Question 4 (20 marks) A stock is priced at $20 per share. The volatility is 30%...

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Question 4 (20 marks) A stock is priced at $20 per share. The volatility is 30% per annum and the risk-free interest rate is 7% per annum. Consider the following information on three-month at-the-money options. Call Put Price $1.37 $1.02 Delta .576 -.424 Gamma .1306 .1306 Theta -.0084 -.0046 Vega .0392 .0392 Rho .0254 -.0237 The theta is measured on a per day basis. The Greeks that measure the change in option price per change in interest rate or volatility are expressed in values per 1 percentage point change. a) Suppose a trader sold 50 calls. Assuming the stock price remained unchanged for one week, estimate the dollar profit on the position. (2 marks) b) Suppose a trader buys 50 put options. Estimate the change in the value of the position if the volatility increased from 30% to 34%. (2 marks) c) Suppose a trader owns 50 calls. Give your best estimation for the value of the trader's position one week later assuming the stock price increased by $3 per share and all else remain the same. Explain your answer. (4 marks) d) Suppose the stock price increase by $2, what is the percentage change in the call option price? Explain your answer. (2 marks) e) Assume that the beta of the stock is 1.30 and the market return is 12% per year. What is the expected rate of return for investing 50 units of this call option? Explain your answers and state your assumptions. (3 marks) f) A trader wants to establish a position which will make money if the volatility increases. Using the call and put given above, establish a position which is immune to stock price changes and have a 0.0404 increase in his position value when volatility becomes 31%. What is the gamma of this position? (4 marks) g) Compare this position in f) to a long straddle established using the call and put given above, and comment on the differences in terms of its sensitivity to changes in stock price and volatility. (3 marks) Question 4 (20 marks) A stock is priced at $20 per share. The volatility is 30% per annum and the risk-free interest rate is 7% per annum. Consider the following information on three-month at-the-money options. Call Put Price $1.37 $1.02 Delta .576 -.424 Gamma .1306 .1306 Theta -.0084 -.0046 Vega .0392 .0392 Rho .0254 -.0237 The theta is measured on a per day basis. The Greeks that measure the change in option price per change in interest rate or volatility are expressed in values per 1 percentage point change. a) Suppose a trader sold 50 calls. Assuming the stock price remained unchanged for one week, estimate the dollar profit on the position. (2 marks) b) Suppose a trader buys 50 put options. Estimate the change in the value of the position if the volatility increased from 30% to 34%. (2 marks) c) Suppose a trader owns 50 calls. Give your best estimation for the value of the trader's position one week later assuming the stock price increased by $3 per share and all else remain the same. Explain your answer. (4 marks) d) Suppose the stock price increase by $2, what is the percentage change in the call option price? Explain your answer. (2 marks) e) Assume that the beta of the stock is 1.30 and the market return is 12% per year. What is the expected rate of return for investing 50 units of this call option? Explain your answers and state your assumptions. (3 marks) f) A trader wants to establish a position which will make money if the volatility increases. Using the call and put given above, establish a position which is immune to stock price changes and have a 0.0404 increase in his position value when volatility becomes 31%. What is the gamma of this position? (4 marks) g) Compare this position in f) to a long straddle established using the call and put given above, and comment on the differences in terms of its sensitivity to changes in stock price and volatility. (3 marks)