Mathematical finance

Mathematical finance is a field of applied mathematics, concerned with financial markets. The subject has a close relationship with the discipline of financial economics, which is concerned with much of the underlying theory. Generally, mathematical finance will derive and extend the mathematical or numerical models suggested by financial economics. Thus, for example, while a financial economist might study the structural reasons why a company may have a certain share price, a financial mathematician may take the share price as a given, and attempt to use stochastic calculus to obtain the fair value of derivatives of the stock (see: Valuation of options).
In terms of practice, mathematical finance also overlaps heavily with the field of computational finance (also known as financial engineering). Arguably, these are largely synonymous, although the latter focuses on application, while the former focuses on modeling and derivation (see: Quantitative analyst). The fundamental theorem of arbitragefree pricing is one of the key theorems in mathematical finance. Many universities around the world now offer degree and research programs in mathematical finance; see Master of Mathematical Finance.
Contents
History: Q versus P
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 hand. One of the main differences is that they use different probabilities, namely the riskneutral probability, denoted by "Q", and the actual 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. Examples of securities being priced are plain vanilla and exotic options, convertible bonds, etc. Once a fair price has been determined, the sellside 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 sellside community.
Derivatives pricing: the Q world Goal "extrapolate the present" Environment riskneutral probability Processes continuoustime martingales Dimension low Tools Ito calculus, PDE’s Challenges calibration Business sellside Quantitative derivatives pricing was initiated by Louis Bachelier in The Theory of Speculation (published 1900), with the introduction of the most basic and most influential of processes, the Brownian motion, and its applications to the pricing of options. However, Bachelier's work hardly caught any attention outside academia.
Main article: Black–ScholesThe 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 of his death 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 a security is arbitragefree, and thus truly fair, only if there exists a stochastic process Pt with constant expected value which describes its future evolution:
(
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 "riskneutral" 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 lowdimensional in nature. Calibration is one of the main challenges of the Q world: once a continuoustime 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 continuoustime Qprocesses are Ito’s stochastic calculus and partial differential equations (PDE’s).Risk and portfolio management: the P world
Risk and portfolio management aims at modelling the 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 "riskneutral" probability "" used in derivatives pricing.
Based on the P distribution, the buyside community takes decisions on which securities to purchase in order to improve the prospective profitandloss profile of their positions considered as a portfolio.Risk and portfolio management: the P world Goal "model the future" Environment real probability Processes discretetime series Dimension large Tools multivariate statistics Challenges estimation Business buyside The quantitative theory of risk and portfolio management started with the meanvariance framework of Harry Markowitz (1952), who caused a shift away from the concept of trying to identify the best individual stock for investment. Using a linear regression strategy to understand and quantify the risk (i.e. variance) and return (i.e. mean) of an entire portfolio of stocks, bonds, and other securities, an optimization strategy was used to choose a portfolio with largest mean return subject to acceptable levels of variance in the return.
Next, breakthrough advances were made with the Capital Asset Pricing Model (CAPM) and the Arbitrage Pricing Theory (APT) developed by Treynor (1962), Mossin (1966), William Sharpe (1964), Lintner (1965) and Ross (1976).
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 portfolioselection work of Markowitz and Sharpe introduced mathematics to the “black art” of investment management.
With time, the mathematics has become more sophisticated. Thanks to Robert Merton and Paul Samuelson, oneperiod models were replaced by continuous time, Brownianmotion models, and the quadratic utility function implicit in mean–variance optimization was replaced by more general increasing, concave utility functions.^{[1]}
Furthermore, in more 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 ^{[2]}Criticism
More sophisticated mathematical models and derivative pricing strategies were then 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 Nassim Nicholas Taleb in his book The Black Swan^{[3]} and Paul Wilmott. 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 2008^{[4]} which addresses some of the most serious concerns.
Bodies such as the Institute for New Economic Thinking are now attempting to establish more effective theories and methods.^{[5]}Mathematical finance articles
Mathematical tools
 Asymptotic analysis
 Calculus
 Copulas
 Differential equations
 Expected value
 Ergodic theory
 Feynman–Kac formula
 Fourier transform
 Gaussian copulas
 Girsanov's theorem
 Itô's lemma
 Martingale representation theorem
 Mathematical models
 Monte Carlo method
 Numerical analysis
 Real analysis
 Partial differential equations
 Probability
 Probability distributions
 Quantile functions
 Radon–Nikodym derivative
 Riskneutral measure
 Stochastic calculus
 Stochastic differential equations
 Stochastic volatility
 Value at risk
 Volatility
Derivatives pricing
 The Brownian Motion Model of Financial Markets
 Rational pricing assumptions
 Risk neutral valuation
 Arbitragefree pricing
 Futures contract pricing
 Options
 Put–call parity (Arbitrage relationships for options)
 Intrinsic value, Time value
 Moneyness
 Pricing models
 Black–Scholes model
 Black model
 Binomial options model
 Monte Carlo option model
 Implied volatility, Volatility smile
 SABR Volatility Model
 Markov Switching Multifractal
 The Greeks
 Finite difference methods for option pricing
 Vanna Volga method
 Trinomial tree
 Optimal stopping (Pricing of American options)
 Interest rate derivatives
 Short rate model
 Hull–White model
 Cox–Ingersoll–Ross model
 Chen model
 LIBOR Market Model
 Heath–Jarrow–Morton framework
 Short rate model
See also
 Computational finance
 Quantitative Behavioral Finance
 Derivative (finance), list of derivatives topics
 Modeling and analysis of financial markets
 International Swaps and Derivatives Association
 Fundamental financial concepts  topics
 Model (economics)
 List of finance topics
 List of economics topics, List of economists
 List of accounting topics
 Statistical Finance
 Brownian model of financial markets
 Master of Mathematical Finance
Notes
 ^ Karatzas, I., Methods of Mathematical Finance, Secaucus, NJ, USA: SpringerVerlag New York, Incorporated, 1998
 ^ Meucci, A., Risk and Asset Allocation, Springer, 2005
 ^ Taleb, N. N. (2007) The Black Swan: The Impact of the Highly Improbable, Random House Trade, ISBN 9781400063512
 ^ http://www.wilmott.com/blogs/paul/index.cfm/2009/1/8/FinancialModelersManifesto
 ^ Gillian Tett (April 15, 2010), Mathematicians must get out of their ivory towers, Financial Times, http://www.ft.com/cms/s/0/cfb9c43a48b711df8af400144feab49a.html
References
 Harold Markowitz, Portfolio Selection, Journal of Finance, 7, 1952, pp. 77–91
 William Sharpe, Investments, PrenticeHall, 1985
 Attilio Meucci, P versus Q: Differences and Commonalities between the Two Areas of Quantitative Finance, GARP Risk Professional, February 2011, pp. 4144
External links
General areas of finance Computational finance · Experimental finance · Financial economics · Financial institutions · Financial markets · Investment management · Mathematical finance · Personal finance · Public finance · Quantitative behavioral finance · Quantum Finance · Statistical finance
Financial risk and financial risk management Categories Financial risk modeling Market portfolio · Riskfree rate · Modern portfolio theory · Risk parity · RAROC · Value at risk · Sharpe ratioBasic concepts Categories: Actuarial science
 Applied mathematics
 Formal sciences
 Mathematical finance
 Mathematical science occupations

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