Brownian Motion
One Period Binomial Model
Priliminary Knowledge:
Binomial Asset Pricing model is a very important tool in finance. In simple words- asset is a item of ownership convertible into cash. In Binomial Asset Pricing model- we model stock prices in discrete time. We assume that- at each step the stock price would change to one of two possible values. Let us start with an example.
Suppose at time t = 0, the initial stock price is- \(S_0 > 0\). There are two positive numbers- \(u\) and \(d\) with- \(0 < d < u\), such that- at the next time period (i.e. t = 1) the stock price would be- either \(u.S_0\) or \(d.S_0\).
There are some simple assumptions here which we will learn along the way. Suppose you have some money \(X_0\). You can invest this money in one of two places- either in a riskfree institution or in a risky institution.
What are riskfree and risky financial institutions ?
The most common example of riskfree institution will be a bank. If you save some \(X_0\) money in a bank with interest rate = r%, after t years you will surely get back- \(X_t=X_0(1+r)^t\) amount of money from the bank. It is guaranteed that tat time point t the bank will return \(X_t\) amount of money to you.
But If you invest that \(X_0\) amount of money in a stock market, there is no guarantee how much money you will receive at time point t. It may be very high; it may be very low; it may even be zero. That is why it is called a risky financial institution.
But why are we learning about risky and riskfree financial institution?
Let us construct an arbitrary case, you will understand it very easily. Suppose you have some money- that you want to invest. You are in a simple world that only gives you two options to invest- a bank, which is a riskfree financial institution. Another option is- the stock market. This stock market has very simple return. Either the invested amount is multiplied by \(u\) or by \(d\), where \(u\) is the upward movement factor and \(d\) is the downward factor.
Now, what should be the relation between \(u\), \(d\) and \(r\)?
We are going to find this logically. Suppose the amount of money you want to invest is- \(X_0\). Without loss of generality we have- \(X_0 > 0\).
If you invest the money at time \(t=0\) in the bank, at time \(t=1\), you will get back \(X_0.(1+r)\) in return. But if you invest the money at time \(t=0\) in the above mentioned stock market- at time \(t=1\) either you will get \(u.X_0\) or \(d.X_0\). We already assumed- \(0<d<u\). Hence we have- \[0 < d.X_0 < u.X_0\]
Now, you probably already know investing in the stock market involves risk taking, because you do not know how much money you will get in return. It may be high as you hope for, but it may also be lower than the amount you have invested.
| Investment chosen | t=0 | t=1 |
| Bank | \(X_0\) | \(X_0(1+r)\) |
| Stock Market | \(X_0\) | \(u.X_0\) or \(d.X_0\) |
Suppose you have- \((1+r) \geq u\): Then we have- \[\begin{aligned} &X_0(1+r) > \max(u.X_0,d.X_0) \\ \implies &X_0(1+r) > u.X_0 \end{aligned}\] So, without taking any risk in the stock market, you are getting more return from the bank. Logically it can not be right.
WHY TAKING RISKS IF YOU HAVE GURANTEED HIGHER RETURNS!!
In that case no one will invest in the stock market. So the inequality should be reversed. i.e. \((1+r) < u\)
Similarly, if we have- \((1+r) \leq d\), even after taking risks by investing in the stock market, you can earn more money than investing in risk free institutions like banks. So no rational person will invest in the bank.
So in this case also,the inequality should be reversed, i.e. \((1+r) > d\). Combining these two inequalities we get- \[0 < d < (1+r) < u\]
Now, this type of simplified stock market model is called Binomial asset pricing model. But why are we reading about Binomial asset pricing model? To know see my next blog.