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. 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%. 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. d) Suppose the stock price increase by $2, what is the percentage change in the call option price? Explain your answer. 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. 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? 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.