Current stock price S is $22. Time to maturity T is six months. Continuously compounded, risk-free interest rate r is 5 percent per annum. European options prices are given in the following table:Strike Price Call Price Put Price K1=$17.50 $5.00 $0.05 K2=$20.00 $3.00 $0.75 K3=$22.50 $1.75 $1.75 K4=$25.00 $0.75 $3.50(a) What is the aim of a long (or bottom) straddle strategy? Create a long straddle by buying a call and put with strike price K3=$22.50 [10 marks] (b) What is the aim of a short (or top) strangle strategy? Create a short strangle by writing a call with strike price K3=$22.50 and a put with strike price K2=$20. [10 marks] Question 2 [10 marks] (a) Why is the binomial model a useful technique for approximating options prices from the Black–Scholes model? [5 marks] (b) Describe some applications and uses of this model. [5 marks] Question 3 [20 marks]Consider the binomial model for an American call and put on a stock whose price is $60. The exercise price for both the put and the call is $45. The standard deviation of the stock returns is 30 percent per annum, and the risk-free rate is 5 percent per annum. The options expire in 90 days. The stock will pay a dividend equal to 3 percent of its value in 50 days. (a) Draw the three-period stock tree and the corresponding trees for the call and the put. [7.5 marks] (b) Compute the price of these options using the three-period trees. [7.5 marks] (c) Explain when, if ever, each option should be exercised. [5 marks]Question 4 [30 marks] Consider a stock with a price of $120 and a standard deviation of 20 percent. The stock will pay a dividend of $5 in 40 days and a second dividend of $5 and 130 days. The current risk-free rate is 5 percent per annum. An American call on this stock has an exercise price of $150 and expires in 100 days. Price the American call option using the Black-Scholes Model. Show all calculations. Question 5 [20 marks] ABC is currently trading at $78 per share. Your previous calculation of the historical volatility for ABC indicated an annual standard deviation of return of 27 percent, but examining the implied volatility of several ABC options reveals an increase in annual volatility to 32 percent. There are two traded options series that expire in 245 days as show in the following table: X = 75 X = 80 Call Put Call Put DELTA 0.6674 -0.3326 0.574 -0.426 GAMMA 0.0176 0.0176 0.019 0.019The options have $75 and $80 strike prices respectively. The current 245-day risk-free interest rate is 4.75 percent per annum, and you hold 2,000 shares of ABC. Construct a portfolio that is DELTA – and GAMMA- neutral using the call options written on ABC. Show all calculations.
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