We might be satisfied with our current formula. However, by reformulating the formula we can obtain more insights from it. Moreover, it will be much easier to deal with other relationships that will be covered in the following sessions.
The goal is to find the minimum number of upward movements so that the max operator in the call option payoff provides positive numbers.
Before we proceed further, recall the binomial theorem first. In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial (x+y)ⁿ into a sum involving terms of the form axᵇ yᶜ, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending on n and b. Some examples are:
The coefficient a in the term of axᵇ yᶜ is known as the binomial coefficient. These coefficients for varying n and b can be arranged to form Pascal’s triangle. These numbers also arise in combinatorics, where the number of different combinations of b elements can be chosen from an n-element set. Therefore it is often pronounced as “n choose b”.
Once we feel comfortable with the binomial coefficients, we are ready to move on to the derivation of the expectation of a random variable. We will practice how to calculate the expectation of a binomial random variable as an example.
The first thing to note is that it doesn’t matter whether we use x, i, j, or k for our index of the derivation. We will use the index k in the following.
Next, we need to guess what is going to happen to each term when we expand the summation. And notice that the term with k = 0 will be zero because it is 0 times something, and that something is not infinity. So we are sure that we can exclude the term with k = 0 without affecting the final result. This kind of technique will be used later in other derivations. If used wisely, it will save a lot of effort.
A quick fact about binomial coefficients might be helpful.
Now, this is the derivation of the expectation of a binomial distribution. I believe that the video is self-explanatory.
The real part of the reformulation is the following.
Binomial animation
