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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] |
Lattice Model (finance)
In finance, a lattice model is a technique applied to the valuation of derivatives, where a discrete time model is required. For equity options, a typical example would be pricing an American option, where a decision as to option exercise is required at "all" times (any time) before and including maturity. A continuous model, on the other hand, such as Black–Scholes, would only allow for the valuation of European options, where exercise is on 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, where because of path dependence in the payoff, Monte Carlo methods for option pricing fail to account for optimal decisions to terminate the derivative by early exercise, though methods now exist for solving this problem. Equity and commodity derivatives In general the approach is to divid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moment (mathematics)
In mathematics, the moments of a function are certain quantitative measures related to the shape of the function's graph. If the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total mass) is the center of mass, and the second moment is the moment of inertia. If the function is a probability distribution, then the first moment is the expected value, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis. The mathematical concept is closely related to the concept of moment in physics. For a distribution of mass or probability on a bounded interval, the collection of all the moments (of all orders, from to ) uniquely determines the distribution (Hausdorff moment problem). The same is not true on unbounded intervals (Hamburger moment problem). In the mid-nineteenth century, Pafnuty Chebyshev became the first person to think systematic ... [...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] |
Aalto University
Aalto University ( fi, Aalto-yliopisto; sv, Aalto-universitetet) is a public research university located in Espoo, Finland. It was established in 2010 as a merger of three major Finnish universities: the Helsinki University of Technology, the Helsinki School of Economics and the University of Art and Design Helsinki. The close collaboration between the scientific, business and arts communities is intended to foster multi-disciplinary education and research. The Finnish government, in 2010, set out to create a university that fosters innovation, merging the three institutions into one. The university is composed of six schools with close to 17,500 students and 4,000 staff members, making it Finland's second largest university. The main campus of Aalto University is located in Otaniemi, Espoo. Aalto University Executive Education operates in the district of Töölö, Helsinki. In addition to the Greater Helsinki area, the university also operates its Bachelor's Programme in Inter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Warwick
The University of Warwick ( ; abbreviated as ''Warw.'' in post-nominal letters) is a public research university on the outskirts of Coventry between the West Midlands (county), West Midlands and Warwickshire, England. The university was founded in 1965 as part of a government initiative to expand higher education. The Warwick Business School was established in 1967, the Warwick Law School in 1968, WMG, University of Warwick, Warwick Manufacturing Group (WMG) in 1980, and Warwick Medical School in 2000. Warwick incorporated Coventry College of Education in 1979 and Horticulture Research International in 2004. Warwick is primarily based on a campus on the outskirts of Coventry, with a satellite campus in Wellesbourne and a central London base at the Shard. It is organised into three faculties—Arts, Science Engineering and Medicine, and Social Sciences—within which there are 32 departments. As of 2021, Warwick has around 29,534 full-time students and 2,691 academic and research ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Derivatives
A journal, from the Old French ''journal'' (meaning "daily"), may refer to: *Bullet journal, a method of personal organization *Diary, a record of what happened over the course of a day or other period *Daybook, also known as a general journal, a daily record of financial transactions *Logbook, a record of events important to the operation of a vehicle, facility, or otherwise *Record (other) *Transaction log, a chronological record of data processing *Travel journal In publishing, ''journal'' can refer to various periodicals or serials: *Academic journal, an academic or scholarly periodical **Scientific journal, an academic journal focusing on science **Medical journal, an academic journal focusing on medicine **Law review, a professional journal focusing on legal interpretation *Magazine, non-academic or scholarly periodicals in general **Trade magazine, a magazine of interest to those of a particular profession or trade **Literary magazine, a magazine devoted to literat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Implied Trinomial Tree
In finance, a lattice model is a technique applied to the valuation of derivatives, where a discrete time model is required. For equity options, a typical example would be pricing an American option, where a decision as to option exercise is required at "all" times (any time) before and including maturity. A continuous model, on the other hand, such as Black–Scholes, would only allow for the valuation of European options, where exercise is on 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, where because of path dependence in the payoff, Monte Carlo methods for option pricing fail to account for optimal decisions to terminate the derivative by early exercise, though methods now exist for solving this problem. Equity and commodity derivatives In general the approach is to divid ... [...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] |
Exotic Option
In finance, an exotic option is an option which has features making it more complex than commonly traded vanilla options. Like the more general exotic derivatives they may have several triggers relating to determination of payoff. An exotic option may also include a non-standard underlying instrument, developed for a particular client or for a particular market. Exotic options are more complex than options that trade on an exchange, and are generally traded over-the-counter. Etymology The term "exotic option" was popularized by Mark Rubinstein's 1990 working paper (published 1992, with Eric Reiner) "Exotic Options", with the term based either on exotic wagers in horse racing, or due to the use of international terms such as "Asian option", suggesting the "exotic Orient". Journalist Brian Palmer used the "successful $1 bet on the superfecta" in the 2010 Kentucky Derby that "paid a whopping $101,284.60" as an example of the controversial high-risk, high-payout exotic bets that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vanilla Option
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] |
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] |
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