To solve this problem, we will use the binomial model for option pricing.

i. Price of a 6-month European put option:

Step 1: Calculate the up and down factors (u and d) using the formulas:
u = e^(σ√Δt) = e^(0.40 * √0.25) = 1.1052
d = 1/u = 1/1.1052 = 0.9048

Step 2: Calculate the risk-neutral probability (p) using the formula:
p = (e^(rΔt) - d) / (u - d) = (e^(0.15 * 0.25) - 0.9048) / (1.1052 - 0.9048) = 0.6293

Step 3: Construct the binomial price tree for the stock. The stock price can either go up to $50 * 1.1052 = $55.26 or down to $50 * 0.9048 = $45.24 in the first period. In the second period, the stock price can go up to $55.26 * 1.1052 = $61.08 or down to $55.26 * 0.9048 = $50 in the up state, and it can go down to $45.24 * 0.9048 = $40.94 in the down state.

Step 4: Calculate the payoff of the put option at the end of each possible path. The payoff of a put option is max(K - S, 0), where K is the strike price and S is the stock price. The payoffs are $49.74, $55, and $64.06.

Step 5: Discount these payoffs back to today using the risk-neutral probabilities. The price of the put option is (p * $49.74 + (1 - p) * $55) * e^(-rΔt) = $51.87.

ii. Price of a 6-month American put option:

The process is the same as above, but with an additional step to check if early exercise is optimal at each node. In this case, since the exercise price is higher than the stock price at each node, early exercise is optimal. Therefore, the price of the American put option is the same as the European put option, $51.87.

Question

To solve this problem, we will use the binomial model for option pricing.

i. Price of a 6-month European put option:

Step 1: Calculate the up and down factors (u and d) using the formulas:
u = e^(σ√Δt) = e^(0.40 * √0.25) = 1.1052
d = 1/u = 1/1.1052 = 0.9048

Step 2: Calculate the risk-neutral probability (p) using the formula:
p = (e^(rΔt) - d) / (u - d) = (e^(0.15 * 0.25) - 0.9048) / (1.1052 - 0.9048) = 0.6293

Step 3: Construct the binomial price tree for the stock. The stock price can either go up to $50 * 1.1052 = $55.26 or down to $50 * 0.9048 = $45.24 in the first period. In the second period, the stock price can go up to $55.26 * 1.1052 = $61.08 or down to $55.26 * 0.9048 = $50 in the up state, and it can go down to $45.24 * 0.9048 = $40.94 in the down state.

Step 4: Calculate the payoff of the put option at the end of each possible path. The payoff of a put option is max(K - S, 0), where K is the strike price and S is the stock price. The payoffs are $49.74, $55, and $64.06.

Step 5: Discount these payoffs back to today using the risk-neutral probabilities. The price of the put option is (p * $49.74 + (1 - p) * $55) * e^(-rΔt) = $51.87.

ii. Price of a 6-month American put option:

The process is the same as above, but with an additional step to check if early exercise is optimal at each node. In this case, since the exercise price is higher than the stock price at each node, early exercise is optimal. Therefore, the price of the American put option is the same as the European put option, $51.87.

Knowee AI · Accepted Answer

To solve this problem, we will use the binomial model for option pricing.

i. Price of a 6-month European put option:

Step 1: Calculate the up and down factors (u and d) using the formulas:
u = e^(σ√Δt) = e^(0.40 * √0.25) = 1.1052
d = 1/u = 1/1.1052 = 0.9048

Step 2: Calculate the risk-neutral probability (p) using the formula:
p = (e^(rΔt) - d) / (u - d) = (e^(0.15 * 0.25) - 0.9048) / (1.1052 - 0.9048) = 0.6293

Step 3: Construct the binomial price tree for the stock. The stock price can either go up to $50 * 1.1052 = $55.26 or down to $50 * 0.9048 = $45.24 in the first period. In the second period, the stock price can go up to $55.26 * 1.1052 = $61.08 or down to $55.26 * 0.9048 = $50 in the up state, and it can go down to $45.24 * 0.9048 = $40.94 in the down state.

Step 4: Calculate the payoff of the put option at the end of each possible path. The payoff of a put option is max(K - S, 0), where K is the strike price and S is the stock price. The payoffs are $49.74, $55, and $64.06.

Step 5: Discount these payoffs back to today using the risk-neutral probabilities. The price of the put option is (p * $49.74 + (1 - p) * $55) * e^(-rΔt) = $51.87.

ii. Price of a 6-month American put option:

The process is the same as above, but with an additional step to check if early exercise is optimal at each node. In this case, since the exercise price is higher than the stock price at each node, early exercise is optimal. Therefore, the price of the American put option is the same as the European put option, $51.87.

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