Binomial Option Pricing Model

The Binomial Option Pricing Model is a powerful and versatile tool used for pricing options, especially American options which can be exercised at any point up to the expiration date. Developed in 1979 by John Cox, Stephen Ross, and Mark Rubinstein, this model provides a systematic method of determining the fair value of an option by modeling the different paths that an underlying asset’s price can take over time.

Fundamental Concepts

Options

Options are financial derivatives that give the holder the right, but not the obligation, to buy or sell an asset at a predetermined price before or at a certain date. The two main types of options are:

• Call Options: Provide the right to buy the underlying asset.
• Put Options: Provide the right to sell the underlying asset.

American vs. European Options

• American Options: Can be exercised at any time before the expiration date.
• European Options: Can only be exercised at the expiration date.

Binomial Tree

The Binomial Option Pricing Model uses a discrete-time model of the varying price of the underlying asset. It creates a binomial tree representing different possible price paths for the asset over the life of the option. Each node in the tree represents a possible price of the underlying asset at a certain point in time, and these nodes are connected by branches representing possible upward or downward movements in the price.

Steps in the Binomial Option Pricing Model

The model is implemented through a step-by-step process:

1. Setting Up the Parameters

• Initial Asset Price (S₀): Current price of the underlying asset.
• Strike Price (K): The price at which the option can be exercised.
• Time to Expiration (T): Duration in years until the option expires.
• Number of Time Steps (N): How many steps (intervals) the time to expiration is divided into.
• Volatility (σ): Standard deviation of the underlying asset’s returns.
• Risk-Free Rate (r): Continuous compounded risk-free interest rate.

2. Calculating Step Size

The time step (( [Delta](../d/delta.html) t )) is calculated as:

[ [Delta](../d/delta.html) t = \frac{T}{N} ]

3. Calculating Up and Down Factors

The up factor (u) and down factor (d) represent the size of an upward and downward price movement within a single time step. They are calculated using the asset volatility (( \sigma )) and time step (( [Delta](../d/delta.html) t )):

[ u = e^{\sigma\sqrt{[Delta](../d/delta.html) t}} ] [ d = e^{-\sigma\sqrt{[Delta](../d/delta.html) t}} ]

4. Calculating Risk-Neutral Probabilities

Risk-neutral probabilities (p and q) denote the probabilities of upward and downward movements in a risk-neutral world. These are calculated using the up factor (u), down factor (d), and risk-free rate (r):

[ p = \frac{e^{r[Delta](../d/delta.html) t} - d}{u - d} ] [ q = 1 - p ]

5. Building the Binomial Tree

Starting with the initial price (S₀), the tree is built by multiplying the price at each node by the up factor (u) and down factor (d).

6. Calculating Option Values at Terminal Nodes

At expiration (final nodes of the tree), the option’s intrinsic value is calculated:

• Call Option: ( \max(S_T - K, 0) )
• Put Option: ( \max(K - S_T, 0) )

7. Discounting Back to the Present Value

The option values are then discounted back through the tree to present value using the risk-neutral probabilities (p and q) and the risk-free rate (r). The value at each node is calculated as:

[ V = e^{-r[Delta](../d/delta.html) t} (p V_u + q V_d) ]

where ( V_u ) and ( V_d ) are the option values at the next time step’s upward and downward nodes, respectively.

8. Adjusting for Early Exercise (for American Options)

For American options, the possibility of early exercise is checked at each node by comparing the discounted option value with the intrinsic value. If early exercise provides a higher value, it is taken.

