Problem A financial institution in Japan has the following three positions in currency options on the United States dollar, Type Position Delta of option Gamma of option Vega of option Type A 150 0.5 0.03 50.0 Type B -100 0.4 0.02 70.0 Type C 50 0.3 0.01 80.0 where the vega is the sensitivity of an option value to the volatility of the underlying asset. (a) What position in the United States dollar would make the portfolio delta neutral? (b) For a European call option on an asset paying a yield q (continuously compounded), the delta, gamma, and vega are given by 4 = e-97 N(d), N'(d)e-97 r= SONT V = S,VTN'(d)e-9T, where So is the current exchange rate (the value of one unit of the foreign currency measured in the domestic currency), T is the option maturity, o is the volatility of the exchange rate, di is given by with strike price K and the risk-free interest rate r, and OVT N'(x)== . Compute the delta, gamma, and vega of the 1-year and 4-year European ATM (at the money) call option on the United States dollar. Here assume that so = 100,0 = 10%(per annum), the U.S. dollar interest rate is 2% per annum (compounded continuously) and the Japanese yen interest rate is 0% per annum (compounded continuously). (c) The institution is now expecting the severely volatile markets in the near future. Therefore, it has decided to make the portfolio not only delta neutral, but also gamma and vega neutral. Then, What positions in the options considered in (b) and in the United States dollar would make the portfolio delta, gamma, and vega neutral? Problem A financial institution in Japan has the following three positions in currency options on the United States dollar, Type Position Delta of option Gamma of option Vega of option Type A 150 0.5 0.03 50.0 Type B -100 0.4 0.02 70.0 Type C 50 0.3 0.01 80.0 where the vega is the sensitivity of an option value to the volatility of the underlying asset. (a) What position in the United States dollar would make the portfolio delta neutral? (b) For a European call option on an asset paying a yield q (continuously compounded), the delta, gamma, and vega are given by 4 = e-97 N(d), N'(d)e-97 r= SONT V = S,VTN'(d)e-9T, where So is the current exchange rate (the value of one unit of the foreign currency measured in the domestic currency), T is the option maturity, o is the volatility of the exchange rate, di is given by with strike price K and the risk-free interest rate r, and OVT N'(x)== . Compute the delta, gamma, and vega of the 1-year and 4-year European ATM (at the money) call option on the United States dollar. Here assume that so = 100,0 = 10%(per annum), the U.S. dollar interest rate is 2% per annum (compounded continuously) and the Japanese yen interest rate is 0% per annum (compounded continuously). (c) The institution is now expecting the severely volatile markets in the near future. Therefore, it has decided to make the portfolio not only delta neutral, but also gamma and vega neutral. Then, What positions in the options considered in (b) and in the United States dollar would make the portfolio delta, gamma, and vega neutral
