You are given the following information: The current price to buy one share of XYZ stock is 500. The stock does not pay dividends. The risk-free interest rate, compounded continuously, is 6%. • A European call option on one share of XYZ stock with a strike price of K that expires in one year costs 66.59. • A European put option on one share of XYZ stock with a strike price of K that expires in one year costs 18.64. Using put-call parity, determine the strike price, K.
You are given the following information: The current price to buy one share of XYZ stock is 500. The stock does not pay dividends. The risk-free interest rate, compounded continuously, is 6%. • A European call option on one share of XYZ stock with a strike price of K that expires in one year costs 66.59. • A European put option on one share of XYZ stock with a strike price of K that expires in one year costs 18.64. Using put-call parity, determine the strike price, K.
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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![### Option Pricing and Put-Call Parity
#### Given Information:
1. **Stock Price:**
- The current price to buy one share of XYZ stock is $500. The stock does not pay dividends.
2. **Interest Rate:**
- The risk-free interest rate, compounded continuously, is 6%.
3. **European Call Option:**
- A European call option on one share of XYZ stock with a strike price of \( K \) that expires in one year costs $66.59.
4. **European Put Option:**
- A European put option on one share of XYZ stock with a strike price of \( K \) that expires in one year costs $18.64.
#### Task:
Using put-call parity, determine the strike price, \( K \).
#### Explanation:
Put-call parity is a principle that defines the relationship between the price of European put and call options of the same class, with the same strike price and expiration date. The formula for put-call parity is:
\[ C + K \cdot e^{-rT} = P + S_0 \]
Where:
- \( C \) is the price of the European call option
- \( K \) is the strike price of the options
- \( r \) is the risk-free interest rate
- \( T \) is the time to maturity (in years)
- \( P \) is the price of the European put option
- \( S_0 \) is the current stock price
Given the values:
- \( C = \$66.59 \)
- \( r = 0.06 \)
- \( T = 1 \)
- \( P = \$18.64 \)
- \( S_0 = \$500 \)
We substitute these values into the put-call parity equation to solve for \( K \):
\[ 66.59 + K \cdot e^{-0.06} = 18.64 + 500 \]
This equation must be solved for \( K \).
#### Detailed Solution:
1. Calculate \( e^{-0.06} \), which is the exponential of \(-0.06\):
\[ e^{-0.06} \approx 0.94176 \]
2. Substitute \( e^{-0.06} \) into the equation:
\[ 66.59 + K \cdot 0.94176 = 18.64 +](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9e925dc7-1400-40ad-98f3-d1989a82945f%2F37d10b12-4312-49a2-9a68-3b7ceb62586f%2Fmcd1m5c.png&w=3840&q=75)
Transcribed Image Text:### Option Pricing and Put-Call Parity
#### Given Information:
1. **Stock Price:**
- The current price to buy one share of XYZ stock is $500. The stock does not pay dividends.
2. **Interest Rate:**
- The risk-free interest rate, compounded continuously, is 6%.
3. **European Call Option:**
- A European call option on one share of XYZ stock with a strike price of \( K \) that expires in one year costs $66.59.
4. **European Put Option:**
- A European put option on one share of XYZ stock with a strike price of \( K \) that expires in one year costs $18.64.
#### Task:
Using put-call parity, determine the strike price, \( K \).
#### Explanation:
Put-call parity is a principle that defines the relationship between the price of European put and call options of the same class, with the same strike price and expiration date. The formula for put-call parity is:
\[ C + K \cdot e^{-rT} = P + S_0 \]
Where:
- \( C \) is the price of the European call option
- \( K \) is the strike price of the options
- \( r \) is the risk-free interest rate
- \( T \) is the time to maturity (in years)
- \( P \) is the price of the European put option
- \( S_0 \) is the current stock price
Given the values:
- \( C = \$66.59 \)
- \( r = 0.06 \)
- \( T = 1 \)
- \( P = \$18.64 \)
- \( S_0 = \$500 \)
We substitute these values into the put-call parity equation to solve for \( K \):
\[ 66.59 + K \cdot e^{-0.06} = 18.64 + 500 \]
This equation must be solved for \( K \).
#### Detailed Solution:
1. Calculate \( e^{-0.06} \), which is the exponential of \(-0.06\):
\[ e^{-0.06} \approx 0.94176 \]
2. Substitute \( e^{-0.06} \) into the equation:
\[ 66.59 + K \cdot 0.94176 = 18.64 +
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