BROWNIAN MOTION AND WIENER PROCESSES
With a solid understanding of random numbers in mind, we can delve deeper into how these topics evolved in finance. The term Brownian motion is often heard in financial classrooms and occasionally in jobs that involve securities analysis. In fact the first time I thoroughly went through Brownian motion was when a managing director at my first job after graduate school was introducing his method for analyzing credit default swaps. Brownian motion is named after the 19th-century Scottish botanist Robert Brown. Brown noticed that when pollen grains were suspended in water they jiggled and moved about in seemingly random fashion. Today, this random motion is known as a Wiener process and is used as a foundation to model many financial products and attributes of financial products, the most common being stock prices and interest rates. Since most of us are comfortable with the idea that prices and rates do not fluctuate in a purely random fashion, we must first assert a few stipulations, which we will discuss later in the section, to make this framework more representative of real-world observables.
Wiener Process
Formally a Wiener process (Wn(t)), otherwise known as a random walk, is defined as follows: at any given time ti a binomial process, a process by which there are only two possible outcomes, will take either a step forward or a step backward with a step size of 1/sqrt(n), where n is the total number of steps taken. Thus the outcome at ti+1 will depend only on that of the prior state, namely at ti. Therefore the value of Wn at any time t can be computed by equation 2.1:
where Xi is a random variable that is defined to be either 1 or −1 with equal probability. We can easily deduce from the above equation that generally the value of a Wiener process at any time t can be computed as equation 2.2:
Here we replaced n with T, the total time, and where we are assuming that time is measured in integral time periods such as days, months, or years. If T is large enough, the Central Limit Theorem tells us that WT(t) is a normally distributed random variable with mean 0 and standard deviation of sqrt(t/T), that is N(0, sqrt(t/T)).
MODEL BUILDER 2.7: Basic Brownian Motion
- This Model Builder will take readers through a simple setup for implementing Brownian motion and allowing users to see the results of as many time steps as desired (within a reasonable limit for Excel/VBA). First, we should create a new workbook since this represents a distinctly different setup than before. Name the workbook MB_2.7_User.xls.
- For labeling purposes enter the following labels in the corresponding cells:
A1: “Brownian Motion”
A3: “N – Steps”
- Next enter a value of 100 in B3 and name the same cell nsteps.
- Switch over to the Visual Basic editor, insert a new module, and then enter the following code to create the function BinaryStepSelection:
Function BinaryStepSelection() As Double If (Rnd() < 0.5) Then BinaryStepSelection = 1 Else BinaryStepSelection = −1 End If End FunctionThis function has a familiar element from prior sections of this chapter: the syntax Rnd. In this situation we are creating a function that returns either a 1 or −1 depending on whether a uniform random variable is greater than or equal to .5 or less than .5.
- The function we just created is used in the following subroutine RandomWalk(), which takes in the user desired number of time steps and runs a basic random walk. Enter the following code in the same module where the BinaryStepSelection function was entered:
Sub RandomWalk() Dim walk() As Double Dim stepSize As Double Dim steps, i As Long steps = Range(“nsteps”).Value stepSize = 1 / steps ReDim walk(1 To steps) walk(1) =BinaryStepSelection() * stepSize For i = 2 To UBound(walk) walk(i) = walk(i-1) + BinaryStepSelection() * stepSize Next i Call CopySimResultsToScratch(walk) Call PlotRandomWalk(steps) End SubYou will notice that the code is quite simple and involves declaring variables, assigning values from the sheet and with basic math, creating an array to store the results, and then looping through each time step and filling the array with the calculated values. Note that two other subroutines are used in the workbook, but these are purely for output purposes. In order to make the best use of text space the code for these subroutines is contained in the completed MB_2.7_Complete.xls workbook on the website.
- Viewing the complete version we can view a random walk from the VBA code on a line graph, which is shown in Figure 2.13.
FIGURE 2.13 A random walk with 1000 steps.
Brownian Motion as a Stochastic Process in the Real World
As mentioned in the start of this section, we can use Brownian motion to represent the fluctuations typically seen in the stock market. Very naively we could assume the stock price, S, with some initial value So, to be purely Brownian with variance, σ. This would give the relation in equation 2.3,
Note that this behaves exactly as the process shown in Figure 2.10 but with an offset of So. Immediately we can see a few drawbacks to this direct approach. First and foremost it should be clear to most readers that a stock price cannot go negative. A second more subtle issue is that financial observables will tend to drift and not simply oscillate around the mean.
Let's try to resolve the first issue by coming up with a scheme in which the stock price is never negative. The following, equation 2.4, would be a simple but effective guess:
By encapsulating the stochastic element as a power of an exponential function, the result will always be positive even if the exponent itself is negative. You may now be wondering why we chose in particular the exponential function. After all any positive number taken to the nth power will always return a value between 0 and infinity. The reason why this is a good choice will soon become evident.
Now we're making progress, but we're still not done. Recall that a Wiener process has a mean of 0 with some constant variance. This means the stock price will simply fluctuate about Soexp(0) = So. What we need to do here is to introduce a drift term, μ, that scales with time. This would force a movement in future stock prices by shifting the mean at each time step (equation 2.5).
