Market Expectations Trough
Derivative Instruments
Stefano Caprioli
2nd Lesson
Euribor Crisis through EONIA
Discounting
1
Overview:
-OIS Discounting as standardized market convention in
order to take into account liquidity, credit and
counterparty risk
-Implications on Collateral Management
-Implications on Derivatives World
-OIS-LIBOR Spread as Key Risk Indicator
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Since the financial crisis, a whole host of new issues and complications have arisen surrounding
the previously uncomplicated area of valuing simple derivative instruments. Liquidity and
credit risk, and basis risk volatility which arose in interest rate markets during the crisis has
meant that Libor rates are no longer appropriate for derivative discounting. New discounting
methods have had to be developed to deal with these issues. There are further difficulties
stemming from how collateral impacts upon the trade, and issues such as collateral
optionality are now sources of much debate in the trading community.
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What is Collateral Management?
Collateral management is the method of granting, verifying, and
giving advice on collateral transactions in order to reduce credit risk in
unsecured financial transactions. The fundamental idea of collateral
management is very simple, that is cash or securities are passed from
one counterparty to another as security for a credit exposure. The form
of collateral is agreed before initiation of the contract. Collateral
agreements are often bilateral. Collateral has to be returned or posted in
the opposite direction when exposure decreases.
Collateral management has many different functions. One of these
functions is credit enhancement, in which a borrower is able to receive
more affordable borrowing rates. Aspects of portfolio risk, risk
management, capital adequacy, regulatory compliance and
operational risk and asset-liability management are also included in
many collateral management situations.
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Dear Counterparty,
We would like to advise you that XXXXX will move to OIS curve for
discounting purposes, with references to its derivatives collateralized
exposure. Change will be effective Monday, 6.
Starting Tuesday, 7, all margin calls will be calculated and managed
accordingly.
Our Sales Force is available for any further information you may require.
With Kind Regards
XXX Collateral Management Desk
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OIS reflects “real” conditions better than Libor Rate since the underlying index for
OIS is based on traded rates rather than a poll.
7
When Banks CDS went to 200 bps, Market Operators started to
distinguish between OIS and Libor.
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Fast topics overview:
LIBOR is the rate at which an individual Contributor Panel Bank could borrow
funds, were it to do so by asking for and then accepting interbank offers in
reasonable market size just prior to 11:00 [am] London time.
In the early 1980s, in order to ensure continued growth in the trading of
numerous new instruments such as interest rate swaps and currency options,
there arose the pressing need to estabilish a uniform benchmark index against
which these instruments could be referenced. LIBOR is not a traded rate, bit
rather the outcome of a poll of bank rates.
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An overnight indexed swap (OIS) is an interest rate swap where the periodic
floating rate of the swap is equal to the geometric average of an overnight index
(i.e., a published interest rate which is also called Overnight Rate or EONIA) over
every day of the payment period. The index is typically an interest rate
considered less risky than the corresponding interbank rate (LIBOR).
OIS rates (or, in particular, the difference or "spread" between LIBOR and OIS
rates) are an important measure of risk and liquidity in the money market. A
higher spread (high Libor) is typically interpreted as indication of a decreased
willingness to lend by major banks, while a lower spread indicates higher liquidity
in the market. As such, the spread can be viewed as indication of banks'
perception of the creditworthiness of other financial institutions and the general
availability of funds for lending purposes.
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Another New Market Feature in Order to Minimize Credit Risk
A lot of instruments are “forward premium”: they paid on the delivery date and
not the spot date. This convention was introduced in October 2009 by the
interbank options market as an alternative to the spot premium. As a result of the
recent crisis it becomes more likely that an option seller may default. Accordingly,
especially for long term options, the forward premium convention reduces the
risk. In addition, another advantage is that all cash flows are exchanged on one
date (on the delivery).
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OIS as Risk free Curve and Forward Premium to minimize Credit and Counterparty Risk
Market
Source:Reuters
12
OIS as Risk free Curve and Forward Premium to minimize Credit and Counterparty Risk
Market
Source:Reuters
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Two Curves and Fx Framework: Academic Approach
In the double–curve case, no arbitrage requires the existence at any time t0 ≤ t ≤ T
of a spot and a forward exchange rate between equivalent amounts of money in the
two curves such that
Where the subscripts f and d stand for forecasting and discounting. Cd(t) is
any cash flow (amount of money) at time t in units of discount curve and
Cf(t) is the corresponding cash flow at time t (the corresponding amount
of money) in units of forecasting curve.
Xfd(t,T) -> xfd(t,T) for t->T
Remember also that, having a single curve, xfd(0)=1
Note that’s the same Foreign Structure in order to build a non arbitrage
free framework whereas d stays for domestic, f for foreign and x is the
spot rate.
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Two Curves and Fx Framework
Considering the “spread ratio” between Fwds (d and f) we have:
So we obtain the following relation:
BAfd (t, T1, T2) =(Pd (t, T1)Xfd (t, T1) − Pd (t, T2)Xfd (t, T2))/[Xfd(t, T2) (Pd (t, T1) − Pd (t, T2))]
Assuming the driftless lognormal martingale for Ff (t,T1,T2) we have:
d Ff (t,T1,T2)= Ff (t,T1,T2)σf(t)dWf
Under Pf(t,T2) Numeraire.
We know that, under Pd(t,T2) Numeraire, Xfd(t,T2) moves according to a
martingala:
d Xfd (t, T2)= Xfd (t, T2) σX(t)dWx
With ρdt=dWxdWf
So, in order to calculate expectations under “discounting numeraire”, we
must add a drift, so we have:
d Ff (t,T1,T2)= μf(t)dt+Ff (t,T1,T2)σf(t)dWf
With μf(t)=-σf(t) σx(t) ρ
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We thus obtain the following expressions:
T
Qd2
Et
F T , T , T   F t , T , T QA
f
1
1
2
f
1
2
fd
( t , T1 ,  f ,  X ,  )
 T1

