Binomial Tree Model

So far we have looked at a lot of Mathematical concepts, let try to apply it to something other than a coin. We will try go through a Binomial Tree model, and try to apply it for a Stock.

So, the Stock will start at a price S0 at t=0, we assume that with probability p it goes up by a factor of uS0→S0u and that it goes down with probability 1−pby a factor of d.

Thus the Stock at t=1 ( which we will denote by S1 ) can be represented by S0u with probability p$p$ and S0d with probability 1−p

Now if we keep repeating the same experiment at each node we end up with something like

In this manner we can keep continuing the experiment, but for now, lets stick to a total of 4$4$ time steps i.e at the end t=4

So, what can we do with a binomial model ? Well we can compute the probability of each stock price node at each step.

At step 1$1$ for example - P(uS0)=p and P(dS0)=1−p at say step 3 we have a few more states, P(u3S0)=p3, P(dS0)=3p(1−p)2 and so on.

We can also compute things like, what is the Expected value of a stock at each time step. At t=2we have

E(S2)=P(u2S0)u2S0+P(S0)S0=P(d2S0)d2S0=S0((up)2+2p(1−p)ud+(d(1−p))2)

Not only this, but the binomial model also adjusts to given information. What it means is that if you know where you are at step 1, the probabilities are now different! How does that happen? Well there is one more thing that changes as the time steps grow, that is the Filtration.

When we write P we actually mean P|F where Fis the given filtration at the step.

So say at step 2 we now the probability of u2S0 is 1/4 but that is not correct, the correct way to say it is given no information i.e F0the probability is 1/4, but if we knew where the stock ends up at step one i.e F1 the probabilities change accordingly. So if say the Stock went up to uS0 at step 1. P|F1(u2S0)=1/2.

Thus, once we create a binomial model for a Stock, we can

1. Tell the probability of each point of Stock - Time on tree
2. Compute the expected value of the Stock at each Time Step
3. Amend the Tree probabilities as we get more information