FRM- Binomial (one & two step) | Option Pricing Strategies | Python implementation

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Timestamp:
00:12 - 01:00 - Pricing an option using binomial trees
01:00 - 04:40 - One step binomial tree
04:01 - 06:23 - Two-step binomial tree

In this video, we will be equipping ourselves with essential mathematical knowledge required to understand how Options pricing models are derived.

In this video lecture, we will discuss a popular technique for pricing an option using Binomial Trees. A Binomial tree is a diagram that flows from one starting node into two nodes and continues the same for n-layers. In this case, the binomial tree represents different possible paths that the underlying might follow in the life span of the option.
An assumption here is that the underlying follows a random walk, i.e. future price movements are independent of past price movements. At each step in the binomial tree there is a defined probability of the underlying either moving up or moving down by a certain percentage amount. There could be many steps in the tree depending on how complex a model we want to create.

Let us start by looking at a simple one step model using an example of a European call option. We are interested in finding the price of a European call option with a strike price of INR 110 expiring in 3 months. Assume that the underlying stock price is valued at INR 100 at present. At the end of the three months, there will be two scenarios, either the stock price moves to INR 120, which makes the option worth INR 10, or the stock price moves to INR 80, in which case the option expires worthless.
We assume two conditions, the first being that arbitrage opportunities do not exist. We will set up a portfolio in such a way that there is no uncertainty about the value of the portfolio at the end of 3 months. The next condition is that because the portfolio has zero risk, the return it earns must be equal to the risk-free interest rate. This means, this portfolio will give us the same returns as a risk-free bank deposit. Let us understand this better with an example. First let us understand the “no arbitrage” assumption, which implies that quantr there should not exist any opportunity to make risk free profit.

Consider a position where we buy delta number of shares of the underlying stock and sell one call option. We will compute the value of delta, which makes this a risk- free portfolio. We will equate the values of the two outcomes after 3 months. In the first case when the stock price is at INR 120 and the value of the option is INR 10, the value of our portfolio is 120 times delta minus 10. In the second case, when the stock price is at INR 80, the value of our long position is 80 delta and the option expires worthless, hence the portfolio has a value of 80 delta. On equating these two cases, 120 delta minus 10 is equal to 80 delta, which implies 40 delta is equal to 10. We see that when delta is equal to 0.25, we can create a risk-free portfolio. You can substitute this value of delta in both the cases and see that the value of our portfolio remains to be INR 20 after three months in either outcomes.

We can extend this analysis to a two-step binomial tree. At each step, we make an assumption, that the price can either go up or down by 20%. The time scale for each step is 3 months and the option has 6 months to expiration. At the end of the first step, the underlying might either be priced at INR 120 or INR 80, this is the same as the example we have seen previously. Now, for computing the option price, we will sequentially compute the price of the option, first at three months in the future at points B and C following which we will then compute the price of the option today at point A.
The value or price of the option at points C, E and F will be 0, as the option will expire out of the money if it takes the bottom path. The price of the option at point B will be computed by considering the branch B-D-E to be a one step binary tree. Once we compute the price at point B, we will consider the A-B-C branch and compute the option price at point A. We will apply our learnings, from the one step binary tree example, twice, to arrive at the option price at point A.
Applying this methodology, the fair value of the option is computed to be INR 10.26. You can try this as an exercise by computing the values yourself.
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