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I'm struggling to come up with an arbitrage argumento to prove the following statement:

Let $B(t,T)$ denote the cost at time $t$ of a risk-free $1$ euro bond, at time $T$. Assume that the interest rate is a deterministic function. Show that the absence of arbitrage requires that:

$B(0,1)$$B(1,2)$ $=$ $B(0,2)$

I believe in order to prove this I have to consider the case where $B(0,1)$$B(1,2)$ $<$ $B(0,2)$ and $B(0,1)$$B(1,2)$ $>$ $B(0,2)$ and show that if this happens there is an arbitrage possibility. However when I was trying to come up with the arbitrage strategy for the case $B(0,1)$$B(1,2)$ $>$ $B(0,2)$ I couldn't get a profit $>$ $0$ from an initial investment of $0$.

I considered short-selling $B(0,1)$$B(1,2)$ (and then sell it at the market) and buy $B(0,2)$ and invest in the bank the remainder of the money. However, at maturity I wouldn't have enough money to buy $B(0,1)$$B(1,2)$ back and return it to the ownwer. I also considered borrowing from the bank $B(0,1)$$B(1,2)$ and investing the amount $B(0,2)$ but I would end uo with a similar issue.

Could you please help me clarify what should be the arbitrage strategy.

Thanks in advance!

gt6989b
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Diogo Fonseca
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1 Answers1

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You are comparing 2 strategies. The LHS involves investing some amount $A$ into a 1-year bond and then reinvesting it for another year. The RHS invests it into the bond for 2 years from the start.

So if LHS > RHS, you sell the LHS on the market and yourself invest into the RHS. At the end of the 2-year period, you keep the difference as the arbitrage.

If, on the other hand, LHS < RHS, you sell the RHS and invest into the LHS, again keeping the difference in price.

Since both strategies are risk-free and have identical incoming and outgoing cash flows, the 2-position strategy has no inflows and no outflows except pocketing the difference in price between 2 positions. In either case, you get risk-free money, which means arbitrage.

gt6989b
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