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Entropic Risk Measure
In financial mathematics (concerned with mathematical modeling of financial markets), the entropic risk measure is a risk measure which depends on the risk aversion of the user through the exponential utility function. It is a possible alternative to other risk measures as value-at-risk or expected shortfall. It is a theoretically interesting measure because it provides different risk values for different individuals whose attitudes toward risk may differ. However, in practice it would be difficult to use since quantifying the risk aversion for an individual is difficult to do. The entropic risk measure is the prime example of a convex risk measure which is not coherent. Given the connection to utility functions, it can be used in utility maximization problems. Mathematical definition The entropic risk measure with the risk aversion parameter \theta > 0 is defined as : \rho^(X) = \frac\log\left(\mathbb ^right) = \sup_ \left\ \, where H(Q, P) = E\left frac\log\frac\right/math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Financial Mathematics
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. 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. Mathematical finance 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, while the former focuses, in addition to analysis, on building tools of implementation for the models. Also related is quantitative investing, which relies on statistical and numerical models (and lately machine learning) as opposed to traditional fundamental analysis when managing portfolios. French mathematician Louis Bachelier's doctoral thesis, defended in 1900, is considered the first scholarly work on mathematical fina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relative Entropy
Relative may refer to: General use *Kinship and family, the principle binding the most basic social units society. If two people are connected by circumstances of birth, they are said to be ''relatives'' Philosophy *Relativism, the concept that points of view have no absolute truth or validity, having only relative, subjective value according to differences in perception and consideration, or relatively, as in the relative value of an object to a person * Relative value (philosophy) Economics *Relative value (economics) Popular culture Film and television * ''Relatively Speaking'' (1965 play), 1965 British play * ''Relatively Speaking'' (game show), late 1980s television game show * ''Everything's Relative'' (episode)#Yu-Gi-Oh! (Yu-Gi-Oh! Duel Monsters), 2000 Japanese anime ''Yu-Gi-Oh! Duel Monsters'' episode *'' Relative Values'', 2000 film based on the play of the same name. *''It's All Relative'', 2003-4 comedy television series *''Intelligence is Relative'', tag line for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Financial Performance Measures
This article comprises a list of measures of financial performance. Return measures * Arithmetic return: average return of different observation periods * Geometric return: return depending only on start date and end date of one overall observation period * Rate of return or return on investment * Total shareholder return: annualized growth in capital assuming that dividends are reinvested Risk measures * Risk measure ** Distortion risk measure ** Tail conditional expectation ** Value at risk ** Convex risk measure *** Entropic risk measure ** Coherent risk measure *** Discounted maximum loss *** Expected shortfall *** Superhedging price *** Spectral risk measure * Deviation risk measure ** Standard deviation or Variance * Mid-range ** Interdecile range ** Interquartile range Risk-adjusted performance measures {{main, Financial ratio * Calmar ratio * Coefficient of variation * Information ratio * Jaws ratio * Jensen's alpha * Modigliani risk-adjusted performance * Roy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entropic Value At Risk
In financial mathematics and stochastic optimization, the concept of risk measure is used to quantify the risk involved in a random outcome or risk position. Many risk measures have hitherto been proposed, each having certain characteristics. The entropic value at risk (EVaR) is a coherent risk measure introduced by Ahmadi-Javid, which is an upper bound for the value at risk (VaR) and the conditional value at risk (CVaR), obtained from the Chernoff inequality. The EVaR can also be represented by using the concept of relative entropy. Because of its connection with the VaR and the relative entropy, this risk measure is called "entropic value at risk". The EVaR was developed to tackle some computational inefficiencies of the CVaR. Getting inspiration from the dual representation of the EVaR, Ahmadi-Javid developed a wide class of coherent risk measures, called g-entropic risk measures. Both the CVaR and the EVaR are members of this class. Definition Let (\Omega,\mathcal,P) be a pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Differential Equation
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 are used to model various phenomena such as stock prices or physical systems subject to thermal fluctuations. Typically, SDEs contain a variable which represents random white noise calculated as the derivative of Brownian motion or the Wiener process. However, other types of random behaviour are possible, such as jump processes. Random differential equations are conjugate to stochastic differential equations. Background Stochastic differential equations originated in the theory of Brownian motion, in the work of Albert Einstein and Smoluchowski. These early examples were linear stochastic differential equations, also called 'Langevin' equations after French physicist Langevin, describing the motion of a harmonic oscillator subject to a random force. The mathematical theory of stocha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Time Consistent
