Introduction:
Stock definition
Monte carlo simulations
Mathematical Background:
Random walks
Brownian motion
Geometric Brownian motion
Introduction:
Stock definition
A stock, also known as equity, represents ownership of a corporation or company. Stock ownership means that the owner is entitled to a proportion of the company’s assets and profits. Most stock trading is done at stock exchanges. Some famous stock exchanges include the New York Stock Exchange, Nasdaq, London Stock Exchange, Shanghai Stock Exchange, among several others. Investors buy stocks of companies that they expect their values to increase, leading to an increase in the stock price. The investors will then sell the stocks at the new inflated price and make a profit. In this study, we will consider the stocks that do not give dividends to investors.
Monte Carlo simulations
Monte Carlo simulation is a statistical technique.
It has received wide applications in several fields, and the Finance field has not been left out.
Random Walk
A random walk, defined, is a process where we cannot predict the next step. We can describe it as similar to a drunkard’s walk whose next step cannot be predicted. In statistics, a random walk is a stochastic process that describes a path that consists of a succession of random steps. In finance, the random walk theory states that assets’ prices, such as equities, do not move in a discernible pattern and cannot predict their short-term price movement. An asset’s price time series is said to follow a random walk if:
Where the increments Are serially independent random variables. The sequence is identically distributed with mean zero and variance , but this is not a necessary assumption.
By recursive substitution, it is shown that;
Thus the current value of depends on the initial value and all the disturbances accruing
between time t = 1 and the current period, t. The first two moments are:
- Mean:
- Variance:
The random-walk model has been widely considered as a statistical model for the movement of logged stock prices. Under such a model, the stock price is not predictable or mean-reverting. The variance is a function of time t. It increases linearly with time. The Geometric Brownian Motion incorporates the concepts of the random walk, as we will discuss more later.
Standard Brownian Motion (Wiener Process)
The Brownian motion is also referred to as the Wiener process.
A stochastic process , is a Brownian motion if:
- has continuous sample paths.
This means that the Wt graph as a function of t doesn’t have any breaks in it.
- For any , the increment is usually distributed,
This property shows that the increments are stationary in that their statistical properties rely on the size of the interval .
- has independent increments, that is, for any sequence of times , we have that the increments are independent random variables.
The fact that a Wiener Process has independent increments implies it is Markovian (in fact, it is ‘strong Markovian’).
Geometric Brownian Motion