Mathematical finance

Serious Information

Mathematical finance is the branch in applied mathematics concerned with financial markets. The subject naturally has a close relationship with the discipline of financial economics, but mathematical finance is narrower in scope and more abstract. A central difference is that while a financial economist might study the structural reasons why a company may have a certain share price, a mathematician may take the share price as a given, and attempt to use stochastic calculus to obtain the fair value of derivatives of the stock.

In probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff ("value"). Thus, it represents the average amount one "expects" to win per bet if bets with identical odds are repeated many times. Note that the value itself may not be expected in the general sense; it may be unlikely or even impossible. A game or situation in which the expected value for the player is zero (no net gain nor loss) is called a "fair game."

Rational pricing is the assumption in financial economics that asset prices (and hence asset pricing models) will reflect the arbitrage-free price of the asset as any deviation from this price will be "arbitraged away". This assumption is useful in pricing fixed income securities, particularly bonds, and is fundamental to the pricing of derivative instruments.

Futures contract

Margin
Although the value of a contract at time of trading should be zero, its price constantly fluctuates. This renders the owner liable to adverse changes in value, and creates a credit risk to the exchange, who always acts as counterparty. To minimise this risk, the exchange demands that contract owners post a form of collateral, in the US formally called performance bond, but commonly known as margin.

Computational finance (also known as financial engineering) is a cross-disciplinary field which relies on mathematical finance, numerical methods and computer simulations to make trading, hedging and investment decisions, as well as facilitating the risk management of those decisions. Utilizing various methods, practitioners of computational finance aim to precisely determine the financial risk that certain financial instruments create.

In economics, a model is a theoretical construct that represents economic processes by a set of variables and a set of logical and quantitative relationships between them. As in other fields, models are simplified frameworks designed to illuminate complex processes.

The LIBOR Markt Modell is an interest rate model used for the pricing of interest rate derivatives, especially for complex derivatives. The model primitives are a set of forward rates. Each forward rate is modeled by a lognormal process, i.e. a Black model. Thus the LIBOR market model may be interpreted as a collection of Black-models considered under a common pricing measure.

Heath-Jarrow-Morton framework is a general framework to model the evolution of interest rates (forward rates in particular). The HJM framework originates from the work of D. Heath, R.A. Jarrow and A. 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.

The key to these techniques is the recognition that the drifts of the no-arbitrage evolution of certain variables can be expressed as functions of their volatilities and the correlations among themselves. In other words, no drift estimation is needed. Models developed according to the HJM framework are different from the so called short-rate models (e.g. the Ho-Lee model) in the sense that HJM-type models capture the full dynamics of the entire forward rate curve, while the short-rate models only capture the dynamics of a point on the curve (the short rate).

The Black model (sometimes known as the Black-76 model) is a variant the Black-Scholes option pricing model. It is widely used in the futures market and interest rate market for pricing bond options. It was first presented in a paper written by Fischer Black in 1976.

The main problem with the Black model is that it does not easily deal with price correlation of multiple options. Each option is considered to be priced independently of other options and the linkages between the prices of different options is not easily incorporated into the model.

Long-Term Capital Management (LTCM) was a hedge fund founded in 1994 The company had developed complex mathematical models to take advantage of fixed income arbitrage deals (termed convergence trades) usually with U.S., Japanese, and European government bonds. The basic idea was that over time the value of long-dated bonds issued a short time apart would tend to become identical. However the rate at which these bonds approached this price would be different, and that more heavily traded bonds such as US Treasury bonds would approach the long term price more quickly than less heavily traded and less liquid bonds.

Initially amazingly successful, it folded in 1998, losing $4.6 billion in less than four months.

Links on websites on subject

http://www.federalreserve.gov/Boarddocs/testimony/2000/20000621.htm Testimony of Chairman Alan Greenspan on Federal Reserve Board's views on the Commodity Futures Modernization Act of 2000

http://www.federalreserve.gov/pubs/feds/2005/200539/200539pap.pdf A paper which claims that

http://www.ise.ie/index.asp The home page of Irish stock exchange

http://www.nasdaq.com/ The homepage of the stock exchange

http://www.cityequities.com/
http://www.moneyweek.com

government_bond information

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