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Valuation Of Options
In finance, a price (premium) is paid or received for purchasing or selling options. The calculation of this premium will require sophisticated mathematics. Premium components This price can be split into two components: intrinsic value, and time value (also called "extrinsic value"). Intrinsic value The ''intrinsic value'' is the difference between the underlying spot price and the strike price, to the extent that this is in favor of the option holder. For a call option, the option is in-the-money if the underlying spot price is higher than the strike price; then the intrinsic value is the underlying price minus the strike price. For a put option, the option is in-the-money if the ''strike'' price is higher than the underlying spot price; then the intrinsic value is the strike price minus the underlying spot price. Otherwise the intrinsic value is zero. For example, when a DJI call (bullish/long) option is 18,000 and the underlying DJI Index is priced at $18,050 then the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finance
Finance refers to monetary resources and to the study and Academic discipline, discipline of money, currency, assets and Liability (financial accounting), liabilities. As a subject of study, is a field of Business administration, Business Administration wich study the planning, organizing, leading, and controlling of an organization's resources to achieve its goals. Based on the scope of financial activities in financial systems, the discipline can be divided into Personal finance, personal, Corporate finance, corporate, and public finance. In these financial systems, assets are bought, sold, or traded as financial instruments, such as Currency, currencies, loans, Bond (finance), bonds, Share (finance), shares, stocks, Option (finance), options, Futures contract, futures, etc. Assets can also be banked, Investment, invested, and Insurance, insured to maximize value and minimize loss. In practice, Financial risk, risks are always present in any financial action and entities. Due ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moneyness
In finance, moneyness is the relative position of the current price (or future price) of an underlying asset (e.g., a stock) with respect to the strike price of a derivative, most commonly a call option or a put option. Moneyness is firstly a three-fold classification: * If the derivative would have positive intrinsic value if it were to expire today, it is said to be in the money (ITM); * If the derivative would be worthless if expiring with the underlying at its current price, it is said to be out of the money (OTM); * And if the current underlying price and strike price are equal, the derivative is said to be at the money (ATM). There are two slightly different definitions, according to whether one uses the current price (spot) or future price (forward), specified as "at the money spot" or "at the money forward", etc. This rough classification can be quantified by various definitions to express the moneyness as a number, measuring how far the asset is in the money or out o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trinomial Tree
The trinomial tree is a Lattice model (finance), lattice-based computational model used in financial mathematics to price option (finance), options. It was developed by Phelim Boyle in 1986. It is an extension of the binomial options pricing model, and is conceptually similar. It can also be shown that the approach is equivalent to the Finite difference method#Explicit method, explicit finite difference methods for option pricing, finite difference method for option pricing. For fixed income and interest rate derivatives see Lattice model (finance)#Interest rate derivatives. Formula Under the trinomial method, the underlying stock price is modeled as a recombining tree, where, at each node the price has three possible paths: an up, down and stable or middle path. These values are found by multiplying the value at the current node by the appropriate factor u\,, d\, or m\, where : u = e^ : d = e^ = \frac \, (the structure is recombining) : m = 1 \, and the corresponding probabilit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Options Pricing Model
In finance, the binomial options pricing model (BOPM) provides a generalizable numerical method for the valuation of options. Essentially, the model uses a "discrete-time" ( lattice based) model of the varying price over time of the underlying financial instrument, addressing cases where the closed-form Black–Scholes formula is wanting, which in general does not exist for the BOPM. The binomial model was first proposed by William Sharpe in the 1978 edition of ''Investments'' (), and formalized by Cox, Ross and Rubinstein in 1979 and by Rendleman and Bartter in that same year. For binomial trees as applied to fixed income and interest rate derivatives see . Use of the model The Binomial options pricing model approach has been widely used since it is able to handle a variety of conditions for which other models cannot easily be applied. This is largely because the BOPM is based on the description of an underlying instrument over a period of time rather than a single point ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice Model (finance)
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 other hand, such as the standard Black–Scholes one, would only allow for the valuation of European options, where exercise is limited to the option's maturity date. For interest rate derivatives lattices are additionally useful in that they address many of the issues encountered with continuous models, such as pull to par. The method is also used for valuing certain exotic options, because of path dependence in the payoff. Traditional Monte Carlo methods for option pricing fail to account for optimal decisions to terminate the derivative by early exercise, but some methods now exist for so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Black Model
