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; 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]

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, 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

The Q world
Goal "extrapolate the present"
Environment risk-neutral probability
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]

(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.

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

The P world
Goal "model the future"
Environment real-world probability
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 "", 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.

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.


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.

Mathematical finance articles

See also Outline of finance: § Financial mathematics; § Mathematical tools; § Derivatives pricing.

Mathematical tools

Derivatives pricing

Portfolio modelling

Outline of finance § Portfolio theory

See also


  1. ^ Johnson, Tim. "What is financial mathematics?". +Plus Magazine. Retrieved 28 March 2014.
  2. ^ "Quantitative Finance". 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. ^ a b Taleb, Nassim Nicholas (2007). The Black Swan: The Impact of the Highly Improbable. Random House Trade. ISBN 978-1-4000-6351-2.
  12. ^ "Financial Modelers' Manifesto". Paul Wilmott's Blog. January 8, 2009. 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


Adjusted present value

The APV was introduced by the italian mathematician Lorenzo Peccati, Professor at the Bocconi University. The method is to calculate the NPV of the project as if it is all-equity financed (so called base case). Then the base-case NPV is adjusted for the benefits of financing. Usually, the main benefit is a tax shield resulted from tax deductibility of interest payments. Another benefit can be a subsidized borrowing at sub-market rates. The APV method is especially effective when a leveraged buyout case is considered since the company is loaded with an extreme amount of debt, so the tax shield is substantial.

Technically, an APV valuation model looks similar to a standard DCF model. However, instead of WACC, cash flows would be discounted at the unlevered cost of equity, and tax shields at either the cost of debt (Myers) or following later academics also with the unlevered cost of equity. APV and the standard DCF approaches should give the identical result if the capital structure remains stable.

Alpha (finance)

Alpha is a measure of the active return on an investment, the performance of that investment compared with a suitable market index. An alpha of 1% means the investment's return on investment over a selected period of time was 1% better than the market during that same period; a negative alpha means the investment underperformed the market. Alpha, along with beta, is one of two key coefficients in the capital asset pricing model used in modern portfolio theory and is closely related to other important quantities such as standard deviation, R-squared and the Sharpe ratio.In modern financial markets, where index funds are widely available for purchase, alpha is commonly used to judge the performance of mutual funds and similar investments. As these funds include various fees normally expressed in percent terms, the fund has to maintain an alpha greater than its fees in order to provide positive gains compared with an index fund. Historically, the vast majority of traditional funds have had negative alphas, which has led to a flight of capital to index funds and non-traditional hedge funds.

It is also possible to analyze a portfolio of investments and calculate a theoretical performance, most commonly using the capital asset pricing model (CAPM). Returns on that portfolio can be compared with the theoretical returns, in which case the measure is known as Jensen's alpha. This is useful for non-traditional or highly focused funds, where a single stock index might not be representative of the investment's holdings.

Compound annual growth rate

Compound annual growth rate (CAGR) is a business and investing specific term for the geometric progression ratio that provides a constant rate of return over the time period. CAGR is not an accounting term, but it is often used to describe some element of the business, for example revenue, units delivered, registered users, etc. CAGR dampens the effect of volatility of periodic returns that can render arithmetic means irrelevant. It is particularly useful to compare growth rates from various data sets of common domain such as revenue growth of companies in the same industry.CAGR is equivalent to the more generic exponential growth rate when the exponential growth interval is one year.

Computational finance

Computational finance is a branch of applied computer science that deals with problems of practical interest in finance. Some slightly different definitions are the study of data and algorithms currently used in finance and the mathematics of computer programs that realize financial models or systems.Computational finance emphasizes practical numerical methods rather than mathematical proofs and focuses on techniques that apply directly to economic analyses. It is an interdisciplinary field between mathematical finance and numerical methods. Two major areas are efficient and accurate computation of fair values of financial securities and the modeling of stochastic price series.

Correlation swap

A correlation swap is an over-the-counter financial derivative that allows one to speculate on or hedge risks associated with the observed average correlation, of a collection of underlying products, where each product has periodically observable prices, as with a commodity, exchange rate, interest rate, or stock index.

Delta neutral

In finance, delta neutral describes a portfolio of related financial securities, in which the portfolio value remains unchanged when small changes occur in the value of the underlying security. Such a portfolio typically contains options and their corresponding underlying securities such that positive and negative delta components offset, resulting in the portfolio's value being relatively insensitive to changes in the value of the underlying security.

