Question: “The actions by speculators affect options prices in options markets”. True or False? Discuss.

1. Stock ABC is currently standing at £1,508. Consider the corresponding stock options that expire in 90 days. The current Treasury Bills have a risk-free annual rate 2 percent. On the market, the calls and puts listed for ABC have an exercise price of £1,550.
   1. Assume there are no dividends paid for stock ABC.
      1. Explain the payoff diagrams for the put & call options for both European options and American options. (7%)
      2. Explain the lowest and highest possible European put prices on Stock ABC. (5%)
      3. Explain what the maximal possible loss are for European calls & puts. Assume that European calls and puts for ABC had the price of £59.54 and £92.91 respectively. Explain and compute the profits for both European calls and puts when Stock ABC stands at 1432 at expiry. (5%)

(v) Explain whether an increase in volatility would have impacts on profits/payoffs diagrams of puts on stock ABC. If you are a hedger, why would you continue to buy such put options with increased uncertainty? Explain. (10%)

1. 2. (i) Assume that European calls and puts for ABC have the price of £59.54 and £92.91 respectively, and there are no dividends for stock ABC. Can you detect arbitrage opportunities for the options? Explain the equations you use and how you can construct a risk-free arbitrage portfolio to earn money. (8%)

(ii) How would your answers change if Stock ABC is a stock index and having 1.5% continuous-time dividend yield per year? Assume the same options prices as in (b)(i). (5%)

1. 3. Assume that European calls and puts for ABC have the prices of £ 74.40 and £ 108.78 respectively, and there are no dividends for stock ABC. For a speculator, what would be possible strategy or (strategies) if the speculator reckons there would be greater fluctuations for Stock ABC during the life of options? Explain the strategy/strategies. Also explain the outcome of the strategies if uncertainty about Stock ABC turns out be lower than expected. (10%)

2. “The actions by speculators affect options prices in options markets”. True or False? Discuss. (Maximum of 500 words.) (20%)
3. Consider a 6-month binomial model in which the underlying asset is trading at £30. The underlying stock can go up 20 percent or down by 15 percent in the 6-month period. The expected required rate of returns for the underlying is valued as 15% per period. The annual risk-free rate is 5 percent and striking price is £32.
   1. Determine the price of a European put option. Explain your method and detail the procedure. (5%)
   2. Determine and explain how to obtain the price of a European put option expiring in two periods – two 6-month periods, by considering a 2-period binomial model. (5%)
   3. Continue with part (a). What would your answer change if the underlying asset pays a £5 dividend at the end of 2 periods? Explain the equations and procedure. (5%)
4. Angus Beef Corp. currently trades at £100 in the market. Calls and puts on Angus Beef are available with an exercise price of £104, The options expire in 240 days and the volatility is estimated as 0.4 (40% per year). Assume the continuous-time risk-free rate is 1% per year.
   1. Compute and explain the values of European call and put options using the BSM model. Assume there are no dividends for Angus Beef. Note that you would need to explain the components/numbers of the BSM formula in more details for the options of Angus Beef. (5%)
   2. As a put writer of Angus Beef, how could you construct a risk-free portfolio to protect your positions? Explain the practicality of such a continuous-time hedging against the fluctuations of Stock Angus Beef. (5%)
   3. Continue with (b). Now 40 days later, Angus Beef stock prices increase to £109, and the volatility is estimated to be 0.45. As a put writer, what would your actions be to protect your positions? Please explain the underlying reasons for your actions. (5%)