A trader decides to protect her portfolio with a put option. The portfolio is wo

Question

A trader decides to protect her portfolio with a put option. The portfolio is worth $150 million and the required put option has a strike price of $145 million with a maturity of 24 weeks. The volatility of the portfolio is 15% and the dividend yield on the portfolio is 3% per annum. The risk-free rate is 4%. Because the option is not available on exchanges, the trader decides to create an option by maintaining a position in the underlying portfolio with the required delta. What percentage of the original portfolio should be sold and invested at the risk-free rate:

a. Initially at time zero?

b. After one week, when value is $145 million?

c. After two weeks when value is $148 million?

Explanation / Answer

Current Portfolio $ 150 million Strike Price $ 145 million Week to maturity 24 weeks standard deviation 15% Dividend yeild 3% r 4% Delta ? a) Put Delta e^-qt(N(d1) - 1) e^-(0.03*24/52)*(N(d1)-1) N(d1) In(S0e^-qt/X) + t(r+ sd^2/2)/sd*t^0.50 In(150e^-0.03*0.461538/145) + 24/52(0.04+0.15^2/2)/0.15*(24/52)^0.50 (0.020055+0.02365)/0.15*0.679366 0.043705/0.10190 0.4289 Put Delta e^-qt(N(0.4289)-1) e^-0.03*24/52(0.666002-1) 0.9862*-0.334 -0.3293908 32.90% of original portfolio should be sold and invested as risk free rate b) After one week, when value is $145 million Put Delta e^-qt(N(d1) - 1) N(d1) In(S0e^-qt/X) + t(r+ sd^2/2)/sd*t^0.50 In(145e^-0.03*0.44231/145) + 23/52(0.04+0.15^2/2)/0.15*(23/52)^0.50 (-0.013269+0.02266)/0.15*0.665062 0.009391/0.099759 0.094137 Put Delta e^-qt(N(0.094137)-1) e^-0.03*23/52(0.5375-1) 0.9868*-0.4625 -0.4564 After one week if value is $145 than sell additional portfolio of 12.7%(0.4564-0.329) c) After two weeks when value is $148 million? Put Delta e^-qt(N(d1) - 1) N(d1) In(S0e^-qt/X) + t(r+ sd^2/2)/sd*t^0.50 In(148e^-0.03*0.423076/145) + 22/52(0.04+0.15^2/2)/0.15*(22/52)^0.50 (0.007786+0.021683)/0.15*0.65044 0.02947/0.0975665 0.302 Put Delta e^-qt(N(0.302)-1) e^-0.03*22/52(0.618674-1) 0.98738*-0.38133 -0.3765 After two week if value is $ 148 million than we will have to buy back 7.99% (-0.3765-(-0.4564))

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