Mathematical finance
Mathematical finance, also known as quantitative finance, is a field of applied mathematics, concerned with mathematical modeling of financial markets. Generally, mathematical finance will derive and extend the mathematical or numerical models without necessarily establishing a link to financial theory, taking observed market prices as input. Mathematical consistency is required, not compatibility with economic theory. 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 corresponding value of derivatives of the stock (see: Valuation of options; Financial modeling). 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.[1]
Mathematical finance also overlaps heavily with the fields of computational finance and financial engineering. The latter focuses on applications and modeling, often by help of stochastic asset models (see: Quantitative analyst), while the former focuses, in addition to analysis, on building tools of implementation for the models. 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.[2]
French mathematician Louis Bachelier is considered the author of the first scholarly work on mathematical finance, published in 1900. 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.
Today many universities offer degree and research programs in mathematical finance.
Contents
1 History: Q versus P
1.1 Derivatives pricing: the Q world
1.2 Risk and portfolio management: the P world
2 Criticism
3 Mathematical finance articles
3.1 Mathematical tools
3.2 Derivatives pricing
3.3 Portfolio modelling
4 See also
5 Notes
6 References
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, namely 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
Goal | "extrapolate the present" |
Environment | risk-neutral probability Q{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 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, the Brownian motion, and its applications to the pricing of options.[3][4] The Brownian motion is derived using the Langevin equation and the discrete random walk.[5] 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.[6]
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 of his death in 1995.[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 a 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:[8]
P0=E0(Pt){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 "Q{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 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.
The main quantitative tools necessary to handle continuous-time Q-processes are Itō’s stochastic calculus, simulation and partial differential equations (PDE’s).
Risk and portfolio management: the P world
Goal | "model the future" |
Environment | real-world probability P{displaystyle mathbb {P} } |
Processes | discrete-time series |
Dimension | large |
Tools | multivariate statistics |
Challenges | estimation |
Business | buy-side |
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 "P{displaystyle mathbb {P} }", as opposed to the "risk-neutral" probability "Q{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.
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.[9] 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.[10]
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.
Criticism
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.[11] 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[12] 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.[13]
In general, modeling the changes by distributions with finite variance is, increasingly, said to be inappropriate.[14] 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.[15] 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.[11]
Mathematical finance articles
- See also Outline of finance: § Financial mathematics; § Mathematical tools; § Derivatives pricing.
Mathematical tools
- Asymptotic analysis
- Calculus
Copulas, including Gaussian- Differential equations
- Expected value
- Ergodic theory
- Feynman–Kac formula
- Fourier transform
- Girsanov theorem
- Itô's lemma
- Martingale representation theorem
- Mathematical models
- Monte Carlo method
- Numerical analysis
- Real analysis
Partial differential equations
- Heat equation
Numerical partial differential equations
- Crank–Nicolson method
- Finite difference method
- Probability
Probability distributions
- Binomial distribution
- Log-normal distribution
- Quantile functions
- Radon–Nikodym derivative
- Risk-neutral measure
- Scenario optimization
