To solve this problem, we will use the Binomial Option Pricing Model. Here are the steps:

Step 1: Calculate the up and down factors
The up (u) and down (d) factors are calculated using the formula:
u = e^(σ√Δt) and d = 1/u
where σ is the volatility, and Δt is the time step.
Given that σ = 20% and Δt = 18/3 = 6 months = 0.5 years, we have:
u = e^(0.2*√0.5) = 1.219
d = 1/1.219 = 0.820

Step 2: Calculate the risk-neutral probability
The risk-neutral probability (p) is calculated using the formula:
p = (e^(rΔt) - d) / (u - d)
where r is the risk-free rate.
Given that r = 5%, we have:
p = (e^(0.05*0.5) - 0.820) / (1.219 - 0.820) = 0.487

Step 3: Construct the binomial price tree
The binomial price tree is a graphical representation of possible intrinsic value of the option at different points in time. It starts with the current stock price and branches for each time step, using the up and down factors.

Step 4: Calculate the option value at each node
The option value at each node is calculated using the formula:
C = max[(pC_u + (1-p)C_d)e^(-rΔt), X - S]
where C_u and C_d are the option values at the up and down nodes, X is the strike price, and S is the stock price.

Step 5: Determine the early exercise gain
The early exercise gain is the difference between the option value with and without the possibility of early exercise.

Step 6: Identify when the gain is made
The gain is made at the point in time when the option value with early exercise exceeds the option value without early exercise.

Please note that this is a simplified explanation and the actual calculations may involve more complex considerations. Also, the values at each node of the binomial tree would need to be calculated to provide a numerical answer.

Question

To solve this problem, we will use the Binomial Option Pricing Model. Here are the steps:

Step 1: Calculate the up and down factors
The up (u) and down (d) factors are calculated using the formula:
u = e^(σ√Δt) and d = 1/u
where σ is the volatility, and Δt is the time step. 
Given that σ = 20% and Δt = 18/3 = 6 months = 0.5 years, we have:
u = e^(0.2*√0.5) = 1.219
d = 1/1.219 = 0.820

Step 2: Calculate the risk-neutral probability
The risk-neutral probability (p) is calculated using the formula:
p = (e^(rΔt) - d) / (u - d)
where r is the risk-free rate. 
Given that r = 5%, we have:
p = (e^(0.05*0.5) - 0.820) / (1.219 - 0.820) = 0.487

Step 3: Construct the binomial price tree
The binomial price tree is a graphical representation of possible intrinsic value of the option at different points in time. It starts with the current stock price and branches for each time step, using the up and down factors.

Step 4: Calculate the option value at each node
The option value at each node is calculated using the formula:
C = max[(pC_u + (1-p)C_d)e^(-rΔt), X - S]
where C_u and C_d are the option values at the up and down nodes, X is the strike price, and S is the stock price.

Step 5: Determine the early exercise gain
The early exercise gain is the difference between the option value with and without the possibility of early exercise.

Step 6: Identify when the gain is made
The gain is made at the point in time when the option value with early exercise exceeds the option value without early exercise.

Please note that this is a simplified explanation and the actual calculations may involve more complex considerations. Also, the values at each node of the binomial tree would need to be calculated to provide a numerical answer.

Knowee AI · Accepted Answer