The Black model (sometimes 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. Black's model can be generalized into a class of models known as log-normal forward models. The Black formula The Black formula is similar to the Black–Scholes formula for valuing stock options except that the spot price of the underlying is replaced by a discounted futures price F. Suppose there is constant risk-free interest rate ''r'' and the futures price ''F(t)'' of a particular underlying is log-normal with constant volatility ''σ''. Then the Black formula states the price for a European call option of maturity ''T'' on a futures contract with strike price ''K'' and delivery date ''T (with T' \geq T) is : c = e^ N(d_1) - KN(d_2)/math> The corresponding ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed-form Expression
In mathematics, an expression or equation is in closed form if it is formed with constants, variables, and a set of functions considered as ''basic'' and connected by arithmetic operations (, and integer powers) and function composition. Commonly, the basic functions that are allowed in closed forms are ''n''th root, exponential function, logarithm, and trigonometric functions. However, the set of basic functions depends on the context. For example, if one adds polynomial roots to the basic functions, the functions that have a closed form are called elementary functions. The ''closed-form problem'' arises when new ways are introduced for specifying mathematical objects, such as limits, series, and integrals: given an object specified with such tools, a natural problem is to find, if possible, a ''closed-form expression'' of this object; that is, an expression of this object in terms of previous ways of specifying it. Example: roots of polynomials The quadratic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asset Pricing
In financial economics, asset pricing refers to a formal treatment and development of two interrelated Price, pricing principles, outlined below, together with the resultant models. There have been many models developed for different situations, but correspondingly, these stem from either General equilibrium theory, general equilibrium asset pricing or Rational pricing, rational asset pricing, the latter corresponding to risk neutral pricing. Investment theory, which is near synonymous, encompasses the body of knowledge used to support the decision-making process of choosing investments, and the asset pricing models are then applied in determining the Required rate of return, asset-specific required rate of return on the investment in question, and for hedging. General equilibrium asset pricing Under general equilibrium theory prices are determined through Market price, market pricing by supply and demand. See, e.g., Tim Bollerslev (2019)"Risk and Return in Equilibrium: The C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic
Stochastic (; ) is the property of being well-described by a random probability distribution. ''Stochasticity'' and ''randomness'' are technically distinct concepts: the former refers to a modeling approach, while the latter describes phenomena; in everyday conversation, however, these terms are often used interchangeably. In probability theory, the formal concept of a '' stochastic process'' is also referred to as a ''random process''. Stochasticity is used in many different fields, including image processing, signal processing, computer science, information theory, telecommunications, chemistry, ecology, neuroscience, physics, and cryptography. It is also used in finance (e.g., stochastic oscillator), due to seemingly random changes in the different markets within the financial sector and in medicine, linguistics, music, media, colour theory, botany, manufacturing and geomorphology. Etymology The word ''stochastic'' in English was originally used as an adjective with the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heston Model
In finance, the Heston model, named after Steven L. Heston, is a mathematical model that describes the evolution of the volatility of an underlying asset. It is a stochastic volatility model: such a model assumes that the volatility of the asset is not constant, nor even deterministic, but follows a random process. Mathematical formulation The Heston model assumes that ''St'', the price of the asset, is determined by a stochastic process, : dS_t = \mu S_t\,dt + \sqrt S_t\,dW^S_t, where the volatility \sqrt is given by a Feller square-root or CIR process, : d\nu_t = \kappa(\theta - \nu_t)\,dt + \xi \sqrt\,dW^_t, and W^S_t, W^_t are Wiener processes (i.e., continuous random walks) with correlation ρ. The value \nu_t, being the square of the volatility, is called the instantaneous variance. The model has five parameters: * \nu_0, the initial variance. * \theta, the long variance, or long-run average variance of the price; as ''t'' tends to infinity, the expected value of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heath–Jarrow–Morton Framework
The Heath–Jarrow–Morton (HJM) framework is a general framework to model the evolution of interest rate curves – instantaneous forward rate curves in particular (as opposed to simple forward rates). When the volatility and drift of the instantaneous forward rate are assumed to be deterministic, this is known as the Gaussian Heath–Jarrow–Morton (HJM) model of forward rates. For direct modeling of simple forward rates the Brace–Gatarek–Musiela model represents an example. The HJM framework originates from the work of David Heath, Robert A. Jarrow, and Andrew Morton in the late 1980s, especially ''Bond pricing and the term structure of interest rates: a new methodology'' (1987) – working paper, Cornell University, and ''Bond pricing and the term structure of interest rates: a new methodology'' (1989) – working paper (revised ed.), Cornell University. It has its critics, however, with Paul Wilmott describing it as "...actually just a big rug for is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Black–Scholes Model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing Derivative (finance), derivative investment instruments. From the parabolic partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of option style, European-style option (finance), options and shows that the option has a ''unique'' price given the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). The equation and model are named after economists Fischer Black and Myron Scholes. Robert C. Merton, who first wrote an academic paper on the subject, is sometimes also credited. The main principle behind the model is to hedge (finance), hedge the option by buying and selling the underlying asset in a specific way to eliminate risk. This type of hedging is called "continuou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
