# Mathematical finance

Last updated

Mathematical finance, also known as quantitative finance and financial mathematics, 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; 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. [1]

## Contents

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.

## 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

 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 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]

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

(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 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 ${\displaystyle \mathbb {P} }$ 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 "${\displaystyle \mathbb {P} }$", 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.

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 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] See also Variance gamma process#Option pricing.

## Notes

1. Johnson, Tim (September 2009). "What is financial mathematics?". +Plus Magazine. Retrieved 28 March 2014.
2. "Quantitative Finance". About.com. Retrieved 28 March 2014.
3. E., Shreve, Steven (2004). Stochastic calculus for finance. New York: Springer. ISBN   9780387401003. OCLC   53289874.
4. Stephen., Blyth (2013). Introduction to Quantitative Finance. Oxford University Press, USA. p. 157. ISBN   9780199666591. OCLC   868286679.
5. B., Schmidt, Anatoly (2005). Quantitative finance for physicists : an introduction. San Diego, Calif.: Elsevier Academic Press. ISBN   9780080492209. OCLC   57743436.
6. Bachelir, Louis. "The Theory of Speculation" . Retrieved 28 March 2014.
7. Lindbeck, Assar. "The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1969-2007". Nobel Prize. Retrieved 28 March 2014.
8. Brown, Angus (1 Dec 2008). "A risky business: How to price derivatives". Price+ Magazine. Retrieved 28 March 2014.
9. Karatzas, Ioannis; Shreve, Steve (1998). Methods of Mathematical Finance. Secaucus, NJ, USA: Springer-Verlag New York, Incorporated. ISBN   9780387948393.
10. Meucci, Attilio (2005). Risk and Asset Allocation. Springer. ISBN   9783642009648.
11. Taleb, Nassim Nicholas (2007). . Random House Trade. ISBN   978-1-4000-6351-2.
12. "Financial Modelers' Manifesto". Paul Wilmott's Blog. January 8, 2009. Archived from the original on September 8, 2014. Retrieved June 1, 2012.
13. Gillian Tett (April 15, 2010). "Mathematicians must get out of their ivory towers". Financial Times .
14. 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.
15. B. Mandelbrot, "The variation of certain Speculative Prices", The Journal of Business 1963

## Related Research Articles

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 prices, interest rates and shares, 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 underpin 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 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 regardless of the risk of the security and its expected return. The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. It is widely used, although often with some adjustments, by options market participants.

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 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 early exercise features.

In financial economics, asset pricing refers to a formal treatment and development of two main pricing principles, outlined below, together with the resultant models. There have been many models developed for different situations, but correspondingly, these stem from 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.

In statistics, stochastic volatility models are those in which the variance of a stochastic process is itself randomly distributed. They are used in the field of mathematical finance to evaluate derivative securities, such as options. The name derives from the models' treatment of the underlying security's volatility as a random process, governed by state variables such as the price level of the underlying security, the tendency of volatility to revert to some long-run mean value, and the variance of the volatility process itself, among others.

The following outline is provided as an overview of and topical guide to finance:

In mathematical finance, the SABR model is a stochastic volatility model, which attempts to capture the volatility smile in derivatives markets. The name stands for "stochastic alpha, beta, rho", referring to the parameters of the model. The SABR model is widely used by practitioners in the financial industry, especially in the interest rate derivative markets. It was developed by Patrick S. Hagan, Deep Kumar, Andrew Lesniewski, and Diana Woodward.

In finance, an option is a contract which gives the buyer the right, but not the obligation, to buy or sell an underlying asset or instrument at a specified strike price prior to or on a specified date, depending on the form of the option. The strike price may be set by reference to the spot price of the underlying security or commodity on the day an option is taken out, or it may be fixed at a discount or at a premium. The seller has the corresponding obligation to fulfill the transaction – to sell or buy – if the buyer (owner) "exercises" the option. An option that conveys to the owner the right to buy at a specific price is referred to as a call; an option that conveys the right of the owner to sell at a specific price is referred to as a put. Both are commonly traded, but the call option is more frequently discussed.

In finance, the Heston model, named after Steven Heston, is a mathematical model describing 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 one that treats volatility as a function of both the current asset level and of time . As such, a local volatility model is a generalisation of the Black-Scholes model, where the volatility is a constant.

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. However, model risk is more and more prevalent in activities other than financial securities valuation, such as assigning consumer credit scores, real-time probability prediction of fraudulent credit card transactions, and computing the probability of air flight passenger being a terrorist. Rebonato in 2002 defines model risk as "the risk of occurrence of a significant difference between the mark-to-model value of a complex and/or illiquid instrument, and the price at which the same instrument is revealed to have traded in the market".

Quantitative analysis is the use of mathematical and statistical methods in finance. Those working in the field are quantitative analysts. Quants tend to specialize in specific areas which may include derivative structuring or pricing, risk management, algorithmic trading and investment management. 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. The resulting strategies may involve high-frequency trading.

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.