# Arbitrage argument with bonds

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:

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

For instance, could you give a detailed explanation of what to do if $$B(0,1)B(1,2) > B(0,2)$$ ?

• What have you tried to solve this problem? – Bob Jansen Mar 31 '20 at 13:35
• I have to suppose two cases: $B(0,1)B(1,2) >B(0,2)$ and $B(0,1)B(1,2) <B(0,2)$ and show that both lead to an arbitrage – Babado Mar 31 '20 at 13:37
• As usual, to make a profit you sell the thing(s) which is most expensive and buy the thing(s) which is cheaper. – noob2 Mar 31 '20 at 15:38
• Yeah ok, but I don't understand in real life, what does it mean for example to sell $B(0,1)B(1,2)$ and buy $B(0,2)$ – Babado Mar 31 '20 at 16:03
• Try thinking of borrowing money instead of selling a bond and of lending money instead of buying a bond – simzoor Mar 31 '20 at 16:08