solve....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 find the no-arbitrage prices of the European call and put options, we will use the one-period binomial option pricing model. Let's outline the steps and calculate the required prices.

### Step-by-Step Solution

1. **Define Variables and Inputs**
   - Initial stock price, $ S_0 = 28 $.
   - Up-move stock price, $ S_u = 32 $.
   - Down-move stock price, $ S_d = 26 $.
   - Strike price of options, $ K = 28 $.
   - Risk-free interest rate for three months, $ r = 3\% $ (or $ 0.03 $).

2. **Calculate the Risk-Neutral Probability $ p $**
   Using the risk-neutral probability formula:
   \[
   p = \frac{(1 + r)S_0 - S_d}{S_u - S_d}
   \]
   Plugging in the values:
   \[
   p = \frac{(1 + 0.03) \cdot 28 - 26}{32 - 26} = \frac{28.84 - 26}{6} = \frac{2.84}{6} \approx 0.4733
   \]
   So, the probability of an up-move $ p \approx 0.4733 $, and the probability of a down-move $ 1 - p \approx 0.5267 $.

3. **Calculate Option Payoffs at Expiration**
   - If the stock price goes up to $ S_u = 32 $:
     - Call option payoff $ C_u = \max(S_u - K, 0) = \max(32 - 28, 0) = 4 $.
     - Put option payoff $ P_u = \max(K - S_u, 0) = \max(28 - 32, 0) = 0 $.
   - If the stock price goes down to $ S_d = 26 $:
     - Call option payoff $ C_d = \max(S_d - K, 0) = \max(26 - 28, 0) = 0 $.
     - Put option payoff $ P_d = \max(K - S_d, 0) = \max(28 - 26, 0) = 2 $.

4. **Calculate the Expected Option Values**
   Using the risk-neutral probabilities:
   - Expected call option value:
     \[
     C_0 = \frac{1}{1 + r} \left( p \cdot C_u + (1 - p) \cdot C_d \right)
     \]
     Substituting values:
     \[
     C_0 = \frac{1}{1.03} \left( 0.4733 \cdot 4 + 0.5267 \cdot 0 \right)
     = \frac{1}{1.03} \times 1.8932 \approx 1.838
     \]

   - Expected put option value:
     \[
     P_0 = \frac{1}{1 + r} \left( p \cdot P_u + (1 - p) \cdot P_d \right)
     \]
     Substituting values:
     \[
     P_0 = \frac{1}{1.03} \left( 0.4733 \cdot 0 + 0.5267 \cdot 2 \right)
     = \frac{1}{1.03} \times 1.0534 \approx 1.023
     \]

### Final Answers
- **No-arbitrage price of the call option**: $ C_0 \approx 1.838 $
- **No-arbitrage price of the put option**: $ P_0 \approx 1.023 $

Would you like further details on any specific part, or do you have questions?

### Related Questions:
1. How does the risk-neutral probability affect the option prices?
2. What would happen to option prices if the interest rate changes?
3. How is the binomial option pricing model different from the Black-Scholes model?
4. What does the no-arbitrage condition imply in financial markets?
5. How would the option prices change if the stock prices had a different volatility?

### Tip:
In options pricing, understanding how to construct a replicating portfolio can be a powerful tool in determining theoretical values of derivatives like calls and puts.

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.

Explore the no-arbitrage prices of a European call and put option using the one-period binomial option pricing model. This problem involves stock price movements, risk-neutral probabilities, and expected option values. Learn how to apply the binomial model to determine the fair value of options in a simple financial market setup.

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