Does this really work? We can prove to ourselves that it does work by computing the expectation value of the exponent (equation 2.6).
since <WT> = 0 and [μ,σ] are constants.
Our simple guesses have taken us quite far. But you may have noticed that while we solved two problems, we have now introduced another, namely that the price is now unbounded and will eventually drift toward infinity. All the issues we have discussed so far are problems from a purely mathematical standpoint. However, in practice it can be overlooked as long as the modeler is aware of their pitfalls. Models with these attributes are popular primarily because they are simple and the solutions to these models are easily tractable and faster to implement. The Black-Scholes model is an example of a popular implementation that is subject to these types of pitfalls. The initial stochastic process from which the solutions are derived may not go negative but it will converge toward infinity. Of course, more complicated models are designed to specifically address these issues, such as the Hull-White model we will go through in Chapter 4. The drawback for the increased accuracy, as you will see, is a far more complicated implementation.
Pricing Derivatives
Now let's explore why we chose the exponential in the first place. For those who have taken finance courses you may recall that the return of a bond that matures with constant interest rate, r, at time, T, is determined by the relation (equation 2.7),
Does this look familiar? If we took the rate as the observable for our stochastic process, then this equation, rewritten as equation 2.8, would look a lot like what we discussed in the last section.
The bond price won't have such a simple form anymore, but that is not the point. What we have just developed is a foundation on which we can build interest rate models that can be used to price various types of financial derivatives.
Brownian Motion as a Differential Process
To further develop our tool sets for simulation, we will introduce Brownian motion as a differential process. Please refer to Appendix A if you are not familiar with partial differential equations or require additional mathematical background. Many physical processes, including those in finance, behave according to slight changes in the underlying variables. For example, interest rates are in reality a continuous process and to properly characterize processes dependent on the rate and time, we have to formulate that process as dependent on the differential change in the rate. With this in mind, we can rewrite equation 2.8 as a differential process (equation 2.9)
Now making generalized statements about r, like the expected value, is a little more difficult because the solution to the equation would have r be dependent on two variables, W and t. Fortunately we can do a basic integration to find the function r(t,W) that satisfies equation 2.9 because each of the dependent variables, t and W, are completely separable. For the case of the regular Brownian motion, we see that equation 2.8 exactly meets our requirements. We would then expect r to be normally distributed with standard deviation σ and mean μt.
Geometric Brownian Motion
While values such as interest rates can be assumed to follow a Brownian motion, other quantities, such as stock prices or asset values, are typically assumed to follow a geometric Brownian motion (GBM), equation 2.10.
It may seem equation 2.10 is very similar to equation 2.9, but the solution and overall implications are very different. The dependent variables are not separable anymore as there is an extra S term in the equation. Fortunately, in this scenario we can use Ito's Lemma to help solve our stochastic differential equation (SDE). The lemma states that the differential for a multivariable function, f(t, r), can be approximated by equation 2.11,
If we were to replace S with r, set f(r) = r, and substitute equation 2.9 for dr, you will see that equation 2.9 is recovered. This result should be somewhat expected since we stated beforehand that equation 2.9 can be directly integrated to produce equation 2.8. However, what should f be for a geometric Brownian motion depicted in equation 2.10?
As a starting point, note that if we divide both sides of equation 2.10 by S, we get a term dS/S, which, when integrated, gives us a natural logarithm and a natural starting point for our guess, that is f(S) = ln(S). Before we continue further we should mention that the differential of a variable, say dt, means we are looking at very small changes in the quantities. As a consequence, the square of a very small number, either dt2 or dtdW, can be assumed for all practical purposes to be zero and freely ignored. The only difference in this particular example (a proof will not be given here) is that dW2 tends toward dt. If we apply f(S) = ln(S) along with our assertions to equation 2.10, we get the following (equations 2.12 and 2.13),
Now we are free to integrate equation 2.14 much like we did with equation 2.10 (see equation 2.14).
In light of this new equation we can say the log of the variable S is now normally distributed with standard deviation of σ and mean of (μ-σ2/2)t. Or more succinctly, a variable S that follows a geometric Brownian motion is said to be lognormally distributed. In finance you routinely hear the term phrase, “we expect this result to be lognormally distributed.” When somebody assumes this, they are not stating a law or fact that something has to be distributed this way. They are merely making a statement that they believe the underlying process is GBM.
Before we close our discussion on GBM processes, we must discuss one more issue because it will be useful when building the Merton-KMV model in Chapter 5. Given that we assume a variable follows a GBM process, how does one compute its volatility or standard deviation? One might be tempted to use Excel's STDEV function on the series of numbers produced from ln(Si)-ln(So). This however will not work even though we expect it be true. The reason for this is because the time term is constantly growing, which results in a larger and larger difference that will severely bias the STDEV result. The key to making a technique like this work is to examine the differential, which is the difference between neighboring points (equation 2.15).
We mentioned earlier in this chapter that the expected value of a Wiener process, W, is normally distributed. We also stated that the expected value of ΔW is also normally distributed. Therefore the relation in equation 2.15 is also normally distributed but without the bias imposed by an increasing time term. This is exactly what we need. Furthermore, this line of reasoning would also apply to regular Brownian motions discussed earlier.