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Good–deal Bounds
Good–deal bounds are price bounds for a financial portfolio which depends on an individual trader's preferences. Mathematically, if A is a set of portfolios with future outcomes which are "acceptable" to the trader, then define the function \rho: \mathcal^p \to \mathbb by :\rho(X) = \inf\left\ = \inf\left\ where A_T is the set of final values for self-financing trading strategies. Then any price in the range (-\rho(X), \rho(-X)) does not provide a good deal for this trader, and this range is called the "no good-deal price bounds." If A = \left\ then the good-deal price bounds are the no-arbitrage price bounds, and correspond to the subhedging and superhedging prices. The no-arbitrage bounds are the greatest extremes that good-deal bounds can take. If A = \left\ where u is a utility function, then the good-deal price bounds correspond to the indifference price In finance, indifference pricing is a method of pricing financial securities with regard to a utility function. The ... [...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 dominat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-financing Trading Strategy
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. Mathematical definition Let h_i(t) denote the number of shares of stock number 'i' in the portfolio at time t , and S_i(t) the price of stock number 'i' in a frictionless market with trading in continuous time. Let : V(t) = \sum_^ h_i(t) S_i(t). Then the portfolio (h_1(t), \dots, h_n(t)) is self-financing if : dV(t) = \sum_^ h_i(t) dS_(t). 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 - ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
No-arbitrage Price 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 arbitrage is not only unrealistic, but also contradicts the possibility of an economic equilibrium. All mathematical models of financial markets have to satisfy a no-arbitrage condition to be realistic models. See also * Box spread * Indifference price In finance, indifference pricing is a method of pricing financial securities with regard to a utility function. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Utility Function
As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosophers such as Jeremy Bentham and John Stuart Mill. The term has been adapted and reapplied within neoclassical economics, which dominates modern economic theory, as a utility function that represents a single consumer's preference ordering over a choice set but is not comparable across consumers. This concept of utility is personal and based on choice rather than on pleasure received, and so is specified more rigorously than the original concept but makes it less useful (and controversial) for ethical decisions. Utility function Consider a set of alternatives among which a person can make a preference ordering. The utility obtained from these alternatives is an unknown function of the utilities obtained from each alternative, not the sum of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indifference Price
In finance, indifference pricing is a method of pricing financial securities with regard to a utility function. The indifference price is also known as the reservation price or private valuation. In particular, the indifference price is the price at which an agent would have the same expected utility level by exercising a financial transaction as by not doing so (with optimal trading otherwise). Typically the indifference price is a pricing range (a bid–ask spread) for a specific agent; this price range is an example of good-deal bounds. Mathematics Given a utility function u and a claim C_T with known payoffs at some terminal time T, let the function V: \mathbb \times \mathbb \to \mathbb be defined by : V(x,k) = \sup_ \mathbb\left \left(X_T + k C_T\right)\right/math>, where x is the initial endowment, \mathcal(x) is the set of all self-financing portfolios at time T starting with endowment x, and k is the number of the claim to be purchased (or sold). Then the indifference b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Finance
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] |
