Practice problems in this post reinforce the following blog post on pricing American options using binomial trees:

found in this companion blog.

_____________________________________________________________________________________

**Practice Problems**

*Practice Problem 1*

The inputs to a binomial tree are:

- The initial stock price is $40.
- The strike price is $45.
- The stock is non-dividend paying.
- The annual standard deviation of the stock return is 0.3.
- The annual risk-free interest rate is 5%.
- The binomial tree has 3 periods.
- The time to expiration 0.5 (6 months).

Price an American put option using this binomial tree.

*Practice Problem 2*

The inputs to a binomial tree are:

- The initial stock price is $50.
- The strike price is $55.
- The stock pays dividends at the annual continuous rate of 6%.
- The annual standard deviation of the stock return is 0.25.
- The annual risk-free interest rate is 4%.
- The binomial tree has 3 periods.
- The time to expiration 1.5 (18 months).

Price an American call option using this binomial tree.

*Practice Problem 3*

The inputs to a binomial tree are:

- The initial stock price is $40.
- The strike price is $45.
- The stock is non-dividend paying.
- The annual standard deviation of the stock return is 0.3.
- The annual risk-free interest rate is 5%.
- The binomial tree has 3 periods.
- The time to expiration 0.25 (3 months).

Price both the American put option using this binomial tree. The European put option using this binomial tree is priced in Example 3 in this previous post.

*Practice Problem 4*

The inputs to a binomial tree are:

- The initial stock price is $50.
- The strike price is $60.
- The stock pays dividends at the annual continuous rate of 5%.
- The annual standard deviation of the stock return is 0.3.
- The annual risk-free interest rate is 2%.
- The binomial tree has 3 periods.
- The time to expiration 2 years.

Price both the American put option using this binomial tree. The European put option using this binomial tree is priced in Example 4 in this previous post.

_____________________________________________________________________________________

_____________________________________________________________________________________

**Answers**

**Practice Problem 1 – pricing 6-month American put**

In the above tree, the option value in bold is a node where early exercise is optimal.

**Practice Problem 1 – pricing 6-month American put – Replicating portfolios**

_____________________________________________________________________________________

**Practice Problem 2 – pricing 1.5-year American call**

In the above tree, the option value in bold is a node where early exercise is optimal.

**Practice Problem 2 – pricing 1.5-year American call – Replicating portfolios**

_____________________________________________________________________________________

**Practice Problem 3 – pricing 3-month European put**

In the above tree, the option values in bold are nodes where early exercise is optimal.

**Practice Problem 3 – pricing 3-month European put – Replicating portfolios**

_____________________________________________________________________________________

**Practice Problem 4 – pricing 2-year American call**

In the above tree, the option value in bold is a node where early exercise is optimal.

**Practice Problem 4 – pricing 2-year American call – Replicating portfolios**

For more information on how to calculate the option prices for these practice problems, refer to The binomial option pricing model, part 5.

_____________________________________________________________________________________