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Martingale Pricing
Martingale pricing is a pricing approach based on the notions of martingale and risk neutrality. The martingale pricing approach is a cornerstone of modern quantitative finance and can be applied to a variety of derivatives contracts, e.g. options, futures, interest rate derivatives, credit derivatives, etc. In contrast to the PDE approach to pricing, martingale pricing formulae are in the form of expectations which can be efficiently solved numerically using a Monte Carlo approach. As such, Martingale pricing is preferred when valuing high-dimensional contracts such as a basket of options. On the other hand, valuing American-style contracts is troublesome and requires discretizing the problem (making it like a Bermudan option) and only in 2001 F. A. Longstaff and E. S. Schwartz developed a practical Monte Carlo method for pricing American options. Measure theory representation Suppose the state of the market can be represented by the filtered probability space,(\Omega,(\math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martingale (probability Theory)
In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all prior values. History Originally, '' martingale'' referred to a class of betting strategies that was popular in 18th-century France. The simplest of these strategies was designed for a game in which the gambler wins their stake if a coin comes up heads and loses it if the coin comes up tails. The strategy had the gambler double their bet after every loss so that the first win would recover all previous losses plus win a profit equal to the original stake. As the gambler's wealth and available time jointly approach infinity, their probability of eventually flipping heads approaches 1, which makes the martingale betting strategy seem like a sure thing. However, the exponential growth of the bets eventually bankrupts its users due to f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Francis Longstaff
Francis A. Longstaff (born August 3, 1956) is an American educator and pioneer in quantitative finance. He serves as the Allstate Professor of Insurance and Finance at the Anderson School of Management, University of California, Los Angeles, and the former Finance Area Chair. His research focuses on fixed income markets, term structure, derivatives, credit risk, computational finance and the role of arbitrage in financial markets. He is known for the Longstaff–Schwartz model a multi-factor short-rate model, and the Longstaff-Schwartz method for valuing American options by Monte Carlo Simulation. He has published over 50 research and practitioner articles, and is a recipient of the Michael Brennan Award. Longstaff was head of Fixed Income Derivative Research at Salomon Brothers from 1995 to 1998, and worked in the research department of the Chicago Board of Trade and for Deloitte and Touche as a management consultant. He is a research associate in the National Bureau of Economi ... [...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] |
Finance Theories
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, distribution, and consumption of money, assets, goods and services (the discipline of financial economics bridges the two). Finance activities take place in financial systems at various scopes, thus the field can be roughly divided into personal, corporate, and public finance. In a financial system, assets are bought, sold, or traded as financial instruments, such as currencies, loans, bonds, shares, stocks, options, futures, etc. Assets can also be banked, invested, and insured to maximize value and minimize loss. In practice, risks are always present in any financial action and entities. A broad range of subfields within finance exist due to its wide scope. Asset, money, risk and investment management aim to maximize value and minimize volatility. Financial analysis is viability, stability, and profitability assessmen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brownian Model Of Financial Markets
The Brownian motion models for financial markets are based on the work of Robert C. Merton and Paul A. Samuelson, as extensions to the one-period market models of Harold Markowitz and William F. Sharpe, and are concerned with defining the concepts of financial assets and markets, portfolios, gains and wealth in terms of continuous-time stochastic processes. Under this model, these assets have continuous prices evolving continuously in time and are driven by Brownian motion processes. This model requires an assumption of perfectly divisible assets and a frictionless market (i.e. that no transaction costs occur either for buying or selling). Another assumption is that asset prices have no jumps, that is there are no surprises in the market. This last assumption is removed in jump diffusion models. Financial market processes Consider a financial market consisting of N + 1 financial assets, where one of these assets, called a ''bond'' or ''money market'', is risk free while the remain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Girsanov Theorem
In probability theory, the Girsanov theorem tells how stochastic processes change under changes in measure. The theorem is especially important in the theory of financial mathematics as it tells how to convert from the physical measure which describes the probability that an underlying instrument (such as a share price or interest rate) will take a particular value or values to the risk-neutral measure which is a very useful tool for evaluating the value of derivatives on the underlying. History Results of this type were first proved by Cameron-Martin in the 1940s and by Igor Girsanov in 1960. They have been subsequently extended to more general classes of process culminating in the general form of Lenglart (1977). Significance Girsanov's theorem is important in the general theory of stochastic processes since it enables the key result that if ''Q'' is a measure that is absolutely continuous with respect to ''P'' then every ''P''-semimartingale is a ''Q''-semimartingale. State ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Surely
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. The concept is analogous to the concept of "almost everywhere" in measure theory. In probability experiments on a finite sample space, there is no difference between ''almost surely'' and ''surely'' (since having a probability of 1 often entails including all the sample points). However, this distinction becomes important when the sample space is an infinite set, because an infinite set can have non-empty subsets of probability 0. Some examples of the use of this concept include the strong and uniform versions of the law of large numbers, and the continuity of the paths of Brownian motion. The terms almost certainly (a.c.) and almost always (a.a.) are also used. Almost never describes the opposite of ''almost surely'': an event that h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Filtered Probability Space
Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter medium are described as ''oversize'' and the fluid that passes through is called the ''filtrate''. Oversize particles may form a filter cake on top of the filter and may also block the filter lattice, preventing the fluid phase from crossing the filter, known as ''blinding''. The size of the largest particles that can successfully pass through a filter is called the effective ''pore size'' of that filter. The separation of solid and fluid is imperfect; solids will be contaminated with some fluid and filtrate will contain fine particles (depending on the pore size, filter thickness and biological activity). Filtration occurs both in nature and in engineered systems; there are biological, geological, and industrial forms. Filtration is also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eduardo Schwartz
Eduardo Saul Schwartz (born 1940) is a professor of finance at SFU's Beedie School of Business, where he holds the Ryan Beedie Chair in Finance. He is also a Distinguished Research Professor at the University of California, Los Angeles. He is known for pioneering research in several areas of finance, particularly derivatives. His major contributions include: the real options method of pricing investments under uncertainty; the Longstaff–Schwartz model - a multi-factor short-rate model; the Longstaff-Schwartz method for valuing American options by Monte Carlo Simulation; the use of Finite difference methods for option pricing. He has been faculty at the University of British Columbia and UCLA, and visiting at the London Business School, the University of California, Berkeley and the Universidad Carlos III in Madrid. His wide-ranging research has focused on different dimensions in asset and securities pricing. Topics in recent years include interest rate models, asset all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bermudan 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Risk-neutral Measure
In mathematical finance, a risk-neutral measure (also called an equilibrium measure, or ''equivalent martingale measure'') is a probability measure such that each share price is exactly equal to the discounted expectation of the share price under this measure. This is heavily used in the pricing of financial derivatives due to the fundamental theorem of asset pricing, which implies that in a complete market, a derivative's price is the discounted expected value of the future payoff under the unique risk-neutral measure. Such a measure exists if and only if the market is arbitrage-free. The easiest way to remember what the risk-neutral measure is, or to explain it to a probability generalist who might not know much about finance, is to realize that it is: # The probability measure of a transformed random variable. Typically this transformation is the utility function of the payoff. The risk-neutral measure would be the measure corresponding to an expectation of the payoff with a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
