1. (20) One of the more complex options strategies is the so-called “iron condor” portfolio, using a combinationsof long and short positions, call and put options, and four different strike prices. In this question we willconsider what would motivate traders to construct such a portfolio and the structure of the payoffs that theyreceive from it. Suppose that stock in Hieroglyph Inc. is currently priced at St = 150 per share, with prices ofvarious options written on this stock given below.Strike Call Price Put Price120 35 5140 20 10160 10 20180 5 35(a) (10) Suppose that you are convinced that this stock is not as volatile as the market believes it to be,but are unsure of the direction of future price movements. Construct a suitable portfolio to match yourbeliefs, and plot the payoff diagram (you may assume a zero risk-free rate).(b) (10) Now suppose that, while you still believe that extreme price movements in either direction areunlikely, you wish to limit your downside risk just in case. Modify your portfolio to achieve this, and plotthe payoff diagram.2. (60) Consider applying the binomial method to price American options. Suppose that stock in the MAGAHat Company is priced at S0 = 100 and consider options that are at the money K = 100. The options maturein T = 1 year, which we divide into n = 3 periods, each of length
t = Tn = 13 . The risk-free rate is r = 0.08and MAGA stock has a volatility of  = 0.45.(a) (10) Calculate the magnitude of upticks u and downticks d, as well as the risk-neutral probability p.(b) (10) Draw the stock price tree and calculate the payoffs to call and put options at each final price.(c) (10) Using backward induction, calculate the price of European call and put options.(d) (10) Verify that put-call parity holds for European options.(e) (10) Find the price of American call and put options. Is it ever worthwhile to exercise early?(f) (10) Does put-call parity hold for American options?3. (20) Pick any stock and look up the current option prices. Pay particular attention to the implied volatility,the volatility which, if plugged into the Black-Scholes pricing formula along with the other known parameters,would yield the price of the option.(a) (10) Choose an expiry date, and plot the implied volatility of call options against the strike price, ignoringany implied volatilities of 0. What do you see, and what might explain this?(b) (10) Now select at least three more expiry dates, and repeat the same plot for each. What happens toimplied volatility as time to maturity increases?