In many countries, undergraduate education is fully subsided by the gov- ernment and entrance is guaranteed for everyone that enrolls. Suppose that the total cost of going to college for an individual is r > 0, which includes housing, food, books, etc. If s students graduate in a given year, the prob- ability of each one finding a job is p(s), where p(s) < 0, and the expected wage, conditional on finding a job, is w. Assume for simplicity that with probability 1 p(s) the payoff from going to college is zero, and that the payoff of not going to college at all is also zero. Note that the cost of going to college, r > 0, and the expected wage w > 0 are independent of s.

a) Characterize the equilibrium number of students that choose to go to college. Assume that after enrollment, the probability of graduating is 1. b) Is the equilibrium you found in (a) socially efficient? Why or why

not? Provide an intuitive explanation. c) Discuss a policy that could be used by the government to achieve

the efficient number of students.

8. You are considering the purchase of a 3-month 41.5-strike American call option on a nondividend-paying stock. You are given: (i) The Black-Scholes framework holds. (ii) The stock is currently selling for 40. (iii) The stock's volatility is 30%. (iv) The current call option delta is 0.5. Determine the current price of the option. (A) 20 - 20.45315.02/d2xex (B) 20 - 16.13815.02/d2xex (C) 20 - 40.45315.02/d2xex (D) 453.20d138.1615.02/2xex (E) 15.02/d453.402xex- 20.453

9. Consider the Black-Scholes framework. A market-maker, who delta-hedges, sells a three-month at-the-money European call option on a nondividend-paying stock. You are given: (i) The continuously compounded risk-free interest rate is 10%. (ii) The current stock price is 50. (iii) The current call option delta is 0.61791. (iv) There are 365 days in the year. If, after one day, the market-maker has zero profit or loss, determine the stock price move over the day. (A) 0.41 (B) 0.52 (C) 0.63 (D) 0.75 (E) 1.11

18. A market-maker sells 1,000 1-year European gap call options, and delta-hedges the position with shares. You are given: (i) Each gap call option is written on 1 share of a nondividend-paying stock. (ii) The current price of the stock is 100. (iii) The stock's volatility is 100%. (iv) Each gap call option has a strike price of 130. (v) Each gap call option has a payment trigger of 100. (vi) The risk-free interest rate is 0%. Under the Black-Scholes framework, determine the initial number of shares in the delta-hedge.

19. Consider a forward start option which, 1 year from today, will give its owner a 1-year European call option with a strike price equal to the stock price at that time. You are given: (i) The European call option is on a stock that pays no dividends. (ii) The stock's volatility is 30%. (iii) The forward price for delivery of 1 share of the stock 1 year from today is 100. (iv) The continuously compounded risk-free interest rate is 8%. Under the Black-Scholes framework, determine the price today of the forward start option. (A) 11.90 (B) 13.10 (C) 14.50 (D) 15.70 (E) 16.80