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Superhedging Price
The superhedging price is a coherent risk measure. The superhedging price of a portfolio (A) is equivalent to the smallest amount necessary to be paid for an admissible portfolio (B) at the current time so that at some specified future time the value of B is at least as great as A. In a complete market the superhedging price is equivalent to the price for hedging the initial portfolio. Mathematical definition If the set of equivalent martingale measures is denoted by EMM then the superhedging price of a portfolio ''X'' is \rho(-X) where \rho is defined by : \rho(X) = \sup_ \mathbb^Q X/math>. \rho defined as above is a coherent risk measure. Acceptance set The acceptance set for the superhedging price is the negative of the set of values of a self-financing portfolio at the terminal time. That is : A = \. Subhedging price The subhedging price is the greatest value that can be paid so that in any possible situation at the specified future time you have a second portfolio worth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coherent Risk Measure
In the fields of actuarial science and financial economics there are a number of ways that risk can be defined; to clarify the concept theoreticians have described a number of properties that a risk measure might or might not have. A coherent risk measure is a function that satisfies properties of monotonicity, sub-additivity, homogeneity, and translational invariance. Properties Consider a random outcome X viewed as an element of a linear space \mathcal of measurable functions, defined on an appropriate probability space. A functional \varrho : \mathcal → \R \cup \ is said to be coherent risk measure for \mathcal if it satisfies the following properties: Normalized : \varrho(0) = 0 That is, the risk when holding no assets is zero. Monotonicity : \mathrm\; Z_1,Z_2 \in \mathcal \;\mathrm\; Z_1 \leq Z_2 \; \mathrm ,\; \mathrm \; \varrho(Z_1) \geq \varrho(Z_2) That is, if portfolio Z_2 always has better values than portfolio Z_1 under almost all scenarios then the risk of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Financial Portfolio
In finance, a portfolio is a collection of investments. Definition The term "portfolio" refers to any combination of financial assets such as stocks, bonds and cash. Portfolios may be held by individual investors or managed by financial professionals, hedge funds, banks and other financial institutions. It is a generally accepted principle that a portfolio is designed according to the investor's risk tolerance, time frame and investment objectives. The monetary value of each asset may influence the risk/reward ratio of the portfolio. When determining asset allocation, the aim is to maximise the expected return and minimise the risk. This is an example of a multi-objective optimization problem: many efficient solutions are available and the preferred solution must be selected by considering a tradeoff between risk and return. In particular, a portfolio A is dominated by another portfolio A' if A' has a greater expected gain and a lesser risk than A. If no portfolio dominates A, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Admissible Portfolio
In finance, an admissible trading strategy or admissible strategy is any trading strategy with wealth almost surely bounded from below. In particular, an admissible trading strategy precludes unhedged short sales of any unbounded assets. A typical example of a trading strategy which is not ''admissible'' is the doubling strategy. Mathematical definition Discrete time In a market with d assets, a trading strategy x \in \mathbb^d is ''admissible'' if x^T \bar = x^T \frac is almost surely bounded from below. In the definition let S be the vector of prices, r be the risk-free rate (and therefore \bar is the discounted price). In a model with more than one time then the wealth process associated with an admissible trading strategy must be uniformly bounded from below. Continuous time Let S=(S_t)_ be a d-dimensional semimartingale market and H=(H_t)_ a predictable stochastic process/trading strategy. Then H is called ''admissible integrand for the semimartingale'' S or just ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Market
In economics, a complete market (aka Arrow-Debreu market or complete system of markets) is a market with two conditions: # Negligible transaction costs and therefore also perfect information, # Every asset in every possible state of the world has a price. In such a market, the complete set of possible bets on future states of the world can be constructed with existing assets without friction. Here, goods are state-contingent; that is, a good includes the time and state of the world in which it is consumed. For instance, an umbrella tomorrow if it rains is a distinct good from an umbrella tomorrow if it is clear. The study of complete markets is central to state-preference theory. The theory can be traced to the work of Kenneth Arrow (1964), Gérard Debreu Gérard Debreu (; 4 July 1921 – 31 December 2004) was a French-born economist and mathematician. Best known as a professor of economics at the University of California, Berkeley, where he began work in 1962, he won the 198 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hedge (finance)
