Mathematical finance

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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

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

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".

Derivatives pricing: the Q world

The Q world

Goal	"extrapolate the present"
Environment	risk-neutral probability ${\displaystyle \mathbb {Q} }$
Processes	continuous-time martingales
Dimension	low
Tools	Itō calculus, PDEs
Challenges	calibration
Business	sell-side

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

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 P_t with constant expected value which describes its future evolution:^[9]

${\displaystyle P_{0}=\mathbf {E} _{0}(P_{t})}$

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 "${\displaystyle \mathbb {Q} }$".

The relationship (1) must hold for all times t: therefore the processes used for derivatives pricing are naturally set in continuous time.

The

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.

The main quantitative tools necessary to handle continuous-time Q-processes are Itô's stochastic calculus, simulation and partial differential equations (PDEs).^[10]

The P world

Goal	"model the future"
Environment	real-world probability ${\displaystyle \mathbb {P} }$
Processes	discrete-time series
Dimension	large
Tools	multivariate statistics
Challenges	estimation
Business	buy-side

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 "${\displaystyle \mathbb {P} }$", as opposed to the "risk-neutral" probability "${\displaystyle \mathbb {Q} }$" 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 Outline of finance § Quantitative investing 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

Much effort has gone into the study of financial markets and how prices vary with time.

Over the years, increasingly sophisticated mathematical models and derivative pricing strategies have been developed, but their credibility was damaged by the 2008 financial crisis.

Contemporary practice of mathematical finance has been subjected to criticism from figures within the field notably by

In general, modeling the changes by distributions with finite variance is, increasingly, said to be inappropriate.