B.5.2 Binomial Option Pricing Model for Canadian Securities Course

B.5.2 Binomial Option Pricing Model

The Binomial Option Pricing Model is a cornerstone in the field of financial derivatives, providing a robust framework for valuing options. This model is particularly valued for its simplicity and flexibility, allowing for the valuation of both European and American options. In this section, we will delve into the mechanics of the binomial model, explore its applications, and provide a comprehensive example to illustrate its practical use.

Understanding the Binomial Model

The binomial model divides the time to expiration of an option into multiple discrete periods, creating a lattice or tree of possible asset price paths. At each node in the tree, the asset price can move up or down by specific factors, \( u \) and \( d \), respectively. This approach allows for a straightforward calculation of option values through backward induction.

Key Components of the Binomial Model

1. Time Division: The time to expiration is divided into \( N \) periods, each of length \( \Delta t = \frac{T}{N} \), where \( T \) is the total time to expiration.

2. Price Movement Factors:

   • Up Factor (\( u \)): The factor by which the asset price increases.
     $$ u = e^{\sigma \sqrt{\Delta t}} $$
   • Down Factor (\( d \)): The factor by which the asset price decreases.
     $$ d = \frac{1}{u} $$
     Here, \( \sigma \) represents the volatility of the asset.

3. Risk-Neutral Probability (\( p \)): The probability of an upward movement in a risk-neutral world.

   $$ p = \frac{e^{r \Delta t} - d}{u - d} $$
   Where \( r \) is the risk-free interest rate.

Building a Binomial Tree

To illustrate the binomial model, we will construct a two-period binomial tree for a stock with the following parameters:

• Initial stock price (\( S_0 \)) = $100
• Volatility (\( \sigma \)) = 20%
• Risk-free rate (\( r \)) = 5%
• Time to expiration (\( T \)) = 1 year

Step-by-Step Example

1. Calculate Time Increment (\( \Delta t \)):

   $$ \Delta t = \frac{T}{N} = \frac{1}{2} = 0.5 $$

2. Determine Up and Down Factors:

   $$ u = e^{0.2 \times \sqrt{0.5}} \approx 1.1487 $$
   $$ d = \frac{1}{u} \approx 0.8706 $$

3. Calculate Risk-Neutral Probability:

   $$ p = \frac{e^{0.05 \times 0.5} - 0.8706}{1.1487 - 0.8706} \approx 0.524 $$

4. Construct the Price Tree:

    graph TD;
	    A(S_0 = $100) --> B(S_u = $114.87);
	    A --> C(S_d = $87.06);
	    B --> D(S_uu = $131.86);
	    B --> E(S_ud = $100);
	    C --> F(S_du = $100);
	    C --> G(S_dd = $75.76);

Calculating Option Values

Using the constructed price tree, we can now calculate the option values at each node using backward induction. For simplicity, let’s assume we are pricing a European call option with a strike price of $100.

Option Valuation at Maturity

• \( C_{uu} = \max(131.86 - 100, 0) = 31.86 \)
• \( C_{ud} = \max(100 - 100, 0) = 0 \)
• \( C_{du} = \max(100 - 100, 0) = 0 \)
• \( C_{dd} = \max(75.76 - 100, 0) = 0 \)

Backward Induction

1. Calculate Option Value at Node \( S_u \):

   $$ C_u = e^{-0.05 \times 0.5} \times (0.524 \times 31.86 + (1 - 0.524) \times 0) \approx 15.97 $$

2. Calculate Option Value at Node \( S_d \):

   $$ C_d = e^{-0.05 \times 0.5} \times (0.524 \times 0 + (1 - 0.524) \times 0) = 0 $$

3. Calculate Option Value at Initial Node \( S_0 \):

   $$ C_0 = e^{-0.05 \times 0.5} \times (0.524 \times 15.97 + (1 - 0.524) \times 0) \approx 8.02 $$

Flexibility of the Binomial Model

One of the significant advantages of the binomial model is its ability to handle American options, which can be exercised at any time before expiration. This flexibility is achieved by checking for early exercise at each node during the backward induction process. If the intrinsic value of the option at a node exceeds the calculated value, early exercise is optimal.

Applicability to Various Options

The binomial model is versatile and can be applied to a wide range of options and underlying assets, including:

• Equity Options: Standard call and put options on stocks.
• Index Options: Options on stock indices.
• Currency Options: Options on foreign exchange rates.
• Exotic Options: Options with complex features, such as barriers or lookbacks.

Conclusion

The Binomial Option Pricing Model is a powerful tool in the arsenal of financial analysts and traders. Its ability to model a wide range of scenarios and accommodate various option types makes it indispensable in the field of options pricing. By understanding and applying the principles outlined in this section, you will be well-equipped to tackle complex option valuation problems.

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