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Stochastic Process

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Stochastic process is a concept that forms the base of a lot of statistical methods used in different fields of study. This is a simple yet very powerful idea that can help us capture the random nature of problems and formulate them in mathematical notation and hence use them in different applications.

This process has many applications and is readily used in the finance, physics and mathematics. Did you know that stochastic probabilities are used in many asset pricing models and also in the heat diffusion equation? The stochastic process is the way of representing and analyzing the random behavior in the stock prices or in heat diffusion.

 

Lognormal Distribution for Asset Pricing (Mathematical Details)

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In order to understand why Lognormal distribution is suitable for stock price returns, let’s first think about stock prices, the following graph is a stock price chart over time:T.

Sample Stock Price

 

Lognormal Distribution for Stock Price Returns

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A lognormal distribution is used as the standard model stock price returns in financial economics. In this article we will go in the depth to why this is so.

Statistical Distributions

Before talking about stock price return as lognormal distribution, I want to give a quick overview of normal and lognormal distributions.

Note: this is not a complete tutorial on normal and lognormal distributions but I do talk about the properties that are important for the purpose of this article.


 

Newton-Raphson Method

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Newton-Raphson Method (a.k.a Newton Method) is a method that can be used to find the root of a equation. In other words, the Newton Method helps us find the the input to a function at which the value of the function will be 0, e.g. if we have a function f(x) = x^{3} - 20, we want to find a value of x where f(x) = 0 or x^{3} - 20 = 0.

The general formula for the newton method is:

 

AHP (Analytical Hierarchy Process)

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Analytical Hierarchy Process (AHP) is a process that helps us pick up one of the options of a list of choices. Each choice has a few parameters attached to it and we can set the weights of each parameter and have AHP pick the best choice from the list of choices. I will show this in the following example.


Analytical Hierarchy Process for College Selection


 


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