No-Arbitrage Pricing of a Backset LIBOR Payment

# Backset LIBOR contract


# Backset LIBOR contract












Below is an extract from Steven Shreve’s BK 2, Chapt 10: Term Structure models. LINK

I am trying to understand Stochastic Calculus from the above book with the help of a Pure Math PhD student. Despite trying to wade through resource in Public domain (Google Search) and a few of the other books I bought, we could not understand the point of arrival of below topic. I have not made any progress on the below topic. Kindly if anyone can refer to a book or an article which details the content below will be highly appreciated. I will be happy if any can also just provide some lead/summarise the topic, I can take it forward from the lead.

Section 10.4.3: Pricing a Back set LIBOR Contract

An interest rate swap is an agreement between two parties A and B that A will make fixed interest rate payments on some “notional amount” to B at regularly spaced dates and B will make variable interest rate payments on the same notional amount on these same dates .The variable rate is often backset LIBOR , defined on one payment date to be the LIBOR set on the previous payment date. The no-arbitrage price of a payment of backset LIBOR on a notional amount of 1 is given by the following theorem.

Theorem 10.4.1 (Price of back set LIBOR). Let 0 ≤ t ≤ T and δ >0 be given. The no-arbitrage price at time t of a contract that pays L(T,T) at time T+δ is \begin{equation} S(t)=\begin{cases} { B(t,T+\delta).L(t,T) \; 0\leqq t \leqq T }.\\ { B(t,T+\delta).L(T,T) \; 0\leqq t \leqq T+\delta} \end{cases} \end{equation}

PROOF : These are two cases to consider . In the first case ,T ≤ t ≤ T +δ, LIBOR has been set at L(T,T) and is known at time t .The value at time t of a contract that pays 1 at time T + δ is B(t,T+δ), so the value at time t of a contract that pays L(T,T) at time T + δ is B(t , T + δ)L(T,T) .

In the second case ,0 ≤ t ≤ T, we note from (10.4.4) that $$ B(t,T+\delta).L(t,T) = \frac{1} {\delta}[B(t,T)-B(t,T+\delta )] $$ We must show that the right-hand side is the value at time t of the backset LIBOR contract. To do this, suppose at time t we have $ \frac{1} {\delta}[B(t,T)-B(t,T+\delta )] $, and we use this capital to set up a portfolio that is : $$\text{long} \frac{1} {\delta} \text{bonds maturing at T};$$ $$\text{short} \frac{1} {\delta} \text{bonds maturing at T+δ} $$ At time T ,we receive $ \frac{1} {\delta}$ from the long position and use it to buy $ \frac{1} {\delta}. \frac{1} {[B(t,T+\delta )]} $ bonds maturing at time T+δ, so that we now have a position of $ \frac{1} {\delta}. \frac{1} {[B(t,T+\delta )]} - \frac{1} {\delta}$ in (T+δ)-maturity bonds. At time T+ δ, this portfolio pays

$$ \frac{1} {\delta}.\frac{1} {B(T,T+\delta} - \frac{1} {\delta} = \frac{B(t,T)-B(t,T+\delta )}{ \delta B(t,T+\delta)} = L(T,T)$$

We conclude that the capital $\frac{1} {\delta}[B(t,T)-B(t,T+\delta )] $ we used at time t to set up the portfolio must be the value at time t of the payment L (T ,T) at time T+δ.