solve step by step 2. (a). Consider a one-period financial market M = (Bt
, St). Assume that the
current stock price is $28, and after three months the stock price may either
rise to $32 or decline to $26 . Assume that the three-months interest rate for
deposits and loans is r = 3%. Find the no arbitrage prices of a call option and
a put option with the same strike price K = $28.


To solve this problem, we’ll use the binomial model for option pricing. Here’s the detailed step-by-step solution:

### Step 1: Define the Parameters
We are given:
- Current stock price, $ S_0 = 28 $
- Stock price after three months, which can either rise to $ S_u = 32 $ or fall to $ S_d = 26 $
- Strike price of the option, $ K = 28 $
- Interest rate for three months, $ r = 3\% $

### Step 2: Calculate the Risk-Neutral Probability
The risk-neutral probability is used to find the expected payoff of the option under a no-arbitrage condition. The formula for the risk-neutral probability $ p $ is:
\[
p = \frac{(1 + r) S_0 - S_d}{S_u - S_d}
\]

However, we need to adjust the interest rate for the three-month period, as the rate $ r = 3\% $ is usually quoted annually. 

#### Convert Annual Interest Rate to Quarterly Interest Rate
If $ r $ is the annual rate, the three-month interest rate $ R $ is:
\[
R = \frac{3\%}{4} = 0.75\% = 0.0075
\]

So, $ 1 + R = 1.0075 $.

Now we can calculate $ p $:
\[
p = \frac{(1 + R) S_0 - S_d}{S_u - S_d} = \frac{(1.0075) \times 28 - 26}{32 - 26}
\]

Calculating each part:
- $ (1.0075) \times 28 = 28.21 $
- $ S_u - S_d = 32 - 26 = 6 $
- $ p = \frac{28.21 - 26}{6} = \frac{2.21}{6} \approx 0.3683 $

Thus, the risk-neutral probability $ p \approx 0.3683 $.

### Step 3: Calculate the Option Payoffs at Expiration

#### For the Call Option
The payoff of a call option at expiration is $ \max(S - K, 0) $.

- If the stock price rises to $ S_u = 32 $: The call payoff is $ \max(32 - 28, 0) = 4 $.
- If the stock price falls to $ S_d = 26 $: The call payoff is $ \max(26 - 28, 0) = 0 $.

#### For the Put Option
The payoff of a put option at expiration is $ \max(K - S, 0) $.

- If the stock price rises to $ S_u = 32 $: The put payoff is $ \max(28 - 32, 0) = 0 $.
- If the stock price falls to $ S_d = 26 $: The put payoff is $ \max(28 - 26, 0) = 2 $.

### Step 4: Calculate the Expected Payoffs in the Risk-Neutral World

#### Call Option Expected Payoff
The expected payoff for the call option is:
\[
\text{Expected Payoff (Call)} = p \times 4 + (1 - p) \times 0 = 0.3683 \times 4 + (1 - 0.3683) \times 0
\]
\[
= 1.4732
\]

#### Put Option Expected Payoff
The expected payoff for the put option is:
\[
\text{Expected Payoff (Put)} = p \times 0 + (1 - p) \times 2 = 0 + (1 - 0.3683) \times 2
\]
\[
= 1.2634
\]

### Step 5: Discount the Expected Payoffs to Present Value

To find the no-arbitrage price of each option, we need to discount the expected payoff to the present value using the risk-free rate.

#### Call Option Price
\[
\text{Call Price} = \frac{\text{Expected Payoff (Call)}}{1 + R} = \frac{1.4732}{1.0075} \approx 1.462
\]

#### Put Option Price
\[
\text{Put Price} = \frac{\text{Expected Payoff (Put)}}{1 + R} = \frac{1.2634}{1.0075} \approx 1.254
\]

### Final Answer
The no-arbitrage prices of the options are:
- **Call Option Price**: Approximately $1.462
- **Put Option Price**: Approximately $1.254

Would you like any further clarification or have additional questions?

### Related Questions:
1. How does the risk-neutral probability affect option pricing in a binomial model?
2. What happens to the option prices if the interest rate increases?
3. How would the option prices change if the stock could rise to $35 instead of $32?
4. What would be the price of a European call option with a different strike price?
5. How is the binomial model applied in multi-period settings?

### Tip:
Always verify the time period of the interest rate in option pricing problems. For instance, if the rate is annual but the option period is shorter, convert the rate accordingly to match the option period.

Consider a one-period financial market M = (Bt, St). Assume that the current stock price is $28, and after three months the stock price may either rise to $32 or decline to $26. Assume that the three-months interest rate for deposits and loans is r = 3%. Find the no-arbitrage prices of a call option and a put option with the same strike price K = $28.

Learn how to calculate the no-arbitrage prices of a call and put option using the binomial model. This problem involves a stock with a price of $28, which can rise to $32 or fall to $26, with a 3% interest rate over three months. The no-arbitrage prices are calculated for both options with a strike price of $28.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation