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Consider a stock in two periods (two years). The stock price goes up by 30% or down by 10% in each period. Current stock price is $100. There is a European put option on the stock with exercise price $110 and time to maturity of two years. The interest rate in each period is 6%. In the template, Date 0 denotes today, Date 1 denotes the end of year 1 and Date 2 denotes the end of year 2. Use the two-period binomial tree model and discrete discounting to find the put option price on Date 0.

Consider a stock in two periods (two years). The stock price goes up by 30% or down by 10% in each period. Current stock price is $100. There is a European put option on the stock with exercise price $110 and time to maturity of two years. The interest rate in each period is 6%. In the template, Date 0 denotes today, Date 1 denotes the end of year 1 and Date 2 denotes the end of year 2. Use the two-period binomial tree model and discrete discounting to find the put option price on Date 0.

Essentials Of Investments
Essentials Of Investments
11th Edition
ISBN:9781260013924
Author:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Publisher:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Chapter1: Investments: Background And Issues
Section: Chapter Questions
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Consider a stock in two periods (two years). The stock price goes up by 30% or down by 10%
in each period. Current stock price is $100. There is a European put option on the stock with
exercise price $110 and time to maturity of two years. The interest rate in each period is 6%. In
the template, Date 0 denotes today, Date 1 denotes the end of year 1 and Date 2 denotes the end
of year 2. Use the two-period binomial tree model and discrete discounting to find the put option
price on Date 0.

# Three-Date Binomial Option Pricing ## Parameters: - **Up (U):** 30% - **Down (D):** -10% - **Initial stock price (S₀):** 100 - **Interest rate (r):** 6% - **Exercise price (K):** 110 - **Time to maturity (T in years):** 3 ## Structure: ### 1. Stock Price and Bond Price - **Stock Price Tree (Dates 0, 1, 2):** - Date 0 to Date 1: Stock price can move up or down. - Date 1 to Date 2: Each branch can again move up or down, creating multiple possible future prices. - **Bond Price Tree (Dates 0, 1, 2):** - Represents possible bond prices at different dates, calculated based on risk-free interest rate. ### 2. Put Option Price - **Put Option Price Tree:** - Follows the same branching as the stock price, indicating potential option values on each date. ### 3. Finding ??? for the Put - **Stock and Bond Price Trees at each stage:** - Stocks and bonds are recalculated to find the value of the put option. ### 4. Put Replicating Portfolio - **Solving equations for coefficients (m, n):** - Determine the amounts in bonds and stocks needed to replicate the option’s payoff using coefficients (m for stocks, n for bonds). - **Coefficient Calculation:** - Separate equation sets for each date to calculate m and n at various stages. ### 5. Sequential Stages: - **Date 0 to Date 1 (Finding ??? for the put):** - Calculate initial put option price given stock and bond values. - **Date 1 to Date 2:** - Update calculations reflecting adjustments in stock and bond scenarios. ### Diagram Explanation: - **Tree Diagrams:** - Vertical columns represent different dates. - Horizontal lines depict potential up/down movements of stock prices or changes in bond prices. - Colored boxes indicate respective prices for stocks, bonds, and put options at each node. This structured approach allows for a comprehensive understanding of how binomial option pricing evolves over multiple dates, incorporating factors such as stock movements, interest rates, and exercise price.
Transcribed Image Text:# Three-Date Binomial Option Pricing ## Parameters: - **Up (U):** 30% - **Down (D):** -10% - **Initial stock price (S₀):** 100 - **Interest rate (r):** 6% - **Exercise price (K):** 110 - **Time to maturity (T in years):** 3 ## Structure: ### 1. Stock Price and Bond Price - **Stock Price Tree (Dates 0, 1, 2):** - Date 0 to Date 1: Stock price can move up or down. - Date 1 to Date 2: Each branch can again move up or down, creating multiple possible future prices. - **Bond Price Tree (Dates 0, 1, 2):** - Represents possible bond prices at different dates, calculated based on risk-free interest rate. ### 2. Put Option Price - **Put Option Price Tree:** - Follows the same branching as the stock price, indicating potential option values on each date. ### 3. Finding ??? for the Put - **Stock and Bond Price Trees at each stage:** - Stocks and bonds are recalculated to find the value of the put option. ### 4. Put Replicating Portfolio - **Solving equations for coefficients (m, n):** - Determine the amounts in bonds and stocks needed to replicate the option’s payoff using coefficients (m for stocks, n for bonds). - **Coefficient Calculation:** - Separate equation sets for each date to calculate m and n at various stages. ### 5. Sequential Stages: - **Date 0 to Date 1 (Finding ??? for the put):** - Calculate initial put option price given stock and bond values. - **Date 1 to Date 2:** - Update calculations reflecting adjustments in stock and bond scenarios. ### Diagram Explanation: - **Tree Diagrams:** - Vertical columns represent different dates. - Horizontal lines depict potential up/down movements of stock prices or changes in bond prices. - Colored boxes indicate respective prices for stocks, bonds, and put options at each node. This structured approach allows for a comprehensive understanding of how binomial option pricing evolves over multiple dates, incorporating factors such as stock movements, interest rates, and exercise price.
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