Zero Coupon Bond

[Jonas Hungerbuehler]

Hello

Could anybody help me out here with the following two questions: for a) I did get that result (bootstrapping) since a ZCB can be viewed as a discount factor e^(-r(T-t)) and with AOA we get that the value portfolios with the same terminal value also have the same value at t=0.

1. Let Bt(T) be the cost at time t of a zero coupon bond with maturity T (in years).

(a) Suppose B0(1), B0(2) and B1(2) are known at time 0 (i.e. interest rates are deterministic). Show that the absence of arbitrage requires B0(1)B1(2) = B0(2).

(b) Now suppose B0(1) and B0(2) are known at time 0 but B1(2) will not be known until time 1. Does the previous result still hold? Show that if we know with certainty that m ≤ B1(2) ≤ M, then we can conclude mB0(1) ≤ B0(2) ≤ MB0(1).

Thanks in advance,
Jonas

 

[yetanotherquant]

In your equation B0(1)B1(2) = B0(2)
B0(1) and B0(2) are bonds and their prices are known but B1(2) is not, it is so to say, a forward discount rate and you need to calculate it.

To understand the algorithm of bootstrapping, look at my QuantLib Notes (from page 27)
I give both toy- and reallife-examples.

 

[Quasar Chunawala]

In your equation B0(1)B1(2) = B0(2)
B0(1) and B0(2) are bonds and their prices are known but B1(2) is not, it is so to say, a forward discount rate and you need to calculate it.

To understand the algorithm of bootstrapping, look at my QuantLib Notes (from page 27)
I give both toy- and reallife-examples.

Hi Vasily,

If you have any other notes on C++, please do share. This is exciting stuff!

Thanks,
Quasar.

 

[IntoDarkness]

idk why school keeps teaching this kind of stuff. in reality B0(1)B1(2) = B0(2) never holds

 

[concerned_quant]

your first proof is just a forward discount factor.

you can prove it as follows:

Assume you agree to get into a contract at time 0 to pay k at time t to receive $1 at time T. What is the price of this contract? Using discounted cashflows...

V0 = -kP(0,t) + P(0,T).

The contract is chosen such that value is 0 at time 0 (it costs nothing to get into this contract). Hence...

kP(0,t) = P(0,T)

or k = P(0,T)/P(0,t) := P(t,T).

for part b)

if m<=P(t,T)<= M, then...

m<= P(0,T)/P(0,t) <= M, or...


P(0,t)m<=P(0,T)<= MP(0,t)

whats so hard??

as for the previous comment...it doesn't hold because of credit risk. you still need to prove it in absence of credit risk.