# Constructing an arbitrage opportunity for a company involving Forwards

Let's say an investor enters a long forward contract on 100 units of underlying assets $$S$$ and maturity $$T$$ = 4 years. The asset $$S$$ pays no dividends and the spot price of one asset is $$S_0$$ = £5. The continuously compounded interest rate is $$r = 0.04%$$.

i. Calculate the forward price $$F$$ (for 100 units) at time $$T = 4$$.

ii. Suppose that the investor agrees to sell the forward contract at $$t$$ = 1 year to a company for a price of $$P_1 =$$ £60, when the stock price is $$S_1$$ = £6. Construct an arbitrage opportunity for the company. Note that the company is only allowed to borrow money, buy Zero Coupon Bonds, buy or short-sell stocks and enter any forward contract on stocks.

So for part (i), I established the forward price was $$F =$$ $$500e^{0.04*4}$$ $$= 586.76$$

For part (ii) I never really know where to start on these sorts of questions. Could someone take me through the steps and logic for something like this? My exam is tomorrow so it would be much appreciated.

In any "arbitrage" situation, you are trying to create a scenario where you sold a rich asset and bought a cheap asset, while keeping the overall position as flat as possible to risk.

Assuming interest rates have not changed from continuously compounded 4%, and the company can borrow at this rate and sell a forward at this rate.

The company is buying the forward from the initial investor for £60 by borrowing the £60 for 3 years (will pay back (60*e^0.04*3)). The company has taken over the obligation to buy the 100 shares for a total of £646.76 (£586.76 from the forward contract + £60 they paid for the forward. They would also sell the stock forward at £676.50 (6*100*e^(0.04*3)).

At maturity of the position, they will buy the stock for £586.76. Sell the same stock for £676.50 from the forward they sold. And repaid the £60 loan with interest for a total of £67.6 (60*e^(0.04*3)). The net profit being a total of £22.14

• Thanks for the reply. What happened to the long position in the forward contract? The answer given to me states: 'No cost at t = 1, sure gain of £22.09 at t = 4.' as the answer. – DPJDPJ Jan 15 '20 at 20:58
• I get your logic, which helps a lot. But in the answer they say, (a) borrow £60 for 3 years; (b) buy the long forward contract for 60 pounds (with future price $F = F_{0,4} = 586.76$); (c) enter a short forward contract on 100 stocks Which is ever so slightly different from your method. Could you help by explaining the difference? Thank you so much, my exam is tomorrow so I am kind of bricking it! – DPJDPJ Jan 15 '20 at 21:01
• I misread your question. I thought your question said they sold a forward at a forward price of £6. I amended my answer to reflect they bought the existing forward (forward price of £586.76) from the investor for £60. Incidentally, the original investor sold the forward too cheap. – AlRacoon Jan 15 '20 at 21:42
• So no cost at t=1 since they bought the forward for £60 which they borrowed, and locked in the £22.14 (diff due to rounding) at t=4. – AlRacoon Jan 15 '20 at 21:48
• You’re brilliant. Thank you. You may have just saved my year at university sir. – DPJDPJ Jan 15 '20 at 23:02