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An automated trading system (ATS), a subset of algorithmic trading, uses a computer program to create buy and sell orders and automatically submits the orders to a market center or exchange. The computer program will automatically generate orders based on predefined set of rules using a trading strategy which is based on technical analysis, advanced statistical and mathematical computati

The Bachelier model is a model of an asset price under Brownian motion presented by Louis Bachelier in his PhD thesis The Theory of Speculation (French: Théorie de la spéculation), published 1900. It is also called the normal model equivalently (as opposed to log-normal model or Black–Scholes model). One early criticism of the Bachelier model is that the probability distribution which h

The Black model (also known as the Black-76 model) is a variant of the Black–Scholes option pricing model. Its primary applications are for pricing options on future contracts, bond options, interest rate cap and floors, and swaptions. It was first presented in a paper written by Fischer Black in 1976.

In mathematical finance, the Black–Derman–Toy model (BDT) is a popular short-rate model used in the pricing of bond options, swaptions and other interest rate derivatives; see Lattice model (finance) § Interest rate derivatives. It is a one-factor model; that is, a single stochastic factor—the short rate—determines the future evolution of all interest rates. It was the first model to com

In mathematical finance, the Black–Scholes equation, also called the Black–Scholes–Merton equation, is a partial differential equation (PDE) governing the price evolution of derivatives under the Black–Scholes model. Broadly speaking, the term may refer to a similar PDE that can be derived for a variety of options, or more generally, derivatives.

In economics, a complete market (a.k.a. Arrow-Debreu market or complete system of markets) is a market with two conditions:

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.

The drawdown is the measure of the decline from a historical peak in some variable (typically the cumulative profit or total open equity of a financial trading strategy).

Fairmat is a free-of-charge multi-platform software that allows to model financial contracts (e.g. a derivative contract) or projects with many contingencies (e.g. a Real Options model) by decomposing it into basic parts. Complex structures and dependencies are modelled using a graphical interface. Virtually any pay-off function and asset class( from interest rate derivatives to equity-l

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.

In statistical mechanics and information theory, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the position or velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. The equation can be generalized to other observables as well. The Fokker–Planck equation h

The fundamental theorems of asset pricing (also: of arbitrage, of finance), in both financial economics and mathematical finance, provide necessary and sufficient conditions for a market to be arbitrage-free, and for a market to be complete. An arbitrage opportunity is a way of making money with no initial investment without any possibility of loss. Though arbitrage opportunities do exis

In economics, incomplete markets are markets in which there does not exist an Arrow–Debreu security for every possible state of nature. In contrast with complete markets, this shortage of securities will likely restrict individuals from transferring the desired level of wealth among states.

Intertemporal portfolio choice is the process of allocating one's investable wealth to various assets, especially financial assets, repeatedly over time, in such a way as to optimize some criterion. The set of asset proportions at any time defines a portfolio. Since the returns on almost all assets are not fully predictable, the criterion has to take financial risk into account. Typicall

Itô calculus, named after Kiyosi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance, in stochastic differential equations, and more recently even in machine learning.

In quantitative finance, a lattice model is a numerical approach to the valuation of derivatives in situations requiring a discrete time model. For dividend paying equity options, a typical application would correspond to the pricing of an American-style option, where a decision to exercise is allowed at the closing of any calendar day up to the maturity. A continuous model, on the othe

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 C

In mathematics, the theory of optimal stopping or early stopping is concerned with the problem of choosing a time to take a particular action, in order to maximise an expected reward or minimise an expected cost. Optimal stopping problems can be found in areas of statistics, economics, and mathematical finance (related to the pricing of American options). A key example of an optimal stop

In financial mathematics, the put–call parity defines a relationship between the price of a European call option and European put option, both with the identical strike price and expiry, namely that a portfolio of a long call option and a short put option is equivalent to (and hence has the same value as) a single forward contract at this strike price and expiry. This is because if the p

In financial economics, rational pricing is the assumption that asset pricing models must reflect the arbitrage-free price of the asset

Realized variance or realised variance (RV, see spelling differences) is the sum of squared returns. For instance, the RV can be the sum of squared daily returns for a particular month, which would yield a measure of price variation over this month. More commonly, the realized variance is computed as the sum of squared intraday returns for a particular day.

In mathematical finance, a risk-neutral measure (also called an equilibrium measure, or equivalent martingale measure) is a probability measure such that each share price is exactly equal to the discounted expectation of the share price under this measure.

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. This field was created and started by the Japanese mathematician Kiyosi Itô during World War II.

A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs have many applications throughout pure mathematics and are used to model various behaviours of stochastic models such as prices of listed company shares, random growth models or physical systems tha

Stochastic Volatility Jump Models (SVJ models) are a class of mathematical models in quantitative finance that combine stochastic volatility dynamics with discontinuous jumps in asset prices. These models aim to more accurately reflect the empirical characteristics of financial markets, particularly those that deviate from the assumptions of classical models such as the Black–Scholes mod

The volatility tax is a mathematical finance term first published by Rick Ashburn, CFA in a 2003 column, and formalized by hedge fund manager Mark Spitznagel, describing the effect of large investment losses (or volatility) on compound returns. It has also been called volatility drag, volatility decay or variance drain. This is not literally a tax in the sense of a levy imposed by a gov