1. Accelerated Return Notes provide payoffs at maturity that depend on the value of an underlying stock and the notional N. Assume the stock pays no dividends.
(a) Draw the payoff diagram for the Accelerated Return Note and explain the payoff profile in your own words. Use a notional of N = 100 and an initial value of the underlying of $50. The maturity of the note is in one year. [ 8 marks ]
(b) If the only options traded on the underlying are European calls, how would you replicate the payoff? [ 6 marks ]
(c) How would you replicate the note if only European puts were available? [ 4 marks ]
(d) Now assume that the underlying stock has a volatility of 35% and the (continuously compounded) risk-free rate is given by r = 1%. What is the price of the note in the Black-Scholes-Merton model? [ 4 marks ]
(e) Note: If you need values for any other parameters to answer the questions below, make reasonable assumptions and justify these. Simulate the payoff of the Accelerated Return Note in the Black-Scholes-Merton model. Use at least 10,000 simulations of the stock price. What is the average return of investing in the note, as well as the standard deviation of the returns. [ 15 marks ]
(f) Using your simulation output, is it more risky to invest into the note than to invest into the stock itself? Justify your answer using your simulation output. [ 4 marks ]
(g) Using your simulation output, what is the probability that the return of the note is 20%. [ 4 marks ]
2. Discuss the empirical evidence that suggests that the Black-Scholes-Merton model is rejected for S&P 500 index options. Provide at least three different stylized empirical facts from the option market to support your answer. Despite being rejected by the data, why is the model still being used extensively in the finance industry?
Note: You can refer to discussions from class and lecture notes, but you need to provide references for any sources you use. You do not have to use your own data. [ 30 marks ]
3. Build a three-step CRR stock price tree using the following information:
• maturity: 3 months, i.e. each step is one month
• annual volatility of the stock: 0.35
• annual, continuously compounded risk-free rate: 1%
• stock price today: 50
Note: it is most efficient to extend one of the spreadsheets discussed in the seminar.
(a) What is the risk-neutral probability in the tree? Explain why the risk-neutral probability and not the real probability is used for option pricing. [ 5 marks ]
(b) Calculate the price of an American put option with strike K = 50. [ 10 marks ]
(c) You realise that during the last step in the tree the firm announces earnings. Assume that the stock is twice as volatile during this last step. What is the price of the American put option from (a) now? [ 10 marks ]
(d) The market is quoting a price of $4.50 for the put. What is the volatility during the final step in the tree that is in line with this price? [ 5 marks ]
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