Question

1. Stock Price Process
b. Assume that the change in the natural logarithm of the stock price follows according to a normal distribution that the annual mean return is 20% with a volatility of 80%.
If the stock is currently selling for $65, what do you think the stock price will range with a 95% probability over the next two months?
What about the continuously compounded rate of change in the stock price?

2. Binomial Trees
b. Consider a 6-month put with a strike price of $42 on a stock whose current price is $40.
Assume that there are two time steps, and in each time step the stock price either moves by 10% or moves down by 10%.
We also suppose that the risk-free rate of interest is 2%.
Compute the value of the put using the recombining binomial tree under the assumption that the option is European and under the assumption that the option is American.

3. Index Options
Suppose that you are managing a $5 million stock portfolio with a beta value of 1.5 but are concerned about the value in 2 months when your performance report is due to customers.
You are now trying to hedge by using the S&P 500 index option.
The option has a money multiplier of $100.
Currently, the index stands at 1500 but could go down to 1200.
The index dividend yield is 6% a year, and the risk-free rate is 0.8% a year.
If your target protection level of the portfolio is $4.5 million, explain your risk management strategy with respect to the option’s strike price and why your strategy will work.
Hint: Use the standard Capital Asset Pricing Model (CAPM) in your answer.

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