Options trading: Understanding the binomial option pricing model

The Binomial Option Pricing Model (BOPM) is one of the most used methods for valuing options, particularly American-style options, which can be exercised at any time before expiration. In 1979, Cox, Ross, and Rubinstein came up with this model to provide an organised method for estimating the price of options by breaking down the time to expiration into discrete intervals.

Understanding the workings of the BOPM

The BOPM functions on the assumption that at each time step, the price of the underlying asset can either increase or decrease by a specific factor. This results in a binomial tree, where each node represents a possible price of the asset at a given time.

• Steps in the model:

1. Construct a binomial tree: Start with the current stock price and create branches for potential future prices at each time step.
   

2. Calculate up and down factors: Define how much the price can rise (up) or fall (down) based on volatility and time intervals.
   

3. Determine option payoffs: At expiration, calculate the payoff for each option based on whether it is in-the-money or out-of-the-money.
   

4. Work backwards: Calculate the present value of expected payoffs at each node, discounting them back to the present using a risk-free interest rate.
   

Key components of the BOPM

• Up factor: The factor by which the stock price increases.
  
• Down factor: The factor by which the stock price decreases.
  
• Risk-neutral probability (q): The probability of an upward movement in a risk-neutral world, calculated as:
  

A simple illustration to understand the concept

Suppose you are looking at a stock priced at ₹100 today. You want to calculate the value of a call option that gives you the right to buy this stock at ₹105 one month from now.

To simplify:

• The stock price can either go up to ₹120 or go down to ₹90 after one month.
• You know these two possible outcomes but are uncertain which will happen.

How the model works:

• Day 0: The stock is ₹100
• Day 30: The stock can be either be ₹120 or ₹90

A call option allows you to buy at ₹105. So, if the stock price goes:

• Up to ₹120: The option’s value will be ₹120 - ₹105 = ₹15 (because you can buy at ₹105 and immediately sell at ₹120).
• Down to ₹90: The option’s value will be ₹0 (because buying at ₹105 would mean a loss, so you wouldn’t exercise the option).
• Assume the chance of the stock moving up or down is 50% each (for ease in calculations).
• The model uses the risk-free rate to calculate how much those future payoffs are worth today.

How does the above activity help?

• By combining the possible payoffs (₹15 and ₹0), their probabilities, and discounting them, the model gives the fair price of the option today.

Understanding the main assumptions behind the BOPM

The model works on various assumptions that form its base or framework. These assumptions play an important role in utilising the model effectively while also understanding the outcomes.

1. Discrete time intervals: It breaks down the time until expiration into a finite number of discrete intervals or time steps. With each time step, the underlying asset’s price can move to one of two possible values: an increase (up) or a decrease (down) by specific factors.

2. Two possible outcomes: At each node in the binomial tree, it is assumed that the price of the underlying asset can only move to one of two outcomes over a given time period. This allows for easier calculations and modelling of price movements.

3. Risk-neutral valuation: The model operates under the assumption of a risk-neutral world, where all investors are indifferent to risk. In this framework, the expected return on the underlying asset is equal to the risk-free rate. This assumption simplifies the calculation of probabilities for price movements and allows for straightforward discounting of expected payoffs.

4. Constant volatility: The model assumes that volatility remains constant over the life of the option. This means that the up and down factors used to calculate potential future prices are derived from a fixed measure of volatility, which may not reflect real market conditions where volatility can fluctuate.

5. No arbitrage opportunities: The BOPM assumes that there is an absence of arbitrage opportunities in the market. This means that it is impossible to create a risk-free profit through simultaneous buying and selling of assets, which ensures that prices remain consistent with theoretical valuations.

6. No transaction costs or taxes: The framework presumes that no taxes or transaction costs are charged with the buying or selling of the underlying asset or the options. This assumption simplifies calculations but may not hold true in real-world trading environments.

7. Perfectly efficient markets: The model assumes that markets are efficient, meaning all available information is reflected in asset prices, and there are no lags in price adjustments based on new information.

Advantages of the Binomial Option Pricing Model

• Flexibility: The BOPM can incorporate various factors such as dividends and changing interest rates, making it adaptable to different market conditions.

• Multi-period analysis: Unlike some models that only evaluate options at expiration, the BOPM assesses multiple points in time, providing a more comprehensive view of potential price movements.

• American options valuation: It excels in valuing American options due to its ability to evaluate exercise opportunities at any point before expiration.

Disadvantages of the Binomial Option Pricing Model

• Computational complexity: For many options, calculations can become cumbersome and time-consuming as the number of periods increases.

• Market dynamics: The model assumes the absence of arbitrage opportunities. In fact, in reality, market conditions may create discrepancies between theoretical prices and market prices.

Binomial Option Pricing Model vs the Black-Scholes model

The Binomial Option Pricing Model and the Black-Scholes Model are two prominent methods used for pricing options. Each model has its own strengths and weaknesses, making them suitable for different types of options and market conditions.

The choice between the BOPM and the Black-Scholes Model is very much determined by the requirements of the specific option being priced. For American options or those with complex features, the BOPM is generally preferred due to its flexibility and ability to model various scenarios. Conversely, for European options where the underlying assumptions are valid, the Black-Scholes model offers a quick and efficient pricing solution.

Read More: What is Black-Scholes Model?

To sum up

The BOPM is often seen as a cornerstone in the world of financial modelling, providing a strong and flexible framework for option pricing. Its step-by-step approach allows traders and investors like you to assess potential price movements, making it particularly effective for pricing American-style options. The model comes with its own set of challenges and assumptions, but it can certainly be said that the flexibility and transparency offered by the BOPM make it an essential tool to navigate dynamic market environments.