A hedge is an investment Position (finance), position intended to offset potential losses or gains that may be incurred by a companion investment. A hedge can be constructed from many types of financial instruments, including stocks, exchange-traded funds, insurance policy, insurance, forward contracts, swap (finance), swaps, option (finance), options, gambles, many types of Over-the-counter (finance), over-the-counter and Derivative (finance), derivative products, and futures contracts. Public futures markets were established in the 19th century to allow transparent, standardized, and efficient hedging of agricultural commodity prices; they have since expanded to include futures contracts for hedging the values of Energy derivative, energy, precious metals, foreign currency, and interest rate fluctuations. Etymology Hedging is the practice of taking a position in one market to offset and balance against the risk adopted by assuming a position in a contrary or opposing market o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalent Martingale Measure
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. 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. A risk-neutral measure is a probability measure The easiest way to remember what the risk-neutral measure is, or to explain it to a probability generalist who might not know much about finance, is to realize that it is: # The probability measure of a transformed random variable. Typically this transformation is the utility function of the payoff. The risk-neutral measure would be the measure cor ... [...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] |
Self-financing Portfolio
In financial mathematics, a self-financing portfolio is a portfolio having the feature that, if there is no exogenous infusion or withdrawal of money, the purchase of a new asset must be financed by the sale of an old one. This concept is used to define for example admissible strategies and replicating portfolios, the latter being fundamental for arbitrage-free derivative pricing. Mathematical definition Discrete time Assume we are given a discrete filtered probability space (\Omega,\mathcal,\_^T,P), and let K_t be the solvency cone (with or without transaction costs) at time ''t'' for the market. Denote by L_d^p(K_t) = \. Then a portfolio (H_t)_^T (in physical units, i.e. the number of each stock) is self-financing (with trading on a finite set of times only) if : for all t \in \ we have that H_t - H_ \in -K_t \; P-a.s. with the convention that H_ = 0. If we are only concerned with the set that the portfolio can be at some future time then we can say that H_ \in -K_0 - ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supremum
In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, and if ''b'' is a lower bound of S, then ''b'' is less than or equal to the infimum of S. Consequently, the term ''greatest lower bound'' (abbreviated as ) is also commonly used. The supremum (abbreviated sup; : suprema) of a subset S of a partially ordered set P is the least element in P that is greater than or equal to each element of S, if such an element exists. If the supremum of S exists, it is unique, and if ''b'' is an upper bound of S, then the supremum of S is less than or equal to ''b''. Consequently, the supremum is also referred to as the ''least upper bound'' (or ). The infimum is, in a precise sense, dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infimum
In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, and if ''b'' is a lower bound of S, then ''b'' is less than or equal to the infimum of S. Consequently, the term ''greatest lower bound'' (abbreviated as ) is also commonly used. The supremum (abbreviated sup; : suprema) of a subset S of a partially ordered set P is the least element in P that is greater than or equal to each element of S, if such an element exists. If the supremum of S exists, it is unique, and if ''b'' is an upper bound of S, then the supremum of S is less than or equal to ''b''. Consequently, the supremum is also referred to as the ''least upper bound'' (or ). The infimum is, in a precise sense, dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in anal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
No-arbitrage Bounds
In financial mathematics, no-arbitrage bounds are mathematical relationships specifying limits on financial portfolio prices. These price bounds are a specific example of good–deal bounds, and are in fact the greatest extremes for good–deal bounds. The most frequent nontrivial example of no-arbitrage bounds is put–call parity for option prices. In incomplete markets, the bounds are given by the subhedging and superhedging prices. The essence of no-arbitrage in mathematical finance is excluding the possibility of "making money out of nothing" in the financial market. This is necessary because the existence of persistent risk free arbitrage Arbitrage (, ) is the practice of taking advantage of a difference in prices in two or more marketsstriking a combination of matching deals to capitalize on the difference, the profit being the difference between the market prices at which th ... opportunities is not only unrealistic, but also contradicts the possibility of an e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
