Suppose the European call and put options with strike price $20 and maturity date in 1 month cost $2.0 and $1.0, respectively. The underlying stock price is $18 and the risk-free continuously compounded interest rate is 8%. (a) Is there an arbitrage opportunity? (b)If yes, how would you implement arbitrage opportunity?
Suppose the European call and put options with strike price $20 and maturity date in 1 month cost $2.0 and $1.0, respectively. The underlying stock price is $18 and the risk-free continuously compounded interest rate is 8%. (a) Is there an arbitrage opportunity? (b)If yes, how would you implement arbitrage opportunity?
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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![### European Call and Put Options Analysis
**Scenario:**
Suppose the European call and put options with a strike price of $20 and a maturity date in 1 month cost $2.0 and $1.0, respectively. The underlying stock price is $18, and the risk-free continuously compounded interest rate is 8%.
1. **Identification of an Arbitrage Opportunity:**
- **Question (a):** Is there an arbitrage opportunity?
- **Answer:** To determine if there is an arbitrage opportunity, we need to investigate if put-call parity holds.
The put-call parity for European options states:
\[
C - P = S - Ke^{-rT}
\]
Where:
- \( C \) = Call option price ($2.0)
- \( P \) = Put option price ($1.0)
- \( S \) = Current stock price ($18)
- \( K \) = Strike price ($20)
- \( r \) = Risk-free interest rate (8% or 0.08)
- \( T \) = Time to maturity (1 month or \( \frac{1}{12} \) year)
Plugging these values into the formula:
\[
2.0 - 1.0 = 18 - 20e^{-0.08 \times \frac{1}{12}}
\]
\[
1.0 = 18 - 20e^{-0.00667}
\]
Using \( e^{-0.00667} \approx 0.99334 \):
\[
1.0 = 18 - 20 \times 0.99334
\]
\[
1.0 = 18 - 19.8668
\]
\[
1.0 \ne -1.8668
\]
Since the put-call parity does not hold, there is an arbitrage opportunity.
2. **Implementation of Arbitrage:**
- **Question (b):** If yes, how would you implement the arbitrage opportunity?
- **Answer:** To exploit the arbitrage opportunity, follow these steps:
- **Step 1:** Buy the put option for $1.0.
- **Step 2:** Short sell the stock for $18.
-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9e925dc7-1400-40ad-98f3-d1989a82945f%2F60db11ed-462e-4a0a-9ec9-29e2e5d751e2%2F3g7gsyj.png&w=3840&q=75)
Transcribed Image Text:### European Call and Put Options Analysis
**Scenario:**
Suppose the European call and put options with a strike price of $20 and a maturity date in 1 month cost $2.0 and $1.0, respectively. The underlying stock price is $18, and the risk-free continuously compounded interest rate is 8%.
1. **Identification of an Arbitrage Opportunity:**
- **Question (a):** Is there an arbitrage opportunity?
- **Answer:** To determine if there is an arbitrage opportunity, we need to investigate if put-call parity holds.
The put-call parity for European options states:
\[
C - P = S - Ke^{-rT}
\]
Where:
- \( C \) = Call option price ($2.0)
- \( P \) = Put option price ($1.0)
- \( S \) = Current stock price ($18)
- \( K \) = Strike price ($20)
- \( r \) = Risk-free interest rate (8% or 0.08)
- \( T \) = Time to maturity (1 month or \( \frac{1}{12} \) year)
Plugging these values into the formula:
\[
2.0 - 1.0 = 18 - 20e^{-0.08 \times \frac{1}{12}}
\]
\[
1.0 = 18 - 20e^{-0.00667}
\]
Using \( e^{-0.00667} \approx 0.99334 \):
\[
1.0 = 18 - 20 \times 0.99334
\]
\[
1.0 = 18 - 19.8668
\]
\[
1.0 \ne -1.8668
\]
Since the put-call parity does not hold, there is an arbitrage opportunity.
2. **Implementation of Arbitrage:**
- **Question (b):** If yes, how would you implement the arbitrage opportunity?
- **Answer:** To exploit the arbitrage opportunity, follow these steps:
- **Step 1:** Buy the put option for $1.0.
- **Step 2:** Short sell the stock for $18.
-
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