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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] |
Volatility (finance)
In finance, volatility (usually denoted by "sigma, σ") is the Variability (statistics), degree of variation of a trading price series over time, usually measured by the standard deviation of logarithmic returns. Historic volatility measures a time series of past market prices. Implied volatility looks forward in time, being derived from the market price of a market-traded derivative (in particular, an option). Volatility terminology Volatility as described here refers to the actual volatility, more specifically: * actual current volatility of a financial instrument for a specified period (for example 30 days or 90 days), based on historical prices over the specified period with the last observation the most recent price. * actual historical volatility which refers to the volatility of a financial instrument over a specified period but with the last observation on a date in the past **near synonymous is realized volatility, the square root of the realized variance, in turn c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Texas At Austin
The University of Texas at Austin (UT Austin, UT, or Texas) is a public university, public research university in Austin, Texas, United States. Founded in 1883, it is the flagship institution of the University of Texas System. With 53,082 students as of fall 2023, it is also the largest institution in the system. The university is a major center for academic research, with research expenditures totaling $1.06 billion for the 2023 fiscal year. It joined the Association of American Universities in 1929. The university houses seven museums and seventeen libraries, including the Lyndon Baines Johnson Library and Museum, Lyndon B. Johnson Presidential Library and the Blanton Museum of Art, and operates various auxiliary research facilities, such as the J. J. Pickle Research Campus and McDonald Observatory. UT Austin's athletics constitute the Texas Longhorns. The Longhorns have won four NCAA Division I National Football Championships, six NCAA Division I National Baseball Champions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Financial Mathematics
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling in the Finance#Quantitative_finance, financial field. In general, there exist two separate branches of finance that require advanced quantitative techniques: Derivative (finance), derivatives pricing on the one hand, and risk management, risk and Investment management#Investment managers and portfolio structures, 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 with the 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 investment ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Margrabe's Formula
In mathematical finance, Margrabe's formula is an option pricing formula applicable to an option to exchange one risky asset for another risky asset at maturity. It was derived by William Margrabe (PhD Chicago) in 1978. Margrabe's paper has been cited by over 2000 subsequent articles. Google Scholar'"cites" page for this article/ref> Formula Suppose ''S1(t)'' and ''S2(t)'' are the prices of two risky assets at time ''t'', and that each has a constant continuous dividend yield ''qi''. The option, ''C'', that we wish to price gives the buyer the right, but not the obligation, to exchange the second asset for the first at the time of maturity ''T''. In other words, its payoff, ''C(T)'', is max(0, ''S1(T) - S2(T))''. If the volatilities of ''Si'' are ''σi'', then \textstyle\sigma = \sqrt, where ''ρ'' is the Pearson's correlation coefficient of the Brownian motions of the ''Si'' 's. Margrabe's formula states that the fair price for the option at time 0 is: :e^S_1(0) N(d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Time Value Of Money
The time value of money refers to the fact that there is normally a greater benefit to receiving a sum of money now rather than an identical sum later. It may be seen as an implication of the later-developed concept of time preference. The time value of money refers to the observation that it is better to receive money sooner than later. Money you have today can be invested to earn a positive rate of return, producing more money tomorrow. Therefore, a dollar today is worth more than a dollar in the future. The time value of money is among the factors considered when weighing the opportunity costs of spending rather than saving or investing money. As such, it is among the reasons why interest is paid or earned: interest, whether it is on a bank deposit or debt, compensates the depositor or lender for the loss of their use of their money. Investors are willing to forgo spending their money now only if they expect a favorable net rate of return, return on their investment in the fut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Forward Contract
In finance, a forward contract, or simply a forward, is a non-standardized contract between two parties to buy or sell an asset at a specified future time at a price agreed on in the contract, making it a type of derivative instrument.John C Hull'', Options, Futures and Other Derivatives (6th edition)'', Prentice Hall: New Jersey, USA, 2006, 3 The party agreeing to buy the underlying asset in the future assumes a long position, and the party agreeing to sell the asset in the future assumes a short position. The price agreed upon is called the ''delivery price'', which is equal to the forward price at the time the contract is entered into. The price of the underlying instrument, in whatever form, is paid before control of the instrument changes. This is one of the many forms of buy/sell orders where the time and date of trade is not the same as the value date where the securities themselves are exchanged. Forwards, like other derivative securities, can be used to hedge ris ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Distribution
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac e^\,. The parameter is the mean or expectation of the distribution (and also its median and mode), while the parameter \sigma^2 is the variance. The standard deviation of the distribution is (sigma). A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Futures Contract
In finance, a futures contract (sometimes called futures) is a standardized legal contract to buy or sell something at a predetermined price for delivery at a specified time in the future, between parties not yet known to each other. The item transacted is usually a commodity or financial instrument. The predetermined price of the contract is known as the ''forward price'' or ''delivery price''. The specified time in the future when delivery and payment occur is known as the ''delivery date''. Because it derives its value from the value of the underlying asset, a futures contract is a Derivative (finance), derivative. Contracts are traded at futures exchanges, which act as a marketplace between buyers and sellers. The buyer of a contract is said to be the Long (finance), long position holder and the selling party is said to be the Short (finance), short position holder. As both parties risk their counter-party reneging if the price goes against them, the contract may involve both ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
European Call Option
In finance, the style or family of an option is the class into which the option falls, usually defined by the dates on which the option may be exercised. The vast majority of options are either European or American (style) options. These options—as well as others where the payoff is calculated similarly—are referred to as "vanilla options". Options where the payoff is calculated differently are categorized as "exotic options". Exotic options can pose challenging problems in valuation and hedging. American and European options The key difference between American and European options relates to when the options can be exercised: * A European option may be exercised only at the expiration date of the option, i.e. at a single pre-defined point in time. * An American option on the other hand may be exercised at any time before the expiration date. For both, the payoff—when it occurs—is given by * \max\, for a call option * \max\, for a put option where K is the strike p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Risk-free Interest Rate
The risk-free rate of return, usually shortened to the risk-free rate, is the rate of return of a hypothetical investment with scheduled payments over a fixed period of time that is assumed to meet all payment obligations. Since the risk-free rate can be obtained with no risk, any other investment having some risk will have to have a higher rate of return in order to induce any investors to hold it. In practice, to infer the risk-free interest rate in a particular currency, market participants often choose the yield to maturity on a risk-free bond issued by a government of the same currency whose risks of default are so low as to be negligible. For example, the rate of return on zero-coupon Treasury bonds (T-bills) is sometimes seen as the risk-free rate of return in US dollars. Theoretical measurement As stated by Malcolm Kemp in chapter five of his book ''Market Consistency: Model Calibration in Imperfect Markets'', the risk-free rate means different things to different pe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Future Contract
In finance, a futures contract (sometimes called futures) is a standardized legal contract to buy or sell something at a predetermined price for delivery at a specified time in the future, between parties not yet known to each other. The item transacted is usually a commodity or financial instrument. The predetermined price of the contract is known as the ''forward price'' or ''delivery price''. The specified time in the future when delivery and payment occur is known as the ''delivery date''. Because it derives its value from the value of the underlying asset, a futures contract is a derivative. Contracts are traded at futures exchanges, which act as a marketplace between buyers and sellers. The buyer of a contract is said to be the long position holder and the selling party is said to be the short position holder. As both parties risk their counter-party reneging if the price goes against them, the contract may involve both parties lodging as security a margin of the value of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
