Question

A futures contract on a share, which pays dividend at a continuously compounded rate of 3%, is written when the share has a price of $790, and the continuously compounded risk-free interest rate is 5%. The contract is priced at $800 and expires in 3 months.

(b) Demonstrate how you could execute an arbitrage transaction and calculate arbitrage profit. [5]

Answer 1

No arbitrage price of the future contract, F = S x e^{(rf - d) x T} where S = Spot rate = $ 790; rf = risk free rate = 5%, d = 3% and T = time to maturity = 3 months = 3 / 12 = 0.25 year

Hence, F = 790 x e^{(5% - 3%) x 0.25} = $ 793.96

As actual price of the future contract, F_a = $ 800 > F, there exists an arbitrage opportunity.

Sl. No.	Arbitrage activity	Cash flow at t = 0	Cash flow on maturity i.e. at t = T = 3 months = 0.25 year
1	Short the future contract today, at $ 800	0	+ 800
2	Borrow $ 790 today	+ 790	- 790 x erf x T = - 790 x e5% x 0.25 = - 799.94
3	Buy a stock today and hold it for 3 months. Use this stock to close the short position in the future contract, at the time of maturity, after three months. Receive the dividend	- 790	+ 790(1- e- d x T) = 790 x (1 - e-3% x 0.25) = 5.90
	Total	0	5.97

Thus we enter into a strategy in which there is no investment at t = 0 but we get a riskless positive cash flow of $ 5.97 after t = 3 months. This is the arbitrage opportunity and this is the arbitrage profit.