Question

Assume that the continuously compounded zeros rates for T=1, 2, 3, 4 (years) are 4.5%, 5.2%,
5.7%, 6.3% respectively. A market maker offers, through forward rate agreements (FRAs), the
following rates; 6.078% for the period between the 1st and the 2nd year, 6.900% for the period
between 2nd and the 3rd year and finally 8.300% for the period between the 3rd and the 4th year.
Evaluate if arbitrage opportunities exist. If such opportunities exist, design a strategy that can
deliver the maximum profit on a principal of £100 million. Assume annual compounding for the
FRA’s interest rate quotes.

Answer 1

Time	Zero	Period	No arbitrage implied forward rate between the period*	FRA offered in the market	Arbitrage
1	4.50%
2	5.20%	1 to 2	6.078%	6.078%	No
3	5.70%	2 to 3	6.930%	6.900%	Yes
4	6.30%	3 to 4	8.437%	8.300%	Yes

*No arbitrage implied forward rate between the period 1 & 2 = e^{5.20% x 2} / e^{4.50% x 1} - 1 = 6.078%

*No arbitrage implied forward rate between the period 2 & 3 = e^{5.70% x 3} / e^{5.20% x 2} - 1 = 6.930% and so on.

Since, the no arbitrage implied forward rate between year 1 and 2 = FRA rate for 1 to 2. Hence, there is no arbitrage between year 1 & 2.

However, the the no arbitrage implied forward rate between year 2 and 3 and that between 3 & 4 are not same as FRA rates for 2 to 3 and 3 to 4 respectively, there is an arbitrage opportunity available in year 2 to 3 and then from 3 to 4.

Arbitrage Strategy:

• Borrow through the forward contracts
• Invest into the zero coupon bonds

Profit = Maturity amount of £100 million through 4 year zero coupon bond - Maturity amount of £100 million through FRAs = 100 x e^{6.3% x 4} - 100 x e^{4.5% x 1} x (1 + 6.078%) x (1 + 6.900%) x (1 + 8.300%) = £ 0.197587 million = £ 197,587