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
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
Problem 1PS
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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.
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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