:: Stochastic character of financial variables ::

• From the trajectories of stock prices (also other financial variables - interest rate, exchange rates, etc.) we see that it cannot be described by a deterministic function. Therefore, stochastic processes are used in the modelling. .
  
  
• Left: trend (stock price, 5 years time frame), right: fluctuations (stock price, time frame of a couple of hours)
  
  
  Source: http://finance.google.com

:: Wiener process and Brownian motion ::

Download [ex02processes.sce] - Scilab file with the code shown below; we will write more

• The basic process, used to derive more complicated ones, is a Wienerov proces. Recall its definition:
  
  Random process {W(t), t0} is called a Wiener process, if

  • the increments W(t+) -  W(t) have a normal distribution with zero expected value and variance equal to ,
  • for every partition 0 = t₀  t₁  ...   t_n, the increments W_ti+1 - W_ti nare independent random variables, with distribution given in the previous points ,
  • W(0)=0,
  • trajectories are continuous.
  In what follow, we done a Wiener process by w.
  
  
• How to obtain a sample path (trajectory) of a Wiener process
  • We will generate an approximation - values at discrete points of the form (time, value of the process) which we join by line segments.
  • The times will be 0,,2, . . ., where is a sufficiently small time step.
  • Value at time 0 is equal to 0.
  • Increment on the interval [k, (k + 1)] is a random variable with zero expected value and variance equal to .
  
  In Scilab we firstly define auxiliary functions:
  
  
  
  Now we can plot a trajectory:
  
  

  Plot several trajectories of a Wiener process into one plot. A sample output:

  
  
  
• By adding a linear trend to a multiple of a Wiener process:
  
  we obtain a process which is known as a Brownian motion.

  If the parameter is zero, the resulting graph is a line. For a nonzero value of , there are random fluctuations added to this linear trend.
  
  In Scilab - for example:
  

  

  Plot several trajectories of a Wiener process into one plot and add the expected value. A sample output:

  
  
  

:: Exercises (1) ::

1. Modify parameters of a Brownian motion and note the influence on typical trajectories. Then, add the processes
   • x₁(t)=w(t)
   • x₂(t)=3*w(t)
   • x₃(t)=5+2*t+w(t)
   • x₄(t)=5+2*t+0.5*w(t)
   • x₅(t)=5-3*t+w(t)
   to their sample paths below:
   
   
   

:: Geometric Brownian motion ::

• Geometric Brownian motion is defined as:
  
  where x₀ is the value of the process at time 0.
  
  
• Simulating in Scilab:
  
• Sample paths of a geometric Brownian motion:
  
  Make a similar plot.
  

:: Modelling stock prices by mean of a geometric Brownian motion ::

• GBM as a model for the stock price S:
  
  
  
• Recall from the lecture: If the stock price S follows a GBM, then for the returns we have
  
  hence the returns are independent random variables with the distribution given above.
  
  

:: Exercises (2) ::

Recall from the probability course the definition and basic properties of a lognormal distribution:

Download [ex02stock.sce] - Scilab file with an outline of solution to the following problems and some useful functions

Suppose that the stock price follows a GBM with parameters  = 0.30,  = 0.25 and that its current price is 150 USD.

1. Plot the probability density function of the stock price in one month. To check your computation, compare it with a histogram of simulated values of the stock price in one month.
2. Compute the probability that the price in one month will be lower than 140 USD?
3. What it expected value of the 1/4-year return? What is the probability of this return to be negative?

:: Estimating the parameters of the GBM from stock prices ::

Download [amzn.txt] and [ex02data.sce] - stock price data in text file and Scilab file with the code given below and the outline of estimation

How to obtain estimates of parameters - we use the procedure from the lecture (cf. the slides) and replicate the results

• Firstly, set the working directory, so that it contains the data.
  
• Read the data to Scilab. These are daily data, so we set the value of the variable dt, measuring the length of a time step in years, to 1/252.
• We define the returns - we create a vector of returns v using the formula above
• We know that these returns have a normal distribution. Furthermore, we know that the expected value of a normal distribution is estimated by a sample mean and the variance is estimated by a sample variance. Therefore we compute the mean and the sample variance of the value in vector v - these will be estimates of the quantities and
  
• Finally, we compute the estimates of the parameters and :
  

:: Practice problems ::

1. Denote by t_M the time, in which the sample path of the Wiener process achieved its maximum on the time interval [0, 1] Plot a histogram by simulating the realizations of the random variable t_M.
   
   A trajectory of a Wiener process and corresponding value of t_M is shown below, together with a sample histogram.
   
   
   
2. Let w be a Wiener process; define B(t) = w(t) - t w(1) for time t in interval [0, 1]. It is known as Brownian bridge.
   • Plot some sample paths of the process and explain its name.
   • Compute its variance at time t and sketch its behaviour as a function of time. To check your computation - what does the result have to be for t=0 and t=1?
   
   
3. More practice problems in the course notes

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