Final answer:
To calculate the value today of a one-year European put option with a strike price of $85, we need to determine the expected value of the option at the end of the year and then discount it back to the present using the continuous compounding interest rate of 5% per year. The value today of the put option is approximately $2.86.
Step-by-step explanation:
To calculate the value today of a one-year European put option with a strike price of $85, we need to determine the expected value of the option at the end of the year and then discount it back to the present using the continuous compounding interest rate of 5% per year.
In the given binomial tree model, the stock can either increase to $92 with a probability of 0.8 or fall to $70 with a probability of 0.2. To find the value of the put option at the end of the year, we compare the strike price of $85 to the potential stock prices.
If the stock increases to $92, the put option is not exercised and its value is $0. If the stock falls to $70, the put option is exercised and its value is $85 - $70 = $15.
We can calculate the expected value of the put option as follows:
Expected value = (Value if stock increases) * (Probability of increase) + (Value if stock falls) * (Probability of fall)\\Expected value = ($0 * 0.8) + ($15 * 0.2) = $3\\
The value today of the put option is then obtained by discounting the expected value back to the present:
Value today = Expected value / (1 + interest rate)^number of periods\\Value today = $3 / (1 + 0.05)^1 = $2.86\\
Therefore, the value today of a one-year European put option with a strike price of $85 is approximately $2.86.