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> See Fourier Transformation

Feynman-Kac Theorem

An extremely useful theorem in mathematics, The Feynman-Kac theorem can be used to derive Kolmogorov's Backward Equation. We can also use the theorem to solve the Black-Scholes partial differential equation.

A special form of the Laplace transformation, a Fourier Transform is a generalised form of a the complex Fourier Series.

If we are given a function continuous function f(x), the Fourier transform is given as:

The inverse Fourier transform is simply:

For discrete functions where there is discrete sampling, the discrete Fourier transform is given as:

Where x and z represent the samples and n is the number of samples within the time period.

The inverse discrete Fourier transform is given as:

A Fast Fourier transform is a discrete algorithm introduced by Cooley and Tukey (1965), which reduces the number of computations from the order of to

See: Cooley, J. W. and Tukey, J. W., 1965, "An Algorithm For the Machine Calculation of Complex Fourier Series", Mathematics of Computation, 19, 90, pp. 297-301.

The greek letter Gamma is a variable often used in finance to represent the rate of change of an option's Delta in relation to the underlying asset price. As the Delta is the first derivative of the option with respect to the underlying price, the Gamma is said to be the 2nd derivatives, or second partial derivative of the option with respect to the underlying asset price.

It is denoted as:

Which represents the gamma as being a change in the delta with respect to a change in underlying price. Or alternatively, it can also be denoted as:

Which represents the gamma as being a the second derivative of a change in the option price with respects to the underlying asset price.

The gamma of a European call / put is shown to vary with time to maturity as below:

A Gaussian distribution is a probability distribution with a mean of u and a standard deviation of 'sigma'. Commonly referred to as the normal distribution and takes the shape of a 'bell curve' as seen below:

Where the x-axis at 0 is the median, mean and mode of the normal distribution. If the standard deviation of this distribution was said to be 5, then it can be said that there is a 68% chance the sample is between -5 and 5. Confidence intervals can be used to determine the % of the population within n standard deviations.

The normal distribution takes the following formula:

General Probability Space

Continuing from Brownian Motion is Geometric Brownian Motion (or GBM). GBM is often used to model movements in commodities, stocks and derivatives

GBM can take either a continuous-time form or a discrete-time form.

A key concept within option pricing, Girsanov's theorem is central to how we arise at the closed form valuation of the Black-Scholes model and beyond.

The general idea behind the Girsanov theorem is that it applies a change of measure, and relates the Wiener process to an alternative probability measure by providing an explicit formula giving the likelihood ratio between the two. Applying Girsanov's theorem, we can go from a real world framework to a risk neutral process by changing the drift term.

Given an integral:

The function k(x,t) is considered as Green's Function and helps us in solving linear differential equations which have predetermined boundaries.

For an example of how the Green's Function can be used to arise at the Black-Scholes European option pricing formula

See this website by Dennis Silverman

In linear algebra a Hilbert Space is a vector space containing an inner / dot product; e.g. the angle between two vectors (X, Y) within a vector space is given by . An inner product for 2 functions x and y comes in the general form of:

Finite:

Where * denotes a number in C space.

Infinite:

Hilbert spaces come in either a finite or an infinite form, with the latter being more common. A common infinite Hilbert space seen within statistics and linear mathematics is the form:

or of Lebesgue integrable functions with either Real or Complex values.

One of its primary uses is to generalize certain linear transformations such as the Fourier Transform

Derived by Japanese mathematician Kiyoshi Ito in 1951, Ito's lemma forms an integral foundation to stochastic calculus. It is used to find the differential of a stochastic function and when compared to normal calculus, the Ito's Lemma is considered as the chain rule of stochastic calculus.

Ito's lemma can be applied to finance in the following way.

Consider the value of a variable x (in our case, let us assume the variable x is the stock price movement).

equation I-1a:

Where dz is a Wiener Process and a and b are functions of x and t. It can be seen that the stock price (x) has a drift value of a and a variance of . What Ito's lemma does is show that a function F of x and t follow the process:

In equation I-1a, a is the drift rate and we can apply the same here. The first part of the right term is the 'a' in I-1a, hence the the drift rate of F is:

and the variance of F is analogous to the term.

or

Ito's lemma can be applied to numerous aspects of derivatives, including Black-Scholes derivation, forward contracts and exotics.

For further details on Ito's Lemma, refer to:

Hull, J. "Options, Futures & Other Derivatives", Chp 11. 2003