian motion is derived using the Langevin equation and the discrete random walk.[6] 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.[7]
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 processPt with constant expected value which describes its future evolution:[9]
(1)
A process satisfying (1) 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 "".
The relationship (1) must hold for all times t: therefore the processes used for derivatives pricing are naturally set in continuous time.
The quants who operate in the Q world of derivatives pricing are specialists with deep knowledge of the specific products they model.
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 (1), a similar relationship is used to define the price of new derivatives.
Risk and portfolio management aims at modeling 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 "", as opposed to the "risk-neutral" probability "" 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 Outline of finance §Quantitative investing for a listing of relevant articles.
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.[11] 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.[12] See Financial risk management §Investment management.
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.[citation needed]
The aftermath of the financial crisis of 2009 as well as the multiple Flash Crashes of the early 2010s resulted in social uproars in the general population and ethical malaises in the scientific community which triggered noticeable changes in Quantitative Finance (QF). More specifically, mathematical finance was instructed to change and become more realistic as opposed to more convenient. The concurrent rise of Big data and Data Science contributed to facilitating these changes. More specifically, in terms of defining new models, we saw a significant increase in the use of Machine Learning overtaking traditional Mathematical Finance models.[13]
Over the years, increasingly sophisticated mathematical models and derivative pricing strategies have been developed, but their credibility was damaged by the financial crisis of 2007–2010. Contemporary practice of mathematical finance has been subjected to criticism from figures within the field notably by Paul Wilmott, and by Nassim Nicholas Taleb, in his book The Black Swan.[14] Taleb claims that the prices of financial assets cannot be characterized by the simple models currently in use, rendering much of current practice at best irrelevant, and, at worst, dangerously misleading. Wilmott and Emanuel Derman published the Financial Modelers' Manifesto in January 2009[15] which addresses some of the most serious concerns. Bodies such as the Institute for New Economic Thinking are now attempting to develop new theories and methods.[16]
In general, modeling the changes by distributions with finite variance is, increasingly, said to be inappropriate.[17] In the 1960s it was discovered by Benoit Mandelbrot that changes in prices do not follow a Gaussian distribution, but are rather modeled better by Lévy alpha-stable distributions.[18] The scale of change, or volatility, depends on the length of the time interval to a power a bit more than 1/2. Large changes up or down are more likely than what one would calculate using a Gaussian distribution with an estimated standard deviation. But the problem is that it does not solve the problem as it makes parametrization much harder and risk control less reliable.[14]
Perhaps more fundamental: though mathematical finance models may generate a profit in the short-run, this type of modeling is often in conflict with a central tenet of modern macroeconomics, the Lucas critique - or rational expectations - which states that observed relationships may not be structural in nature and thus may not be possible to exploit for public policy or for profit unless we have identified relationships using causal analysis and econometrics.[19] Mathematical finance models do not, therefore, incorporate complex elements of human psychology that are critical to modeling modern macroeconomic movements such as the self-fulfilling panic that motivates bank runs.
↑ For a survey, see "Financial Models", from Michael Mastro (2013). Financial Derivative and Energy Market Valuation, John Wiley & Sons. ISBN978-1118487716.
↑ Karatzas, Ioannis; Shreve, Steve (1998). Methods of Mathematical Finance. Secaucus, New Jersey, US: Springer-Verlag New York, Incorporated. ISBN9780387948393.
↑ Svetlozar T. Rachev; Frank J. Fabozzi; Christian Menn (2005). Fat-Tailed and Skewed Asset Return Distributions: Implications for Risk Management, Portfolio Selection, and Option Pricing. John Wiley and Sons. ISBN978-0471718864.
Financial economics is the branch of economics characterized by a "concentration on monetary activities", in which "money of one type or another is likely to appear on both sides of a trade". Its concern is thus the interrelation of financial variables, such as share prices, interest rates and exchange rates, as opposed to those concerning the real economy. It has two main areas of focus: asset pricing and corporate finance; the first being the perspective of providers of capital, i.e. investors, and the second of users of capital. It thus provides the theoretical underpinning for much of finance.
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the parabolic partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price given the risk of the security and its expected return. The equation and model are named after economists Fischer Black and Myron Scholes. Robert C. Merton, who first wrote an academic paper on the subject, is sometimes also credited.
In mathematical finance, a risk-neutral measure is a probability measure such that each share price is exactly equal to the discounted expectation of the share price under this measure. This is heavily used in the pricing of financial derivatives due to the fundamental theorem of asset pricing, which implies that in a complete market, a derivative's price is the discounted expected value of the future payoff under the unique risk-neutral measure. Such a measure exists if and only if the market is arbitrage-free.
Rational pricing is the assumption in financial economics that asset prices – and hence asset pricing models – will reflect the arbitrage-free price of the asset as any deviation from this price will be "arbitraged away". This assumption is useful in pricing fixed income securities, particularly bonds, and is fundamental to the pricing of derivative instruments.
