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
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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 +
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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