MATHEMATICAL FINANCE II MATH 357- Winter 2019 Homework # 5

1) Consider the European option with payoff f(S(T)) = S(T)n,

where n {0,1,2,}.

a) Find the fair price of this European option at time t = 0. b) Find the Delta of this European option.

2) Consider a stock with current price S(0) = 58 Canadian dollars. Suppose that the interest rate of the risk-free asset is r = 5%. You buy a put option with strike pice K = 60 Canadian dollars and N(d1(0)) = 0.7. The price of the put option is P E (0) = 0.4127222 Canadian dollars.

a) How many shares of the stock would you buy or sell to have a delta-neutral position?

b) What would be your investment in the risk-free asset if you want the initial value of your portfolio to be zero? That is, you want V (0) = 0. Assume A(0) = 1.

3) Calculate the daily 95% VaR of a portfolio whose daily return is normally distributed with a mean of 1% and a standard deviation of 0.5%. The current value of the portfolio is 2 million Canadian dollars.

4) Suppose that you are investing in two stocks. You hold 60% of your portfolio in stock A with a mean monthly return rate of 1% and standard deviation 2%, and you hold 40% of your portfolio in stock B with a mean monthly return rate of 0.6% and standard deviation 1%. The correlation coefficient between the two stocks is 0.25. Assume normal distribution for the return rates of the two stocks. Calculate the monthly VaR at 99% level for your portfolio if the portfolio value today is 1 million Canadian dollars.