Question

2. Let B(t,T) be the cost at time t of a risk-free dollar at time T. (a) Suppose B(0,1), B(0,2) and B(1,2) are all known at time 0 (i.e. interest rates are deterministic). Show that the absence of arbitrage requires B(0, 1)B(1,2) B(0,2) (b) Now suppose B(0, 1) and B(0, 2) are known at time 0 but B(1,2) will not be known until time 1. What goes wrong with your argument for (a)? Show that if we know with certainty that m s B(1,2) S M then we can still conclude mB(0,1) S B(0,2) s MB (0, 1)

Answer 1

Please don't get scared by the notations and symbols. The two questions are very simple, otherwise.

Before answering the question, let's understand what B(t,T) is.

B(t,T) is the cost at time t of buying an instrument that gives me $ 1 at T

That means, at point of time t, I will have to pay B(t,T) to buy an instrument that will give me $ 1 at the end of T.

Let' snow look at the question

Part (a)

Situation 1: I intend to buy an instrument today i.e. at t=0, that pays me $ 1 after 2 years. So T = 2

Since, B(t,T) is the cost at time t of buying an instrument that gives me $ 1 at T

Hence, B(0,2) should be the cost at t=0 to buy an instrument that pays me $ 1 at the end of T = 2 years

Situation 2: The similar instrument can also be replicated through a two step process:

Step 1: At the end of 1 year i.e. at t=1, I buy an instrument that pays me $ 1 at the end of year 2 i.e T = 2

Since, B(t,T) is the cost at time t of buying an instrument that gives me $ 1 at T; the price of this instrument at t=1 should be B(1,2)

Step 2: In order to get $ B(1,2) at t=1, let's say i need to buy an instrument today i.e. at t=0 that gives me $ B(1,2) at the end of year 1 i.e.T=1

Since, B(t,T) is the cost at time t of buying an instrument that gives me $ 1 at T; the price of the required instrument at t=0 that will give me $ 1 at T=1 will be B(0,1)

Now, B(0,1) is the price today to get $ 1 at T=1

Thus, in order to get $ 1 at T = 1, I need to pay B(0,1) today.

Hence, in order to get $ B(1,2) at T=1, i need to buy an instrument at a price of B(0,1) x B(1,2) today

Now payoff in both the situation at the end of T=2 is $ 1

Hence, the price of both the instruments today should be same.

Hence, B(0,2) = price of the instrument in situation 1 = B(0,1) x B(0,2) = price of the instrument in situation 2

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

Part (b)

Sub part 1: If B(1,2) is not known at t=0 then the equation in part(a) will not be true. Since the B(1,2) is uncertain and risky today, different term structure theories will govern the expected value of B(1,2) say given by E[B(1,2)] and the above equation will then read as:

B(0,2) = B(0,1)E[B(1,2)]

Sub part 2:

From part(a): B(0,1)B(1,2) = B(0,2)

Hence, B(1,2) = B(0,2) / B(0,1)

We have

$m\leq B(1,2)\leq M$

$m\leq \frac{B(0,2)}{B(0,1)}\leq M$

$m B(0,1))\leq \ B(0,2)\leq M B(0,1)$