For all problems in this section, use the binomial tree model. Unless otherwise stated, assume no arbitrage.
A stock is currently priced at $55.00. The risk free rate is 4.6% per annum with continuous compounding. In 9 months, its price will be either $64.90 or $46.20.
(a) Using the binomial tree model, compute the price of an American call with strike price $52.78 expiring in 9 months.
(b) Now, compute the price of an American put option with strike price $52.78 expiring in 9 months.
Please answer asap
To compute the prices of the American call and put options using the binomial tree model, follow these steps:
Step 1: Calculate the up factor (u) and down factor (d) of the stock price.
The up factor (u) can be calculated using the formula:
u = e^(σ√(Δt))
where:
σ is the annual standard deviation of the stock returns
Δt is the length of each time step in the binomial tree (in this case, 9 months)
The down factor (d) can be calculated as the reciprocal of the up factor:
d = 1/u
Step 2: Calculate the risk-neutral probability (p).
The risk-neutral probability (p) can be calculated using the formula:
p = (e^(rΔt) - d) / (u - d)
where:
r is the risk-free interest rate
Δt is the length of each time step in the binomial tree (in this case, 9 months)
d and u are the calculated up and down factors from Step 1
Step 3: Construct the binomial tree.
Start by representing the current stock price (S₀) at the root of the tree. Then, using the up and down factors, calculate the stock prices at each node in the tree.
Step 4: Calculate the option values at each node of the tree.
For each node, calculate the option value (Call or Put) based on the maximum of the exercise value (stock price - strike price) or the expected value discounted back one time step.
Step 5: Backward induction.
Starting from the last node (9 months from now), calculate the value of the options at each node by discounting the expected value from the nodes above it.
Step 6: Determine the option price.
The option price will be the value of the option at the root of the tree (current time).
(a) Calculating the price of an American call option:
Using the provided values:
Stock price (S₀) = $55.00
Strike price (K) = $52.78
Risk-free interest rate (r) = 4.6% per annum (continuous compounding)
Time to expiration (Δt) = 9 months
Calculate u, d, and p using the formulas above. Then construct the binomial tree and calculate the option values at each node. Finally, use backward induction to determine the option price at the root of the tree.
(b) Calculating the price of an American put option:
Follow the same procedure as in part (a), but calculate the option values at each node based on the minimum of the exercise value (strike price - stock price) or the expected value discounted back one time step. Determine the option price at the root of the tree using backward induction.