You own 10,000 shares of AIG. The current market price is $55. You are interested in knowing the ``worst case value'' W of this portfolio in three years, where we define the worst case value W by Pr[S₃W]=0.005 (i.e., 0.5%). Assume that the expected return and volatility of the stock price are 6% and 30%, respectively.

You bought a binary option with a $1000 pay off if the price of AIG in 3 is greater than $60 in six months. The risk-free rate is 3%. What is the value of the option? What is the real-world probability of a payoff?