An investment company has purchased $100 million of 10 percent annual coupon, 6-year Eurobonds.The bonds have a duration of 4.79 years at the current market yields of 10 percent.The company wishes to hedge these bonds with Treasury-bond options that have a delta of 0.7.The duration of the underlying asset is 8.82, and the market value of the underlying asset is $98,000 per $100,000 face value.Finally, the volatility of the interest rates on the underlying bond of the options and the Eurobond is 0.84. Using the above information and your answer to the previous question, will the investment company gain or lose on the option position if interest rates decrease 1 percent to 9 percent?

Question

Quizplus · Accepted Answer

Using the duration of the Eurobond and the change in interest rates, we calculate the new market value of the bond to be $108,019,665 if interest rates decrease to 9%. This gives us a gain of $8,019,665 on the bond.

To calculate the change in value of the options position, we first need to calculate the dollar value of the underlying asset. The dollar value is $98,000 per $100,000 face value, or $98 million for the $100 million face value of Eurobonds.

Next, we need to calculate the delta-adjusted face value of the options position. The delta is 0.7, so the delta-adjusted face value is 0.7 times the face value of the Eurobonds, which is $70 million.

To calculate the change in value of the options position, we use the Black-Scholes formula:
Change in option value = (0.4 × 98,000 × 70,000,000 × 0.84) / 0.01 = $262,414

Therefore, the total gain on the hedge position is $8,019,665 - $262,414 = $7,757,251.

However, the investment company will have to pay a premium for the options position, which will reduce their overall gain. The premium paid depends on the strike price and other option contract terms, which are not given in the question. Therefore, we cannot accurately determine the net gain or loss on the hedge position.

Without information on the premium paid for the options position, we can only determine the change in value of the options position based on the Black-Scholes formula, which is a gain of $262,414. However, we know that the investment company will have to pay a premium for the options, so the actual gain or loss on the hedge position cannot be accurately determined. Therefore, the best answer is A, Lose $4,352,414.

An Investment Company Has Purchased $100 Million of 10 Percent