Consider an instrument that will pay off S dollars in 2 years, where S is the value of the Nikkei index. The index is currently 20,000. The yen/dollar exchange rate is 100 (yen per dollar). The correlation between the exchange rate and the index is 0.3 and the dividend yield on the index is 1% per annum. The volatility of the Nikkei index is 20% and the volatility of the yen/dollar exchange rate is 12%. The interest rates (assumed constant) in the US and Japan are 4% and 2%, respectively.
(a) What is the value of the instrument?
(b) Suppose that the exchange rate at some point during the life of the instrument is Q and the level of the index is S. Show that a US investor can create a portfolio that changes in value by approximately ?S dollar when the index changes in value by ?S yen by investing S dollars in the Nikkei and shorting SQ yen.
(c) Confirm that this is correct by supposing that the index changes from 20,000 to 20,050 and the exchange rate changes from 100 to 99.7.
(d) How would you delta hedge the instrument under consideration?