Find the no-arbitrage options price for a European call option using a simplified stock model. Stock model: The current spot price for a non-dividend-paying stock is S0 = 24. In this simplified model, we assume that at the end of one month there are only two possible prices S1 = 28 or S1 = 20. We also assume that the probability of the stock price equaling 28 is P(S1 = 28) = .75 and P(S1 = 20) = .25. European call option: We consider a European call option with strike price K = 23 and expiration in one month. Risk-free rate: Assume that the one-month risk-free rate is 1%. This is equivalent to an annualized 12% rate compounded monthly. If you borrow or loan $D, then after one month it is worth $D(1.01).

Questions: Show all of your work and explain each step.

1. What are the possible payoffs of the European call option at expiration? Remember that there are only two possible stock prices: 28 and 20.

2. Consider a portfolio with A shares of stock and B dollars invested at the risk free rate. Both A and B are allowed to be fractions and may be positive or negative. Negative values correspond to shorting a stock or borrowing money. Write down formulas for the possible values of this portfolio in one month. Your answer will include the variables A and B as well as the numbers 20, 28, and 1.01.

3. Find the values of A and B that guarantee that the values of #1 and #2 are always equal to each other. Notice that the .75 number is irrelevant.

4. What is the value today of a portfolio with A shares of stock and B dollars, using the values of A and B that you found in #3?

5. Explain using a no-arbitrage argument what is the initial value of the European call option using this model.

6. What is an explanation for why the .75 number does not affect this analysis?