K - O
Otherwise known as Kolmogorov's strong law of large numbers. Considering a sequence of independent variables:
With finite expected values.
Kolmogorov's criterion is said to hold if one of the following 2 criteria are met:
1) The variables have identical distributions
2) The variance of is finite and:
The Laplace transform is a mathematical process which helps us to simplify complex differential equations; notably in the case of partial differential equations. To solve partial differential equation incorporates a so-called inital-value problem, which increases in complexity with increased dimensions.
We can simplify matters by applying the Laplace transform to a PDE in order to convert the PDE into an algebraic form which can be solved with relative ease.
To attain the answer to the PDE, we then apply an inverse Laplace transform to the solution of the algebraic equation(s).
To illustrate the Laplace transform, we consider an arbitrary function f(t). This function can denote a number of things including a Black-Scholes PDE or even a hyperbolic PDE related to black-holes. The Laplace transform of the function f(t) is given by the following:
Where L is the Laplace operator of f(t). The term on the furthest right denotes an integral with limits between 0 and infinity.
To obtain the solution to the original PDE, we apply the inverse Laplace transform to F(x) as follows:
Already we have mentioned the Laplace transform's applicability to PDEs. The transform becomes particularly useful when we price exotic options as many of the pricing formulae involve the use of PDEs which are troublesome to solve directly.
When we consider probability spaces, the Lebesgue measure becomes a key part of it.
The Lebesgue integral as defined by Shreve (1997), is a generalisation of the standard Riemann integral.
Those of you familiar with a normal distribution might know that it is represented by a "bell curve" which peaks at the mean value and drops down on either side, where we can use standard deviation rules to determine the proportion of values lie within x standard deviations.
A lognormal distribution can be useful in finance because of the property that the distribution cannot take on negative values. The majority of professionals and academics will argue that this is representative of the markets as stocks cannot have a value less than 0!
A stochastic process where the future expected value of a value (stock price) is dependent only on the current value. Weak efficient market efficiency concludes that past information is incorporated into the present, hence our decision making process is only determined by the value today.
To give an example, consider a stock which has a value of $100 today. The expected change in value in 1 time period is equal to - Which is a standard normal distribution with mean of 0 and standard deviation of 1.
We can find Markov processes in many applications and the concept is useful within numerous areas of finance.
A martingale is a stochastic process which has no drift. A key property which a martingale has is that its expected value at a future time T is the same as its value today.
The idea behind mean reversion is that a variable will eventually revert back to it's mean or average value. For example, if a stock price has steadily increasing over a period of time, it will eventually decrease back to it's mean value.
The Monte Carlo method, encompasses any technique of statistical sampling employed to approximate solutions to quantitative problems. Stanislaw Ulam, John von Neuman and Nicholas Metropolis are the 3 individuals who are credited with inventing this concept in 1946, with Ulam and Metropolis later publishing the first paper on MonteCarlo simulation several years later in 1949. The method was named after the casinos of the same name.
To consider MonteCarlo simulation, it is useful to think of it as a general technique of numerical integration, which is what makes MCS a very reliable method to get approximations of option prices which follow a partial differential equation (PDEs)
Without delving too deep into the mathematics behind the MonteCarlo method and its integrals, we can take the example of an arbitrary experiment.
Consider an experiment where you are attempting to estimate a value for a variable such as the volume of a balloon, which is inherently dependent on several other variables.
If we proceed to undertake a series of experiments to estimate the volume of a balloon if blow it up for 1 minute at a certain pressure (PSI).
The issue we are faced is that if we do the experiment 5 times, we are bound to obtain mixed results due to the standard error which arises, causing a degree of bias.
Then, consider doing the experiment 1000 times in the same way and finding the tendency of the results. We find that the bias is substantially reduced and that the standard error tends towards 0, giving us a more accurate measurement of the volume of the balloon.
While increasing the number of samples is one method of reducing the standard error of the experiements, doing so can be computationally expensive. A better solution is to employ some technique of variance reduction, which incorporates additional information about the analysis directly into the estimator. This allows them to make the Monte Carlo estimator more deterministic, and hence have a lower standard error.
In the context of derivatives, we go back to what we mentioned earlier related to PDEs. Many of the option pricing models found today are based upon a PDE which can be adapted to the uses of MCS. We can undertake many iteration counts of the integral and find a good approximation of the price of an option. More on Monte Carlo simulation shortly.
In statistics, multivariate, as opposed to univariate or bivariate is a generalisation, which merely shows that something is dependent on 'several' variables as opposed to univariate (1) and bivariate (2)
Numéraires refer to normalised asset prices. They are useful in calculating the relative price of one thing to another by applying a nominal value of $1 to a particular asset (the numéraire). A basic example can be illustrated by considering that we can buy 1 hamburger for $1.50. Instead of saying that 1 hamburger costs $1.50, we can say that for $1.00, we can buy 2/3rds of a hamburger.
In finance, the omega can mean a number of things. In derivatives, the greek letter omega is often used to describe the duration of an option.
In the context of stock prices, this is a represented by the following stochastic differential equation (SDE):
Where is the drift constant,is the volatility of the process and dz is a Wiener process.
Barndorff-Nielsen and Shephard (2001) proposed the application of the Ornstein-Uhlenbeck process in finance, where the process models the volatility of an asset. The model is non-Gaussian in nature and governed by a Levy process.
In providing a stochastic volatility model for stock prices, the Ornstein-Uhlenbeck process can be extremely useful in determining option prices which reflect a stochastic process.
In linear algebra, 2 vectors are said to be orthogonal if they are perpendicular to each other on the same plane.