Multi-Period Binomial Models: Adapting the Valuation Process

Imagine trying to predict the price of a stock option. In the simplest scenario, you might think about just one period of time, say, tomorrow. This is the essence of a single-period binomial model. It’s like saying, “Okay, let’s assume the stock price can only go up or down by tomorrow, and then we’ll figure out the option’s value based on those two possibilities.”

In a single-period model, the valuation process is quite direct. You figure out what the option would be worth if the stock price goes up, and what it would be worth if the stock price goes down. These are the option’s payoffs in each scenario. Then, using something called risk-neutral probabilities, which are essentially the probabilities of the stock going up or down in a way that removes risk from the equation for valuation purposes, you calculate a weighted average of these payoffs. This weighted average, discounted back to today, gives you the option’s fair price. It’s like calculating a simple expected value, but with these special risk-neutral probabilities.

Now, real life isn’t just about tomorrow. Stock prices move over time, often in many steps. This is where the multi-period binomial model comes in. Instead of just one time step, we consider multiple steps, like days, weeks, or months. Think of it like climbing a staircase instead of just taking one giant leap. At each step of the staircase, representing a period of time, the stock price can either go up or down. This creates a branching tree of possibilities.

The crucial difference in valuation with a multi-period model is that we can’t just jump to the end and calculate a single expected value. Instead, we have to work backward, step by step, from the final period to the present. Imagine you are at the top of the staircase, and you need to figure out the value at each step as you descend.

Let’s say we are considering an option that expires in three months, and we are breaking this down into three one-month periods. In the final month, month three, we know the option’s payoff for every possible stock price at that point, just like in the single-period model, it’s either in the money or out of the money.

To value the option at the end of month two, we look at each possible stock price at that time. For each of these prices, we consider the two possibilities for the next period, month three, an upward movement and a downward movement. We know the value of the option in month three for both of these scenarios because we’ve already calculated the payoffs at expiration. We then calculate the risk-neutral expected value of these two possible month three option values, discounted back to the end of month two. This gives us the option’s value at the end of month two for that specific stock price.

We repeat this process for every possible stock price at the end of month two. Once we have the option values for all possible scenarios at the end of month two, we move back to the end of month one. Again, for each possible stock price at the end of month one, we look at the two possible scenarios for month two, an upward movement and a downward movement. We now know the option’s value for both of these scenarios because we just calculated them for month two. We calculate the risk-neutral expected value of these two month two option values, discounted back to the end of month one. This gives us the option’s value at the end of month one for that specific stock price.

Finally, we repeat this one last time to get to today, period zero. We look at the initial stock price, consider the two possibilities for month one, upward or downward movement. We know the option’s value for both of these scenarios because we just calculated them for month one. We calculate the risk-neutral expected value of these two month one option values, discounted back to today. This final value is the option’s price in the multi-period binomial model.

Essentially, the multi-period model breaks down the time to expiration into smaller intervals and values the option step by step, working backward from expiration. This iterative process is crucial. It allows us to capture the evolving nature of uncertainty over time and provides a more realistic and nuanced valuation compared to the simpler, single-period approach. While the single-period model gives a basic understanding, the multi-period model offers a richer and more adaptable framework for pricing options in a world where things change over time.