Sure, let's break down each question:

1. The possible stock values in one year’s time, ST, can be calculated as follows: If the stock goes up by 25%, the value will be $100.1 * 1.25 = $125.125. If the stock goes down by 20%, the value will be $100.1 * 0.8 = $80.08.

2. The payoffs of a European call option with a strike price, X, of $100 and time-to-expiration of 1 year, T = 1, are as follows: If ST = $125.125, the payoff is $125.125 - $100 = $25.125. If ST = $80.08, the payoff is $0, because the strike price is higher than the stock price.

3. The hedge ratio or delta, ∆, is the ratio of the change in the price of the call option to the change in the price of the underlying stock. It can be calculated as (∆ST / ∆S0). In this case, ∆ = ($25.125 - $0) / ($125.125 - $80.08) = 0.6667.

4. The portfolio value in one year’s time, PT, is the value of the stock minus the value of the call option. Since the portfolio is designed to be risk-free, PT = ST - ∆ * S0 = $125.125 - 0.6667 * $100.1 = $58.41 if the stock goes up, and $80.08 - 0.6667 * $100.1 = $13.36 if the stock goes down.

5. The arbitrage-free value of the portfolio today, P0, is the present value of PT, discounted at the risk-free rate. P0 = PT / e^(rT) = $58.41 / e^(0.055*1) = $55.43 if the stock goes up, and $13.36 / e^(0.055*1) = $12.68 if the stock goes down.

6. The premium of the call option today, c0, is the present value of the expected payoff, discounted at the risk-free rate. c0 = E(cT) / e^(rT) = ($25.125 * p + $0 * (1-p)) / e^(0.055*1), where p is the risk-neutral probability that the stock price increases.

7. The risk-neutral probability that the stock price increases, p, is given by (e^(rT) - d) / (u - d), where u is the up factor (1.25) and d is the down factor (0.8). So, p = (e^(0.055*1) - 0.8) / (1.25 - 0.8) = 0.6523.

8. The expected value of the stock in one year’s time, E(ST), under the risk-neutral probabilities, is given by E(ST) = u * S0 * p + d * S0 * (1-p) = 1.25 * $100.1 * 0.6523 + 0.8 * $100.1 * (1 - 0.6523) = $105.06.

9. The continuous rate at which the stock price would have to grow to end up at the expected value is given by the formula r = ln(E(ST) / S0) / T = ln($105.06 / $100.1) / 1 = 0.0487, or 4.87%.

10. The expected value of the call option in one year’s time, E(cT), under the risk-neutral probabilities, is given by E(cT) = max(u * S0 - X, 0) * p + max(d * S0 - X, 0) * (1-p) = max(1.25 * $100.1 - $100, 0) * 0.6523 + max(0.8 * $100.1 - $100, 0) * (1 - 0.6523) = $16.28.

11. The continuous rate at which the call price would have to grow to end up at the expected value is given by the formula r = ln(E(cT) / c0) / T = ln($16.28 / c0) / 1. To calculate this, we first need to find c0 from question 6.

Question

Sure, let's break down each question:

1. The possible stock values in one year’s time, ST, can be calculated as follows: If the stock goes up by 25%, the value will be $100.1 * 1.25 = $125.125. If the stock goes down by 20%, the value will be $100.1 * 0.8 = $80.08.

2. The payoffs of a European call option with a strike price, X, of $100 and time-to-expiration of 1 year, T = 1, are as follows: If ST = $125.125, the payoff is $125.125 - $100 = $25.125. If ST = $80.08, the payoff is $0, because the strike price is higher than the stock price.

3. The hedge ratio or delta, ∆, is the ratio of the change in the price of the call option to the change in the price of the underlying stock. It can be calculated as (∆ST / ∆S0). In this case, ∆ = ($25.125 - $0) / ($125.125 - $80.08) = 0.6667.

4. The portfolio value in one year’s time, PT, is the value of the stock minus the value of the call option. Since the portfolio is designed to be risk-free, PT = ST - ∆ * S0 = $125.125 - 0.6667 * $100.1 = $58.41 if the stock goes up, and $80.08 - 0.6667 * $100.1 = $13.36 if the stock goes down.

5. The arbitrage-free value of the portfolio today, P0, is the present value of PT, discounted at the risk-free rate. P0 = PT / e^(rT) = $58.41 / e^(0.055*1) = $55.43 if the stock goes up, and $13.36 / e^(0.055*1) = $12.68 if the stock goes down.

6. The premium of the call option today, c0, is the present value of the expected payoff, discounted at the risk-free rate. c0 = E(cT) / e^(rT) = ($25.125 * p + $0 * (1-p)) / e^(0.055*1), where p is the risk-neutral probability that the stock price increases.

7. The risk-neutral probability that the stock price increases, p, is given by (e^(rT) - d) / (u - d), where u is the up factor (1.25) and d is the down factor (0.8). So, p = (e^(0.055*1) - 0.8) / (1.25 - 0.8) = 0.6523.

8. The expected value of the stock in one year’s time, E(ST), under the risk-neutral probabilities, is given by E(ST) = u * S0 * p + d * S0 * (1-p) = 1.25 * $100.1 * 0.6523 + 0.8 * $100.1 * (1 - 0.6523) = $105.06.

9. The continuous rate at which the stock price would have to grow to end up at the expected value is given by the formula r = ln(E(ST) / S0) / T = ln($105.06 / $100.1) / 1 = 0.0487, or 4.87%.

10. The expected value of the call option in one year’s time, E(cT), under the risk-neutral probabilities, is given by E(cT) = max(u * S0 - X, 0) * p + max(d * S0 - X, 0) * (1-p) = max(1.25 * $100.1 - $100, 0) * 0.6523 + max(0.8 * $100.1 - $100, 0) * (1 - 0.6523) = $16.28.

11. The continuous rate at which the call price would have to grow to end up at the expected value is given by the formula r = ln(E(cT) / c0) / T = ln($16.28 / c0) / 1. To calculate this, we first need to find c0 from question 6.

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