Find the expected gain (or loss) for a holder of a European put option with strike price $65 to be exercised in 10 months if the stock price on the exercise date may turn out to be $58, $61, or $72 with probability , , and , respectively, given that the option is bought for $5.9, financed by a loan at 7% compounded continuously.
Find the expected gain (or loss) for a holder of a European put option with strike price $65 to be exercised in 10 months if the stock price on the exercise date may turn out to be $58, $61, or $72 with probability , , and , respectively, given that the option is bought for $5.9, financed by a loan at 7% compounded continuously.
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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![**Question 5: Expected Gain or Loss for a European Put Option**
**Problem Statement:**
Determine the expected gain (or loss) for a holder of a European put option with a strike price of $65. This option is to be exercised in 10 months. On the exercise date, the stock price could be either $58, $61, or $72. The probabilities associated with these stock prices are:
- $58 with probability \( \frac{1}{4} \)
- $61 with probability \( \frac{1}{4} \)
- $72 with probability \( \frac{1}{2} \)
The option was purchased for $5.9, which was financed through a loan at an annual interest rate of 7%, compounded continuously.
**Detailed Solution:**
1. **Calculating the Payoff for Each Outcome:**
The payoff of a European put option is given by \( \max(K - S_T, 0) \) where \( K \) is the strike price and \( S_T \) is the stock price at maturity.
- If \( S_T = \$58 \):
\[
\text{Payoff} = \max(65 - 58, 0) = \$7
\]
- If \( S_T = \$61 \):
\[
\text{Payoff} = \max(65 - 61, 0) = \$4
\]
- If \( S_T = \$72 \):
\[
\text{Payoff} = \max(65 - 72, 0) = \$0
\]
2. **Expected Payoff:**
The expected payoff \( E[\text{Payoff}] \) is calculated using the probabilities:
\[
E[\text{Payoff}] = \left(\frac{1}{4} \times 7\right) + \left(\frac{1}{4} \times 4\right) + \left(\frac{1}{2} \times 0\right) = \frac{7}{4} + \frac{4}{4} + 0 = \frac{11}{4} = 2.75
\]
3. **Financing Cost:**
The initial cost of the option is \$5.9, financed by a loan at 7% compounded](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9e925dc7-1400-40ad-98f3-d1989a82945f%2F5f4b28bf-69e6-490b-899e-ec0db60545ed%2F7trpn8n.png&w=3840&q=75)
Transcribed Image Text:**Question 5: Expected Gain or Loss for a European Put Option**
**Problem Statement:**
Determine the expected gain (or loss) for a holder of a European put option with a strike price of $65. This option is to be exercised in 10 months. On the exercise date, the stock price could be either $58, $61, or $72. The probabilities associated with these stock prices are:
- $58 with probability \( \frac{1}{4} \)
- $61 with probability \( \frac{1}{4} \)
- $72 with probability \( \frac{1}{2} \)
The option was purchased for $5.9, which was financed through a loan at an annual interest rate of 7%, compounded continuously.
**Detailed Solution:**
1. **Calculating the Payoff for Each Outcome:**
The payoff of a European put option is given by \( \max(K - S_T, 0) \) where \( K \) is the strike price and \( S_T \) is the stock price at maturity.
- If \( S_T = \$58 \):
\[
\text{Payoff} = \max(65 - 58, 0) = \$7
\]
- If \( S_T = \$61 \):
\[
\text{Payoff} = \max(65 - 61, 0) = \$4
\]
- If \( S_T = \$72 \):
\[
\text{Payoff} = \max(65 - 72, 0) = \$0
\]
2. **Expected Payoff:**
The expected payoff \( E[\text{Payoff}] \) is calculated using the probabilities:
\[
E[\text{Payoff}] = \left(\frac{1}{4} \times 7\right) + \left(\frac{1}{4} \times 4\right) + \left(\frac{1}{2} \times 0\right) = \frac{7}{4} + \frac{4}{4} + 0 = \frac{11}{4} = 2.75
\]
3. **Financing Cost:**
The initial cost of the option is \$5.9, financed by a loan at 7% compounded
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