You are considering an investment in the 'market portfolio', proxied by the S&P 500; you begin your analysis by assuming that over the course of the next year, the market will make 2 (semi-annual) jumps. You believe that in each semi-annual period, the risk-neutral probability of an 'up' jump is 0.52; the risk-neutral probability of a 'down' jump is 0.48. The riskless interest rate over each semi-annual interval is 0.8%.

You believe that in a great (boom) year, the S&P 500 will be worth 3316.60; in an average year, the S&P 500 will be worth 3028.20, and in a recession, the market be worth 2595.60.

a) Given these beliefs, what is the no-arbitrage value of the S&P 500?

b) Consider a change to this world, where the (u,u) value increases by 2% (to 3382.93) and the (d,d) value decreases by 2% (to 2543.69). This increases the volatility of the market outcome. If your beliefs change in no other way, what will happen to the no-arbitrage value of the S&P 500 calculated in this way?

c) Do you believe that it is reasonable to assume that the volatility of the market may increase, and your beliefs change in no other way? Briefly explain your