Consider an option-based portfolio insurance (OBPI) strategy designed to set a capital guarantee to S0= $100 investment in the S&P500 over T=10 year horizon. Assume a constant continuously compounded interest rate r=5% and a constant volatility of 30%

1. Calculate the number n of call options that need to be purchased to implement the strategy assuming that the option price is given by the Black-Scholes-Merton formula. (Note: assume that it is OK to purchase a non-integer number of options.)

2. Write the payoff P10 at date T=10 of the insurance strategy.

3. How much money will be left to the investor (1) in a scenario where the S&P500 goes from $100 to $50 over 10 years and (ii) in a scenario where the S&P500 goes from $100 to $150 over 10 years.