Question 2. (A Breeden-Litzenberger Exercise) (35') Let's consider a hypothetical equity index. For simplicity, assume that the index price level, denoted as St, 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 S0=17. (2.1) What is the payoff diagram for a European call option on the equity index that expires in exactly three months (T=0.25) with a strike price of 20 ? What about European calls with the same T and strike prices 21 and 22 ? Please plot these three payoff diagrams in the same graph with MS Excel. Use a scatter plot and do not interpolate. (2.2) Now, you would like to examine the so-called Arrow securities that pay off in T. An Arrow security for ST=n refers to a security with the following payoff: Payoff=1,ifST=n;Payoff=0,ifST=n. European calls with various strikes are traded on the market. Can you replicate an Arrow security for ST=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 C(K,T).) (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 ST=1,2,,50 and present these fifty Arrow prices in a table. Also, plot these prices against ST in a bar graph. (Hint: What is the call price with K=0 ? What about K=50 ? You may need these call prices. It shouldn't 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 ST ! Let's give it a shot: a) What is the price of a security that pays off one dollar if ST=17, three dollars if ST=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