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: = \frac{1}{1.03} \times 1.8932 \approx 1.838$$
   • Expected put option value: $P_0 = \frac{1}{1 + r} \left( p \cdot P_u + (1 - p) \cdot P_d \right)$ Substituting values: = \frac{1}{1.03} \times 1.0534 \approx 1.023$$

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.