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In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all prior values. Stopped Brownian motion is an example of a martingale. It can model an even coin-toss ...
Martingale pricing is a pricing approach based on the notions of martingale and risk neutrality. The martingale pricing approach is a cornerstone of modern quantitative finance and can be applied to a variety of derivatives contracts, e.g. options , futures , interest rate derivatives , credit derivatives , etc.
If no equivalent martingale measure exists, arbitrage opportunities do. In markets with transaction costs, with no numéraire, the consistent pricing process takes the place of the equivalent martingale measure. There is in fact a 1-to-1 relation between a consistent pricing process and an equivalent martingale measure.
In a discrete (i.e. finite state) market, the following hold: [2] The First Fundamental Theorem of Asset Pricing: A discrete market on a discrete probability space (,,) is arbitrage-free if, and only if, there exists at least one risk neutral probability measure that is equivalent to the original probability measure, P.
In mathematics and information theory of probability, a sigma-martingale is a semimartingale with an integral representation. Sigma-martingales were introduced by C.S. Chou and M. Emery in 1977 and 1978. [ 1 ]
A discrete-time version of the theorem is given below, with 0 denoting the set of natural integers, including zero.. Let X = (X t) t∈ 0 be a discrete-time martingale and τ a stopping time with values in 0 ∪ {∞}, both with respect to a filtration (F t) t∈ 0.
Source code of game engine was released on GitHub under GPLv3 on November 28, 2022. [80] Age of Pirates 2: City of Abandoned Ships: 2009 2021 Windows Role-playing game: Akella: Source code of Storm Engine released on GitHub under GPLv3 in a 2021 and support Sea Dogs: To Each His Own and Age of Pirates 2: City of Abandoned Ships. [81] [82 ...
Financial modeling is the task of building an abstract representation (a model) of a real world financial situation. [1] This is a mathematical model designed to represent (a simplified version of) the performance of a financial asset or portfolio of a business, project, or any other investment.