Time consistency in the context of finance is the property of not having mutually contradictory evaluations of risk at different points in time. This property implies that if investment A is considered riskier than B at some future time, then A will also be considered riskier than B at every prior time. Time consistency and financial risk Time consistency is a property in financial risk related to dynamic risk measures. The purpose of the time the consistent property is to categorize the risk measures which satisfy the condition that if portfolio (A) is riskier than portfolio (B) at some time in the future, then it is guaranteed to be riskier at any time prior to that point. This is an important property since if it were not to hold then there is an event (with probability of occurring greater than 0) such that B is riskier than A at time t although it is certain that A is riskier than B at time t+1. As the name suggests a time inconsistent risk measure can lead to inconsisten ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Risk Measure
In financial mathematics, a conditional risk measure is a random variable of the financial risk (particularly the downside risk) as if measured at some point in the future. A risk measure can be thought of as a conditional risk measure on the trivial sigma algebra. A dynamic risk measure is a risk measure that deals with the question of how evaluations of risk at different times are related. It can be interpreted as a sequence of conditional risk measures. A different approach to dynamic risk measurement has been suggested by Novak. Conditional risk measure Consider a portfolio's returns at some terminal time T as a random variable that is uniformly bounded, i.e., X \in L^\left(\mathcal_T\right) denotes the payoff of a portfolio. A mapping \rho_t: L^\left(\mathcal_T\right) \rightarrow L^_t = L^\left(\mathcal_t\right) is a conditional risk measure if it has the following properties for random portfolio returns X,Y \in L^\left(\mathcal_T\right): ; Conditional cash invariance ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acceptance Set
In financial mathematics, acceptance set is a set of acceptable future net worth which is acceptable to the regulator. It is related to risk measures. Mathematical Definition Given a probability space (\Omega,\mathcal,\mathbb), and letting L^p = L^p(\Omega,\mathcal,\mathbb) be the Lp space in the scalar case and L_d^p = L_d^p(\Omega,\mathcal,\mathbb) in d-dimensions, then we can define acceptance sets as below. Scalar Case An acceptance set is a set A satisfying: # A \supseteq L^p_+ # A \cap L^p_ = \emptyset such that L^p_ = \ # A \cap L^p_- = \ # Additionally if A is convex then it is a convex acceptance set ## And if A is a positively homogeneous cone then it is a coherent acceptance set Set-valued Case An acceptance set (in a space with d assets) is a set A \subseteq L^p_d satisfying: # u \in K_M \Rightarrow u1 \in A with 1 denoting the random variable that is constantly 1 \mathbb-a.s. # u \in -\mathrmK_M \Rightarrow u1 \not\in A # A is directionally closed in M with A + u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolutely Continuous
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of calculus— differentiation and integration. This relationship is commonly characterized (by the fundamental theorem of calculus) in the framework of Riemann integration, but with absolute continuity it may be formulated in terms of Lebesgue integration. For real-valued functions on the real line, two interrelated notions appear: absolute continuity of functions and absolute continuity of measures. These two notions are generalized in different directions. The usual derivative of a function is related to the '' Radon–Nikodym derivative'', or ''density'', of a measure. We have the following chains of inclusions for functions over a compact subset of the real line: : '' absolutely continuous'' ⊆ ''uniformly continuous'' = '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Utility Maximization Problem
Utility maximization was first developed by utilitarian philosophers Jeremy Bentham and John Stuart Mill. In microeconomics, the utility maximization problem is the problem consumers face: "How should I spend my money in order to maximize my utility?" It is a type of optimal decision problem. It consists of choosing how much of each available good or service to consume, taking into account a constraint on total spending (income), the prices of the goods and their preferences. Utility maximization is an important concept in consumer theory as it shows how consumers decide to allocate their income. Because consumers are rational, they seek to extract the most benefit for themselves. However, due to bounded rationality and other biases, consumers sometimes pick bundles that do not necessarily maximize their utility. The utility maximization bundle of the consumer is also not set and can change over time depending on their individual preferences of goods, price changes and incr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Risk Measure
In financial mathematics, a risk measure is used to determine the amount of an asset or set of assets (traditionally currency) to be kept in reserve. The purpose of this reserve is to make the risks taken by financial institutions, such as banks and insurance companies, acceptable to the regulator. In recent years attention has turned towards convex and coherent risk measurement. Mathematically A risk measure is defined as a mapping from a set of random variables to the real numbers. This set of random variables represents portfolio returns. The common notation for a risk measure associated with a random variable X is \rho(X). A risk measure \rho: \mathcal \to \mathbb \cup \ should have certain properties: ; Normalized : \rho(0) = 0 ; Translative : \mathrm\; a \in \mathbb \; \mathrm \; Z \in \mathcal ,\;\mathrm\; \rho(Z + a) = \rho(Z) - a ; Monotone : \mathrm\; Z_1,Z_2 \in \mathcal \;\mathrm\; Z_1 \leq Z_2 ,\; \mathrm \; \rho(Z_2) \leq \rho(Z_1) Set-valued In a situation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
