Introduction

In the world of option pricing, understanding how the value of an option evolves over time is crucial. Binomial trees provide a powerful tool for modeling this evolution by breaking time into discrete steps. This chapter explores the concept of convergence in binomial models, shedding light on how option values stabilize and become more accurate as more time periods are added. We will delve into the mathematics behind this convergence and illustrate it through practical examples. This understanding is pivotal for both investors and financial analysts, as it impacts decision-making processes and risk assessment.

Convergence in Binomial Models

The binomial model divides time into discrete steps or periods. At each step, the underlying asset’s price can move up or down by a certain factor. As more periods are added to the model, the granularity of price movements increases, and the model approaches a continuous-time model like the Black-Scholes model.

The key principle behind convergence in binomial models is that as you add more time periods, the option’s calculated value approaches its true theoretical value. This means that the approximation becomes increasingly accurate as time is subdivided further.

Mathematical Explanation

The formula for the value of an option at each node in a binomial tree is based on risk-neutral probabilities. In a one-step model, for example, the option value at expiration is simply its intrinsic value. However, as you move backward through the tree to earlier time periods, you need to calculate the expected value of the option by considering both up and down movements.

The formula for the option value at a node $(S)$ is:

$$S=e^{-r \Delta t}\left(\pi_{u} S_u+ \pi_{d} S_d\right)$$

Where:

• $r$ is the risk-free interest rate.
• $\Delta t$ is the length of each time step.
• $\pi_{u}$ is the risk-neutral probability of an up move.
• $\pi_{d}$ is the risk-neutral probability of a down move.
• $S_u$ is the option value if the price moves up.
• $S_d$ is the option value if the price moves down.

As you add more time steps, you refine the calculation by considering more possible price paths. This increasingly granular view of possible outcomes leads to convergence.

Example: Assume you have a European call option on a stock with a current price of USD 100, a strike price of USD 105, a risk-free rate of 5%, and one time step of 1 year. In this case:

• $\pi_u$ (the probability of an up move) is 0.6 (assuming a 60% chance of the stock price going up)
• $\pi_d$ (the probability of a down move) is 0.4 (40% chance of the stock price going down)
• $\Delta t$ (the time step) is 1 year

Using the formula, you can calculate the option value at expiration (T) and then work backward to find the option value at earlier time periods.