In the last article, I introduced you briefly to various option pricing models and their comparisons. In this article we will delve deeper into one such pricing model – possibly the easiest of them – called Binomial Option Pricing Model.
Before we go too deep into it, let’s remember ourselves what the objective is here.
We are trying to determine the fair value / theoretical price of an option contract. Since the expiry of the option is in the future, we do not know what the underlying price of the asset at the time of the expiry will be. But we can make some educated guess about how much the asset can go up or down in a sufficiently small timeframe.
For example, we may not know how much the underlying stock will be after a month, but we may be able to guess a maximum limit of how much the stock can move in a day.
So, the question is – can we use that information to estimate the Option price? Let’s see.
Binomial Pricing Model
The binomial option pricing model is a discrete model that uses a series of possible outcomes (or "nodes") to determine the price of an option. It is based on the idea of constructing a binomial tree, which is a graphical representation of the possible outcomes of the underlying asset over a given time period. The binomial tree is constructed such that at each node, the underlying asset can either go up or down by a certain factor, known as the "up factor" or "down factor."
Generating the Binomial Tree
To illustrate the binomial option pricing model, let's consider a simple example.
Suppose we have a call option on a stock with a strike price of $50 and an expiration date 2 months from now. The current price of the stock is $40, and we assume that over the next month, the stock can either go up by 20% or down by 20% with equal probability. We can construct a binomial tree to represent the possible outcomes of the stock price over the next 2 months:
From the binomial tree, we can see that there are 4 possible outcomes for the stock price at the end of the period:
$16, $25.6, $38.4, and $57.6.
We can then use these outcomes to calculate the value of the call option at each node. For example, at the node where the stock price is $25.6, the call option has a value of $0, since it is out-of-the-money (i.e., the strike price is higher than the stock price). At the node where the stock price is $57.6, the call option has a value of $7.6, since it is in-the-money (i.e., the stock price is higher than the strike price).
We can then use the values at each node to determine the fair value of the call option using a technique known as backward induction. This involves working backwards from the final nodes to the initial node, using the following formula:
Option value at node N = (Probability of up * Option value at up node) + (Probability of down * Option value at down node)
For example, at the final nodes (Month 2), the option value at the up node (where the stock price is $57.6) is $7.6, and the option value at the down node (where the stock price is $38.4) is $0 (Remember the strike price is $50).
Using the formula above, we can calculate the option value at the Month 1 up node (where the stock price is $48) as follows:
Option value at Month 1 up node = (0.5 * $7.6) + (0.5 * $0) = $3.8
We can then repeat this process to determine the value of the option at the initial node (Month 0):
Option value at Month 0 = (0.5 * $3.8) + (0.5 * $0) = $1.9
Thus, the fair value of the call option according to the binomial model is $1.9.
Assumptions of Binomial Option Pricing Model
As you can see from the model above, the model continues to assume the same up factor and down factor throughout each step of the binomial tree. This up/down factor is known as Volatility and the constancy of the volatility is a major drawback of this model. But there are more. Here are some of them:
- Constant volatility: The volatility of the underlying asset is assumed to be constant over the life of the option.
- No dividends: The underlying asset is assumed to pay no dividends over the life of the option.
- No transaction costs: It is assumed that there are no transaction costs involved in buying or selling the underlying asset or the option.
- Risk-free interest rate: The risk-free interest rate is assumed to be known and constant over the life of the option.
- Constant risk-free rate: The risk-free rate is assumed to be constant over the life of the option.
That's all for now, next we will learn how to Build the computer program to calculate option pricing using Binomial Pricing Model.