Stochastic calculus
- Brownian motion
- Lévy process
- Stochastic differential equation
- Stochastic optimization
- Stochastic volatility
- Value at risk
Volatility
- ARCH model
- GARCH model
Derivatives pricing
- The Brownian model of financial markets
Rational pricing assumptions
- Risk neutral valuation
Arbitrage-free pricing
- Forward Price Formula
- Futures contract pricing
Swap valuation
- Currency swap #Valuation and Pricing
- Interest rate swap #Valuation and pricing
- Variance swap #Pricing and valuation
- Options
Put–call parity (Arbitrage relationships for options)
Intrinsic value, Time value
- Moneyness
- Pricing models
- Black–Scholes model
- Black model
Binomial options model
- Implied binomial tree
- Edgeworth binomial tree
- Monte Carlo option model
Implied volatility, Volatility smile
- Local volatility
Stochastic volatility
- Constant elasticity of variance model
- Heston model
- SABR volatility model
- Markov switching multifractal
- The Greeks
- Finite difference methods for option pricing
- Vanna–Volga pricing
Trinomial tree
- Implied trinomial tree
- Garman-Kohlhagen model
- Lattice model (finance)
- Margrabe's formula
- Pricing of American options
- Barone-Adesi and Whaley
- Bjerksund and Stensland
- Black's approximation
- Least Square Monte Carlo
- Optimal stopping
- Roll-Geske-Whaley
Interest rate derivatives
Black model
- caps and floors
- swaptions
- Bond options
Short-rate models
- Rendleman–Bartter model
- Vasicek model
- Ho–Lee model
- Hull–White model
- Cox–Ingersoll–Ross model
- Black–Karasinski model
- Black–Derman–Toy model
- Kalotay–Williams–Fabozzi model
- Longstaff–Schwartz model
- Chen model
Forward rate-based models
LIBOR market model (Brace–Gatarek–Musiela Model, BGM)
Heath–Jarrow–Morton Model (HJM)
- Valuation adjustments
- Credit valuation adjustment
- XVA
Portfolio modelling
Outline of finance § Portfolio theory
See also
|
- Brownian model of financial markets
- Computational finance
Derivative (finance), list of derivatives topics
- Economic model
- Financial economics
- Financial engineering
- Financial modeling § Quantitative finance
- International Swaps and Derivatives Association
- Index of accounting articles
- List of economists
- Master of Quantitative Finance
- Outline of economics
- Outline of finance
- Quantitative behavioral finance
- Statistical finance
- Technical analysis
- XVA
Notes
^ Johnson, Tim. "What is financial mathematics?". +Plus Magazine. Retrieved 28 March 2014..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
^ "Quantitative Finance". About.com. Retrieved 28 March 2014.
^ E., Shreve, Steven (2004). Stochastic calculus for finance. New York: Springer. ISBN 9780387401003. OCLC 53289874.
^ Stephen., Blyth, (2013). Introduction to Quantitative Finance. Oxford University Press, USA. p. 157. ISBN 9780199666591. OCLC 868286679.
^ B., Schmidt, Anatoly (2005). Quantitative finance for physicists : an introduction. San Diego, Calif.: Elsevier Academic Press. ISBN 9780080492209. OCLC 57743436.
^ Bachelir, Louis. "The Theory of Speculation". Retrieved 28 March 2014.
^ Lindbeck, Assar. "The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1969-2007". Nobel Prize. Retrieved 28 March 2014.
^ Brown, Angus (1 Dec 2008). "A risky business: How to price derivatives". Price+ Magazine. Retrieved 28 March 2014.
^ Karatzas, Ioannis; Shreve, Steve (1998). Methods of Mathematical Finance. Secaucus, NJ, USA: Springer-Verlag New York, Incorporated. ISBN 9780387948393.
^ Meucci, Attilio (2005). Risk and Asset Allocation. Springer. ISBN 9783642009648.
^ ab Taleb, Nassim Nicholas (2007). The Black Swan: The Impact of the Highly Improbable. Random House Trade. ISBN 978-1-4000-6351-2.
^ "Financial Modelers' Manifesto". Paul Wilmott's Blog. January 8, 2009. Retrieved June 1, 2012.
^ Gillian Tett (April 15, 2010). "Mathematicians must get out of their ivory towers". Financial Times.
^ 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. ISBN 978-0471718864.
^ B. Mandelbrot, "The variation of certain Speculative Prices", The Journal of Business 1963
References
Harold Markowitz, "Portfolio Selection", The Journal of Finance, 7, 1952, pp. 77–91
William F. Sharpe, Investments, Prentice-Hall, 1985- Attilio Meucci, P versus Q: Differences and Commonalities between the Two Areas of Quantitative Finance, GARP Risk Professional, February 2011, pp. 41–44
Nicole El Karoui, The Future of Financial Mathematics, ParisTech Review, September 2013