A related term, delta hedging is the process of setting or keeping the delta of a portfolio as close to zero as possible. In practice, maintaining a zero delta is very complex because there are risks associated with re-hedging on large movements in the underlying stock's price, and research indicates portfolios tend to have lower cash flows if re-hedged too frequently.

Efficient frontier

This article is about a financial mathematical concept. For other frontiers described as efficient, see Production possibilities frontier and Pareto frontier.

In modern portfolio theory, the efficient frontier (or portfolio frontier) is an investment portfolio which occupies the 'efficient' parts of the risk-return spectrum. Formally, it is the set of portfolios which satisfy the condition that no other portfolio exists with a higher expected return but with the same standard deviation of return. The efficient frontier was first formulated by Harry Markowitz in 1952.

Enterprise value

Enterprise value (EV), total enterprise value (TEV), or firm value (FV) is an economic measure reflecting the market value of a business. It is a sum of claims by all claimants: creditors (secured and unsecured) and shareholders (preferred and common). Enterprise value is one of the fundamental metrics used in business valuation, financial modeling, accounting, portfolio analysis, and risk analysis.

Enterprise value is more comprehensive than market capitalization, which only reflects common equity. Importantly, EV reflects the opportunistic nature of business and may change substantially over time because of both external and internal conditions. Therefore, financial analysts often use a comfortable range of EV in their calculations.

Exotic option

In finance, an exotic option is an option which has features making it more complex than commonly traded vanilla options. Like the more general exotic derivatives they may have several triggers relating to determination of payoff. An exotic option may also include non-standard underlying instrument, developed for a particular client or for a particular market. Exotic options are more complex than options that trade on an exchange, and are generally traded over the counter (OTC).

Financial engineering

Financial engineering is a multidisciplinary field involving financial theory, methods of engineering, tools of mathematics and the practice of programming. It has also been defined as the application of technical methods, especially from mathematical finance and computational finance, in the practice of finance. Despite its name, financial engineering does not belong to any of the fields in traditional professional engineering even though many financial engineers have studied engineering beforehand and many universities offering a postgraduate degree in this field require applicants to have a background in engineering as well. In the United States, the Accreditation Board for Engineering and Technology (ABET) does not accredit financial engineering degrees. In the United States, financial engineering programs are accredited by the International Association of Quantitative Finance.Financial engineering draws on tools from applied mathematics, computer science, statistics and economic theory.

In the broadest sense, anyone who uses technical tools in finance could be called a financial engineer, for example any computer programmer in a bank or any statistician in a government economic bureau. However, most practitioners restrict the term to someone educated in the full range of tools of modern finance and whose work is informed by financial theory. It is sometimes restricted even further, to cover only those originating new financial products and strategies. Financial engineering plays a key role in the customer-driven derivatives business which encompasses quantitative modelling and programming, trading and risk managing derivative products in compliance with the regulations and Basel capital/liquidity requirements.

Finite difference methods for option pricing

Finite difference methods for option pricing are numerical methods used in mathematical finance for the valuation of options. Finite difference methods were first applied to option pricing by Eduardo Schwartz in 1977.In general, finite difference methods are used to price options by approximating the (continuous-time) differential equation that describes how an option price evolves over time by a set of (discrete-time) difference equations. The discrete difference equations may then be solved iteratively to calculate a price for the option. The approach arises since the evolution of the option value can be modelled via a partial differential equation (PDE), as a function of (at least) time and price of underlying; see for example Black–Scholes PDE. Once in this form, a finite difference model can be derived, and the valuation obtained.The approach can be used to solve derivative pricing problems that have, in general, the same level of complexity as those problems solved by tree approaches.

Malliavin calculus

In probability theory and related fields, Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to stochastic processes. In particular, it allows the computation of derivatives of random variables. Malliavin calculus is also called the stochastic calculus of variations.

Malliavin calculus is named after Paul Malliavin whose ideas led to a proof that Hörmander's condition implies the existence and smoothness of a density for the solution of a stochastic differential equation; Hörmander's original proof was based on the theory of partial differential equations. The calculus has been applied to stochastic partial differential equations as well.

The calculus allows integration by parts with random variables; this operation is used in mathematical finance to compute the sensitivities of financial derivatives. The calculus has applications in, for example, stochastic filtering.

