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Korn–Kreer–Lenssen Model
The Korn–Kreer–Lenssen model (KKL model) is a discrete trinomial model proposed in 1998 by Ralf Korn, Markus Kreer and Mark Lenssen to model illiquid securities and to value financial derivatives on these. It generalizes the binomial Cox-Ross-Rubinstein model in a natural way as the stock in a given time interval can either rise one unit up, fall one unit down or remain unchanged. In contrast to Black–Scholes or Cox-Ross-Rubinstein model the market consisting of stock and cash is not complete yet. To value and replicate a financial derivative an additional traded security related to the original security needs to be added. This might be a Low Exercise Price Option (or short LEPO). The mathematical proof of arbitrage free pricing is based on martingale representations for point processes pioneered in the 1980s and 1990 by Albert Shiryaev, Robert Liptser and Marc Yor. The dynamics is based on continuous time linear birth–death processes and analytic formulae for option p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trinomial Tree
The trinomial tree is a lattice-based computational model used in financial mathematics to price 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 explicit 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 probabilities are: : p_u = \left(\frac\right)^2 \, : p_d = \left(\frac\right)^2 \, : p_m = 1 - (p_u + p_d) \,. In the above formulae ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birth–death Process
The birth–death process (or birth-and-death process) is a special case of continuous-time Markov process where the state transitions are of only two types: "births", which increase the state variable by one and "deaths", which decrease the state by one. The model's name comes from a common application, the use of such models to represent the current size of a population where the transitions are literal births and deaths. Birth–death processes have many applications in demography, queueing theory, performance engineering, epidemiology, biology and other areas. They may be used, for example, to study the evolution of bacteria, the number of people with a disease within a population, or the number of customers in line at the supermarket. When a birth occurs, the process goes from state ''n'' to ''n'' + 1. When a death occurs, the process goes from state ''n'' to state ''n'' − 1. The process is specified by birth rates \_ and death rates \_. Recu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Options (finance)
In finance, an option is a contract which conveys to its owner, the ''holder'', the right, but not the obligation, to buy or sell a specific quantity of an underlying asset or instrument at a specified strike price on or before a specified date, depending on the style of the option. Options are typically acquired by purchase, as a form of compensation, or as part of a complex financial transaction. Thus, they are also a form of asset and have a valuation that may depend on a complex relationship between underlying asset price, time until expiration, market volatility, the risk-free rate of interest, and the strike price of the option. Options may be traded between private parties in ''over-the-counter'' (OTC) transactions, or they may be exchange-traded in live, public markets in the form of standardized contracts. Definition and application An option is a contract that allows the holder the right to buy or sell an underlying asset or financial instrument at a specified strike ... [...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] |
Financial Models
Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of Production (economics), production, Distribution (economics), distribution, and Consumption (economics), consumption of money, assets, goods and services (the discipline of financial economics bridges the two). Finance activities take place in Financial system, financial systems at various scopes, thus the field can be roughly divided into Personal finance, personal, Corporate finance, corporate, and public finance. In a financial system, assets are bought, sold, or traded as Financial instrument, financial instruments, such as Currency, currencies, Loan, loans, Bond (finance), bonds, Share (finance), shares, Stock, stocks, Option (finance), options, Futures contract, futures, etc. Assets can also be Bank, banked, Investment, invested, and Insurance, insured to maximize value and minimize loss. In practice, Financial risk, risks are alway ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Implementation
In the software development process, a reference implementation (or, less frequently, sample implementation or model implementation) is a program that implements all requirements from a corresponding specification. The reference implementation often accompanies a technical standard, and demonstrates what should be considered the "correct" behavior of any other implementation of it. Characteristics and examples Reference implementations of algorithms, for instance cryptographic algorithms, are often the result or the input of standardization processes. In this function they are often dedicated to the public domain with their source code as public domain software. Examples are the first CERN's httpd, Serpent cipher, base64 variants, and SHA-3. The Openwall Project maintains a list of several algorithms with their reference source code in the public domain. A reference implementation may or may not be production quality. For example, the Fraunhofer reference implementation of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Valuation Of Options
In finance, a price (premium) is paid or received for purchasing or selling options. This article discusses the calculation of this premium in general. For further detail, see: for discussion of the mathematics; Financial engineering for the implementation; as well as generally. Premium components This price can be split into two components: intrinsic value, and time 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 (bu ... [...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. 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. As a consequence, it is used to value Am ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trinomial Tree
The trinomial tree is a lattice-based computational model used in financial mathematics to price 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 explicit 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 probabilities are: : p_u = \left(\frac\right)^2 \, : p_d = \left(\frac\right)^2 \, : p_m = 1 - (p_u + p_d) \,. In the above formulae ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marc Yor
Marc Yor (24 July 1949 – 9 January 2014) was a French mathematician well known for his work on stochastic processes, especially properties of semimartingales, Brownian motion and other Lévy processes, the Bessel processes, and their applications to mathematical finance. Background Yor was a professor at the Paris VI University in Paris, France, from 1981 until his death in 2014. He was a recipient of several awards, including the Humboldt Prize, the Montyon Prize,Official biography at the French Academy website and was awarded the by the French Republic. He was a member of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Illiquid Securities
In business, economics or investment, market liquidity is a market's feature whereby an individual or firm can quickly purchase or sell an asset without causing a drastic change in the asset's price. Liquidity involves the trade-off between the price at which an asset can be sold, and how quickly it can be sold. In a liquid market, the trade-off is mild: one can sell quickly without having to accept a significantly lower price. In a relatively illiquid market, an asset must be discounted in order to sell quickly. Money, or cash, is the most liquid asset because it can be exchanged for goods and services instantly at face value. Overview A liquid asset has some or all of the following features: It can be sold rapidly, with minimal loss of value, anytime within market hours. The essential characteristic of a liquid market is that there are always ready and willing buyers and sellers. It is similar to, but distinct from, market depth, which relates to the trade-off between quantit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Albert Shiryaev
Albert Nikolayevich Shiryaev (russian: Альбе́рт Никола́евич Ширя́ев; born October 12, 1934) is a Soviet and Russian mathematician. He is known for his work in probability theory, statistics and financial mathematics. Career He graduated from Moscow State University in 1957. From that time until now he has been working in Steklov Mathematical Institute. He earned his candidate degree in 1961 (Andrey Kolmogorov was his advisor) and a doctoral degree in 1967 for his work "On statistical sequential analysis". He is a professor of the department of mechanics and mathematics of Moscow State University, since 1971. Shiryaev holds a 20% permanent professorial position at the School of Mathematics, University of Manchester. He has supervised more than 50 doctoral dissertations and is the author or coauthor of more than 250 publications. In 1970 he was an Invited Speaker with talk ''Sur les equations stochastiques aux dérivées partielles'' at the Internationa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
