Lets consider a hypothetical equity index. For simplicity, assume that the index price level, denoted as t , can only take positive integer values. That is, it is only possible for the index price to take values from {1, 2, 3, ..., n, ..., 50}. The index level cannot be zero or negative. Historically, the index price has never exceeded 50; therefore, investors do not believe it is possible for the index to rise beyond 50. The current spot price 0 = 17

2.2) Now, you would like to examine the so-called Arrow securities that pay off in T. An Arrow security for % = refers to a security with the following payoff: Payoff = 1, if % = ; Payoff = 0, if % . European calls with various strikes are traded on the market. Can you replicate an Arrow security for % = 21 using only European calls? What is the price of such a security, expressed as a formula of involved call prices? (Hint: You only need options mentioned in (2.1). The price of a call with strike K and time-to-expiration T can be denoted as (, ).

(2.3) The current prices of European calls with T maturity are shown in [F2023 Midterm.xlsx]. Please calculate the prices of Arrow securities for % = 1, 2, ..., 50 and present these fifty Arrow prices in a table. Also, plot these prices against % in a bar graph. (Hint: What is the call price with K=0? What about K=50? You may need these call prices. It shouldnt take more than a few seconds to figure out these two call prices.)

(2.4) Using what you just learned from (2.3), you can now price any derivative with payoffs contingent on % ! Lets give it a shot: a) What is the price of a security that pays off one dollar if % = 17, three dollars if % = 40, and zero otherwise? (Hint: Replicate it with Arrow securities.) b) What is the price of a put option with strike 30? c) What is the price of a risk-free zero-coupon bond that pays one dollar at T? What is the continuously compounded risk-free interest rate in APR in this economy? F2023 Midterm.xlsx