Margrabe's formula

In mathematical finance, Margrabe's formula is an option pricing formula applicable to an option to exchange one risky asset for another risky asset at maturity. It was derived by William Margrabe (PhD Chicago) in 1978. Margrabe's paper has been cited by over 1500 subsequent articles.

Over-the-counter (finance)

Over-the-counter (OTC) or off-exchange trading is done directly between two parties, without the supervision of an exchange. It is contrasted with exchange trading, which occurs via exchanges. A stock exchange has the benefit of facilitating liquidity, providing transparency, and maintaining the current market price. In an OTC trade, the price is not necessarily published for the public.

OTC trading, as well as exchange trading, occurs with commodities, financial instruments (including stocks), and derivatives of such products. Products traded on the exchange must be well standardized. This means that exchanged deliverables match a narrow range of quantity, quality, and identity which is defined by the exchange and identical to all transactions of that product. This is necessary for there to be transparency in trading. The OTC market does not have this limitation. They may agree on an unusual quantity, for example. In OTC, market contracts are bilateral (i.e. the contract is only between two parties), and each party could have credit risk concerns with respect to the other party. The OTC derivative market is significant in some asset classes: interest rate, foreign exchange, stocks, and commodities.In 2008 approximately 16 percent of all U.S. stock trades were "off-exchange trading"; by April 2014 that number increased to about 40 percent. Although the notional amount outstanding of OTC derivatives in late 2012 had declined 3.3% over the previous year, the volume of cleared transactions at the end of 2012 totalled US$346.4 trillion. "The Bank for International Settlements statistics on OTC derivatives markets showed that notional amounts outstanding totalled $693 trillion at the end of June 2013... The gross market value of OTC derivatives – that is, the cost of replacing all outstanding contracts at current market prices – declined between end-2012 and end-June 2013, from $25 trillion to $20 trillion."

Returns-based style analysis

Returns-based style analysis is a statistical technique used in finance to deconstruct the returns of investment strategies using a variety of explanatory variables. The model results in a strategy’s exposures to asset classes or other factors, interpreted as a measure of a fund or portfolio manager’s style. While the model is most frequently used to show an equity mutual fund’s style with reference to common style axes (such as large/small and value/growth), recent applications have extended the model’s utility to model more complex strategies, such as those employed by hedge funds. Returns based strategies that use factors such as momentum signals (e.g., 52-week high) have been popular to the extent that industry analysts incorporate their use in their Buy/Sell recommendations.

Snell envelope

The Snell envelope, used in stochastics and mathematical finance, is the smallest supermartingale dominating a stochastic process. The Snell envelope is named after James Laurie Snell.

Stochastic calculus

Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. It is used to model systems that behave randomly.

The best-known stochastic process to which stochastic calculus is applied is the Wiener process (named in honor of Norbert Wiener), which is used for modeling Brownian motion as described by Louis Bachelier in 1900 and by Albert Einstein in 1905 and other physical diffusion processes in space of particles subject to random forces. Since the 1970s, the Wiener process has been widely applied in financial mathematics and economics to model the evolution in time of stock prices and bond interest rates.

The main flavours of stochastic calculus are the Itô calculus and its variational relative the Malliavin calculus. For technical reasons the Itô integral is the most useful for general classes of processes, but the related Stratonovich integral is frequently useful in problem formulation (particularly in engineering disciplines). The Stratonovich integral can readily be expressed in terms of the Itô integral. The main benefit of the Stratonovich integral is that it obeys the usual chain rule and therefore does not require Itô's lemma. This enables problems to be expressed in a coordinate system invariant form, which is invaluable when developing stochastic calculus on manifolds other than Rn.

The dominated convergence theorem does not hold for the Stratonovich integral, consequently it is very difficult to prove results without re-expressing the integrals in Itô form.

Trinomial tree

The trinomial tree is a lattice based computational model used in financial mathematics to price options. It was developed by Phelim Boyle in 1986. It is an extension of the binomial options pricing model, and is conceptually similar. It can also be shown that the approach is equivalent to the explicit finite difference method for option pricing. For fixed income and interest rate derivatives see Lattice model (finance) #Interest rate derivatives.

Valuation of options

In finance, a price (premium) is paid or received for purchasing or selling options. This price can be split into two components.

These are:

Intrinsic value

Time value

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