Binomial Option Pricing Model:

Introduction:

"The Binomial Option Pricing Model is a mathematical model used to calculate the theoretical price of financial options, such as stock options."

It was developed by Cox, Ross, and Rubinstein in the 1970s and is based on the concept of constructing a binomial tree to represent the possible price movements of the underlying asset over time.

How binomial option pricing model works?

Here's how the model works:

• Binomial Tree: The model assumes that the price of the underlying asset can move in two discrete states at each time step. Typically, one state represents an upward movement, and the other represents a downward movement. The number of time steps is determined by the user.
• Option Valuation: At each node of the tree, the model calculates the option's value by considering the possible future price movements. This involves calculating the expected value of the option at each node, which is a combination of the option's intrinsic value (the difference between the current asset price and the option's strike price) and the expected value of the option in the future.
• Risk-Neutral Probability: To calculate the expected value, the model assumes a risk-neutral probability, which is the probability of an up or down movement that makes the option's value equal to the expected risk-free return. This risk-neutral probability is used to weight the different possibilities at each node.
• Backward Induction: The model works backward through the tree, calculating the option's value at each node until it reaches the initial node (current time).
• Option Price: The option's theoretical price is then determined by the value at the initial node, which represents the option's present value.

Example:

Let's work through a simple example of the Binomial Option Pricing Model to calculate the theoretical price of a European call option. In this example, we'll assume the following parameters:

Current stock price (S0) = $100

Strike price (K) = $105

Time to expiration (T) = 1 year

Risk-free interest rate (r) = 5%

Volatility (σ) = 20%

Number of time steps (n) = 3

We will use a three-step binomial tree to value the option.

Step 1: Calculate Up and Down Movements

The up and down movements can be calculated using the following formulas:

u = e^(σ√Δt)

d = 1/u

Where Δt is the time to each step. In this case, Δt = T/n = 1/3.

u = e^(0.20 * √(1/3)) ≈ 1.1265

d = 1/u ≈ 0.8874

Step 2: Calculate Risk-Neutral Probability

The risk-neutral probability (p) can be calculated as follows:

p = (e^(rΔt) - d) / (u - d)

p = (e^(0.05 * 1/3) - 0.8874) / (1.1265 - 0.8874)

p ≈ 0.5132

Step 3: Create the Binomial Tree

Construct a three-step binomial tree starting from the current stock price:

(S0 = $100)

/ \

(uS0 = $112.65) (dS0 = $88.74)

/ \ / \

(uuS0 = $126.77)

Step 4: Calculate Option Values at Each Node

Starting from the final nodes and working backward:

At node uuS0, the option's value is max(uuS0 - K, 0) = max($126.77 - $105, 0) = $21.77.

At nodes uS0 and dS0, calculate the option values using the risk-neutral pricing formula:

For uS0:

Option value = e^(-rΔt) [p Option value(up) + (1 - p) * Option value(down)]

Option value = e^(-0.05 1/3) [0.5132 * $21.77 + 0] ≈ $9.99

For dS0:

Option value = e^(-rΔt) [p Option value(up) + (1 - p) * Option value(down)]

Option value = e^(-0.05 1/3) [0.5132 * $0 + 0] = $0

Step 5: Calculate the Option Price

The option's theoretical price is the value at the initial node (S0):

Option price = e^(-r T) [p Option value(up) + (1 - p) * Option value(down)]

Option price = e^(-0.05 1) [0.5132 * $9.99 + 0] ≈ $4.76

So, the theoretical price of the European call option is approximately $4.76.

Assumptions:

The Binomial Option Pricing Model, like any financial model, relies on certain assumptions to make its calculations. The key assumptions of the Binomial Option Pricing Model are as follows:

1. Two-State World: The model assumes that the price of the underlying asset can move in only two discrete states at each time step, typically representing an upward and a downward movement. This simplification makes the model more tractable.
2. Constant Volatility: It assumes that the volatility of the underlying asset's returns remains constant over the option's life. In reality, volatility can change, but the model assumes it to be fixed.
3. Risk-Neutral World: The model operates under the assumption of a risk-neutral world, meaning it assumes that investors are risk-neutral and, therefore, the expected return from holding the option is equal to the risk-free rate. This assumption is necessary for the model's calculation of the option's value at each node.
4. No Dividends: The model assumes that the underlying asset does not pay any dividends during the option's life. In cases where the underlying asset pays dividends, an adjusted version of the model (such as the binomial dividend model) is used.
5. European Options: The original Binomial Option Pricing Model was developed for European options, which can only be exercised at expiration. While the model can be adapted for American options, it assumes European-style options in its simplest form.

Types of binomial option pricing model:

Binomial option pricing is a widely used method for valuing options, especially in the context of financial derivatives. There are two primary types of binomial option pricing models: the Cox-Ross-Rubinstein (CRR) model and the Jarrow-Rudd model. These models are based on the binomial tree approach and are used to calculate the value of options at different time points leading up to their expiration.

Cox-Ross-Rubinstein (CRR) Model:

The CRR model, also known as the binomial tree model, was developed by John Cox, Stephen Ross, and Mark Rubinstein in 1979.

It assumes that the price of the underlying asset can move up or down in discrete, evenly spaced time intervals.

The CRR model uses two parameters: the up factor (U) and the down factor (D), which represent the expected percentage increase and decrease in the underlying asset's price over each time step.

The option's value is calculated at each node of the binomial tree by considering the probability of moving up or down and discounting the expected future cash flows.

The CRR model is widely used for its simplicity and ease of implementation.

Jarrow-Rudd Model:

The Jarrow-Rudd model is a modification of the CRR model and was proposed by Robert Jarrow and Andrew Rudd.

It adjusts the probabilities of up and down movements to make the risk-neutral valuation more consistent with continuous-time models like the Black-Scholes model.

In the Jarrow-Rudd model, the risk-neutral probability of an up movement is set to match the risk-free interest rate, which makes it more suitable for valuing options in an arbitrage-free market.

Conclusion:

The binomial option pricing model is a valuable tool for valuing options by modeling the discrete price movements of the underlying asset. It provides a flexible and intuitive approach to estimate option prices, incorporating time, volatility, and interest rates. However, it's less precise than continuous-time models like Black-Scholes and can be computationally intensive for complex options.