LIBOR, OIS, and their derivatives

Fixed income markets

Fixed income are the biggest financial market.

Market participants have adopted the convention that a rally refers to falling interest rates, while a sell off refers to rising rates.

principal v.s. notional principal: pricipal is the real money your pay.
e.g. you pay 0 for a 100 swap

There are a lot of conventions, do not discuss them, just follow them

LIBOR

ICE LIBOR

“At what rate could you borrow funds, were you to do so by asking for and then accepting interbank offers in a reasonable market size just prior to 11 am?”

FRAs (Forward Rate Agreements)

FRAs are OTC transactions.

In a FRA transaction, counterparty A agrees to pay counterparty B LIBOR settling t years from now applied to a specified notional amount (say, $100 mm). In exchange, counterparty B pays counterparty A a pre-agreed interest rate (say, 3:05%) applied to the same notional.

Because of their granular structure, they allow one to hedge the exposure or express the view on rates move on specific days.
e.g.

an FOMC FRA is a FRA which fixes the day after a Federal Open Markets Committee meeting.
a turn switch allows to hedge or express the view on the rate spikes idiosyncratic to the month-, quarter- or year-end.

Eurodollar futures

In many ways, Eurodollar futures are similar to FRAs, except that their terms, such as contract sizes and settlement dates are standardized.

The Merc makes daily “mark to market” adjustments to the margin account in order to reﬂect the changes in values of each of the individual contracts.

Unlike a FRA, the Eurodollar future quoted price is linear in the underlying rate. convention
- -> why does it not adopt compound interest?
- -> what does FRA use?

At any time, 44 Eurodollar contracts are listed on the Merc.

Bloomberg snapshot of the Eurodollar futures market

ED: eurodollar
H: March
7: 2017
white, red, green name stand for different pack

Swaps

spot starting swap
forward starting swaps
what’s the difference between swaps and FRAs
FRAs is single period swap

OIS ( overnight indexed swap)

OIS is better than LIBOR, because it’s based on sth.

OIS is based on the overnight deposit.

LIBOR/OIS basis swaps

why it’s downward sloping
It’s the fact, I don’t know

SOFR

SOFR is based on the volume weighted average of the Treasury repo rate, the rate on overnight loans collateralized by US Government Treasuries.

The volume is very big so it’s hard to manipulate. (~ 1 trillion)

Fed Fund futures

SOFR is based on the volume weighted average of the Treasury repo rate, the rate on overnight loans collateralized by US Government Treasuries.

Valuation of swaps

Multicurve paradigm

why don’t we use LIBOR also as the index rate
Because we have a huge amount of LIBOR futures in the market

Zero Coupon Bond

Parameter

valuation date
Settlement date
Maturity date

Replication of $P(t,S,T)$ :

Buy 1 $P(t,t,T)$
Sell $P(t,t,S)$ , shares: $x$ , $xP(t,t,S) = P(t,t,T)$ , so $x= P(t,t,T)/P(t,t,S)$
The value of this forward contract should be zero. After the above two steps, the cash flow is:

Time	Cash Flow
t	0
S	-x
T	1

\Rightarrow P(t,t,T) = x= \frac{P(t,t,T)}{P(t,t,S)}

Forward Rate

P(t,S,T) = \exp\left(-\int_S^Tf(t,s)ds\right)

The PV of a swap is the difference between the PVs of the ﬁxed and ﬂoating legs (in this order!): formula

The value of swap rate is independent of the valuation date.

DVO1 = dollar value of 1.

Receive swap, long swap, means you receive fix.

You can trade swap only if you can post a margin.

the value of the swap rate is independent of the valuation date (even though the PV’s of the individual legs of the swap are clearly not).

Building the OIS / LIBOR multicurve

Only use 3 month LIBOR

Bootstrapping is useless