Example Calculation

Let’s go through an example to see how the Binomial Option Pricing Model would work in practice:

Parameters

• Initial asset price (S₀): $100
• Strike price (K): $100
• Time to expiration (T): 1 year
• Number of steps (N): 3
• Volatility (σ): 30% (0.30)
• Risk-free rate (r): 5% (0.05)

Step 1: Calculate the time step

[ [Delta](../d/delta.html) t = \frac{1}{3} = 0.3333 \text{ years} ]

Step 2: Calculate up and down factors

[ u = e^{0.30 \sqrt{0.3333}} = 1.1832 ] [ d = e^{-0.30 \sqrt{0.3333}} = 0.8453 ]

Step 3: Calculate risk-neutral probabilities

[ p = \frac{e^{0.05 \cdot 0.3333} - 0.8453}{1.1832 - 0.8453} = 0.5223 ] [ q = 1 - 0.5223 = 0.4777 ]

Step 4: Build the Binomial Tree

Node Prices

• ( S(0,0) = 100 )
• ( S(1,1) = 100 \cdot 1.1832 = 118.32 )
• ( S(1,0) = 100 \cdot 0.8453 = 84.53 )
• ( S(2,2) = 118.32 \cdot 1.1832 = 140.03 )
• ( S(2,1) = 100 \cdot 1.1832 \cdot 0.8453 = 100 )
• ( S(2,0) = 84.53 \cdot 0.8453 = 71.45 )
• ( S(3,3) = 140.03 \cdot 1.1832 = 165.79 )
• ( S(3,2) = 100 \cdot (1.1832)^2 \cdot (0.8453) = 118.32 )
• ( S(3,1) = 100 \cdot 1.1832 \cdot (0.8453)^2 = 84.53 )
• ( S(3,0) = 71.45 \cdot 0.8453 = 60.37 )

Option Values at Terminal Nodes

• For call option: ( \max(S_T - K, 0) )
• ( V(3,3) = \max(165.79 - 100, 0) = 65.79 )
• ( V(3,2) = \max(118.32 - 100, 0) = 18.32 )
• ( V(3,1) = \max(84.53 - 100, 0) = 0 )
• ( V(3,0) = \max(60.37 - 100, 0) = 0 )

Step 5: Discount Back to Present Value

• For node ( (2,2) ): [ V(2,2) = e^{-0.05 \cdot 0.3333} (0.5223 \cdot 65.79 + 0.4777 \cdot 18.32) = 42.41 ]
• For node ( (2,1) ): [ V(2,1) = e^{-0.05 \cdot 0.3333} (0.5223 \cdot 18.32 + 0.4777 \cdot 0) = 9.23 ]
• For node ( (2,0) ): [ V(2,0) = e^{-0.05 \cdot 0.3333} (0.5223 \cdot 0 + 0.4777 \cdot 0) = 0 ]

Continue discounting back to today’s value for nodes (1,1), (1,0), and finally (0,0).

Step 6: Adjusting for Early Exercise

For American options, you check each node to see if early exercise is advantageous. If so, the value at the node is adjusted to reflect this possibility.

Final Option Price

After discounting through all time steps and accounting for possible early exercise, the calculated value at the initial node ( (0,0) ) will give the price of the option.

Advantages of the Binomial Model

1. Flexibility: Can price American options which allow for early exercise.
2. Accuracy: Provides a more precise valuation by considering different paths and scenarios.
3. Simplicity: Easier to implement compared to other complex models like Monte Carlo simulations.

Limitations

1. Computational Intensity: High number of time steps can result in large computational effort.
2. Assumption of Constant Volatility: Real-world volatility can change over time.

Practical Applications

The Binomial Option Pricing Model is widely used within financial institutions for pricing and managing option-related products. Its flexibility and adaptability make it a vital tool in the following areas:

1. Risk Management: Banks and financial institutions use the model to hedge exposure to options and other derivatives.
2. Employee Stock Options: Companies use the model to value employee stock options for financial reporting purposes.
3. Corporate Finance: Used to evaluate the embedded options in corporate debt and other securities.

Example Companies Utilizing Binomial Model

• Goldman Sachs: Goldman Sachs employs advanced financial models, including binomial models, for asset pricing and risk management.
• Morgan Stanley: Morgan Stanley uses the Binomial Option Pricing Model for assessing option pricing and financial derivatives.

The Binomial Option Pricing Model remains a cornerstone of quantitative finance, offering a robust method for option valuation that balances computational feasibility with pricing accuracy.