Monte Carlo methods are used in corporate finance and mathematical finance to value and analyze (complex) instruments, portfolios and investments by simulating the various sources of uncertainty affecting their value, and then determining the distribution of their value over the range of resultant outcomes. This is usually done by help of stochastic asset models. The advantage of Monte Carlo methods over other techniques increases as the dimensions of the problem increase.
In mathematical finance, a Monte Carlo option model uses Monte Carlo methods to calculate the value of an option with multiple sources of uncertainty or with complicated features. The first application to option pricing was by Phelim Boyle in 1977. In 1996, M. Broadie and P. Glasserman showed how to price Asian options by Monte Carlo. An important development was the introduction in 1996 by Carriere of Monte Carlo methods for options with early exercise features.
In financial economics, asset pricing refers to a formal treatment and development of two interrelated pricing principles, outlined below, together with the resultant models. There have been many models developed for different situations, but correspondingly, these stem from either general equilibrium asset pricing or rational asset pricing, the latter corresponding to risk neutral pricing.
Financial modeling is the task of building an abstract representation of a real world financial situation. This is a mathematical model designed to represent the performance of a financial asset or portfolio of a business, project, or any other investment.
In finance, a price (premium) is paid or received for purchasing or selling options. This article discusses the calculation of this premium in general. For further detail, see: Mathematical finance § Derivatives pricing: the Q world for discussion of the mathematics; Financial engineering for the implementation; as well as Financial modeling § Quantitative finance generally.
In finance, a lattice model is a technique applied to the valuation of derivatives, where a discrete time model is required. For equity options, a typical example would be pricing an American option, where a decision as to option exercise is required at "all" times before and including maturity. A continuous model, on the other hand, such as Black–Scholes, would only allow for the valuation of European options, where exercise is on the option's maturity date. For interest rate derivatives lattices are additionally useful in that they address many of the issues encountered with continuous models, such as pull to par. The method is also used for valuing certain exotic options, where because of path dependence in the payoff, Monte Carlo methods for option pricing fail to account for optimal decisions to terminate the derivative by early exercise, though methods now exist for solving this problem.
The following outline is provided as an overview of and topical guide to finance:
In finance, an option is a contract which conveys to its owner, the holder, the right, but not the obligation, to buy or sell a specific quantity of an underlying asset or instrument at a specified strike price on or before a specified date, depending on the style of the option. Options are typically acquired by purchase, as a form of compensation, or as part of a complex financial transaction. Thus, they are also a form of asset and have a valuation that may depend on a complex relationship between underlying asset price, time until expiration, market volatility, the risk-free rate of interest, and the strike price of the option. Options may be traded between private parties in over-the-counter (OTC) transactions, or they may be exchange-traded in live, public markets in the form of standardized contracts.
In finance, a volatility swap is a forward contract on the future realised volatility of a given underlying asset. Volatility swaps allow investors to trade the volatility of an asset directly, much as they would trade a price index. Its payoff at expiration is equal to
In finance, the Heston model, named after Steven L. Heston, is a mathematical model that describes the evolution of the volatility of an underlying asset. It is a stochastic volatility model: such a model assumes that the volatility of the asset is not constant, nor even deterministic, but follows a random process.
A local volatility model, in mathematical finance and financial engineering, is an option pricing model that treats volatility as a function of both the current asset level and of time . As such, it is a generalisation of the Black–Scholes model, where the volatility is a constant. Local volatility models are often compared with stochastic volatility models, where the instantaneous volatility is not just a function of the asset level but depends also on a new "global" randomness coming from an additional random component.
In finance, model risk is the risk of loss resulting from using insufficiently accurate models to make decisions, originally and frequently in the context of valuing financial securities.
Damiano Brigo is a mathematician known for research in mathematical finance, filtering theory, stochastic analysis with differential geometry, probability theory and statistics, authoring more than 130 research publications and three monographs. From 2012 he serves as full professor with a chair in mathematical finance at the Department of Mathematics of Imperial College London, where he headed the Mathematical Finance group in 2012–2019. He is also a well known quantitative finance researcher, manager and advisor in the industry. His research has been cited and published also in mainstream industry publications, including Risk Magazine, where he has been the most cited author in the twenty years 1998–2017. He is often requested as a plenary or invited speaker both at academic and industry international events. Brigo's research has also been used in court as support for legal proceedings.
Quantitative analysis is the use of mathematical and statistical methods in finance and investment management. Those working in the field are quantitative analysts (quants). Quants tend to specialize in specific areas which may include derivative structuring or pricing, risk management, investment management and other related finance occupations. The occupation is similar to those in industrial mathematics in other industries. The process usually consists of searching vast databases for patterns, such as correlations among liquid assets or price-movement patterns.
Financial correlations measure the relationship between the changes of two or more financial variables over time. For example, the prices of equity stocks and fixed interest bonds often move in opposite directions: when investors sell stocks, they often use the proceeds to buy bonds and vice versa. In this case, stock and bond prices are negatively correlated.
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