QA fd ( t , T1 ,  f ,  X ,  )  exp     f ( u ) X ( u )  ( u ) du 
 t

Where QAfd is the multiplicative adjustment you have to take into
account.
Market Standard is to assume ρ=0, so we have Qafd=1 and you just
need of a strip of σf that match market Cap/Flor Prices.
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Example: Interest Rate Cap for Market Pratictioners
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Let L(t,T, ) be the Forward LIBOR at time t for the accrual period [T,T+ ].
 Is expressed in terms of fractions of a year.
A caplet is a call option on L(t,T, ):
After 2008 crisis we have P(t,T+ ) derived by OIS Curve, while L(t,T, ) is
observed on the Market or derived by Libor Curve.
Market pratictioners call Libor derived curves “Forecasting Curves” and OIS
derived Curve discounting Curve. We use different market quotes in order
to build different forecasting curves: for instance we have EUR 6M
Forecasting Curve that’s based only on 6 months tenor instruments, EUR 3M
on 3 months tenor instruments and so on.
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Using Black Model we have (for Cap):
n 1
 (t , T ) 
n 1

  i ( t )    P ( t , Ti 1 ) L ( t , Ti ,  ) N ( d 1 )  KN ( d 2 )
i0
Ti
Ti

i0
Another important change is the Forward Premium convention: dealers
start to exchange payment flows at each caplet expiry in order to minimize
Lost Given Default embedded into the Cap or Floor.
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New market conventions have also impact on Banking Profit and Losses
since, moving from Euribor to OIS discounting curve, you realize a Present
Value variation on your Book.
It’s interesting to note how implied volatilities change their levels according
to discounting curve you choice, so Brokers are forced to show premiums
also for markets like Cap & Floor or Swaptions where, before 2008, they
pubblished only volatilities.
Diff Flat Vol Eur Disc - Flat Vol OIS Disc on Cap & Floor Euribor 6M
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Swap Rates
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We can maintain the same notation in order to deal with Double Curve
Swap Rates, moving in the same manner:
n
S f (t , T ) A f (t , S ) 
P
f
( t , Ti ) f (T i 1 , Ti ) F f ( t , Ti 1 , T i )
i 1
Where the Annuity is:
m
A f (t , S ) 
P
f
( t , S j ) f ( S j 1 , S j )
j 1
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If we choice Af(t,S) as numeraire we have:
dS f ( t , T , S )  S f ( t , T , S ) v f ( t , T , S ) dW
f
Where vf is the Volatility and dWf is a Brownian Motion. We have:
m
P
d
j 1
m
( t , S j ) d ( S j 1 , S j ) X
fd
( t , S j )  x fd ( t )  P f ( t , S j ) f ( S j 1 , S j )  x fd ( t ) A f ( t , S )
j 1
That’s a non arbitrage condition on Annuities.
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We can define a “Swap Forward Exchange Rate” Yfd(t,S) such that:
x fd ( t ) A f ( t , S )  Y fd ( t , S ) A d ( t , S )
or :
Y fd ( t , S ) 
x fd ( t ) A f ( t , S )
Ad ( t , S )
Yfd(t,S) is a martingale under “Discounting Curve Annuity”, so we have:
dY fd ( t , S )  Y fd ( t , S ) v Y ( t , S ) dW Y
 dt  dW f dW Y
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Again we make a change of numeraire in order to obtain forecasting Swap Rate
dynamic under discounting swap measure:
dS f ( t , T , S )  S f ( t , T , S )  f ( t , T , S ) dt  v f S f ( t , T , S ) dW
f
 f (t , T , S )   v f (t , T , S ) vY (t , T , S ) 
We thus obtain the following expressions:
S
E
Qd
d
S T
f
0

, T , S   S f ( t , T , S ) QA
fd
(t , T , S , v f , vY ,  )
T0

QA fd ( t , T , S , v f , v Y )  exp    v f (  , T , S ) v Y (  , T , S )  
 t

Also in this case market pratictioners moved to QAfd=1 in order to semplify pricing
activities.
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Interest Derivatives continue to “bet” on Euribor but you have to discount with
OIS Curve: you have to use two curves:
-Forecasting Curve (Euribor Based)
-Discounting Curve (OIS Based)
Market
Source:Reuters
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Interest Derivatives continue to “bet” on Euribor but you have to discount with
OIS Curve: you have to use two curves:
Market
Source:Reuters
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OIS as Risk free Curve and Forward Premium to minimize Credit and Counterparty Risk
Market
Source:Reuters
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CONCLUSIONS
We showed two remarkable market standards in order to minimize
credit, liquidity and counterparty risks for interest rate derivatives:
-Collateral Management by OIS Discount Curve
-Forward Premiums
OIS Discounting determines a new way to price IR Derivatives, since
we must add new assumptions on drift you have moving from one
to two curves.
The key is to understand if it’s better a centralized clearing house or
a “one-to-one” collateralized agree both for risk management and
front office purposes or, on other hands, both for regulators and
market operators.
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“Clearing houses around the world generally performed well in the
highly stressed financial environment of the recent crisis. However,
we should not take for granted that we will be as lucky in the future.
Past crises, including the financial panic of 1907 and the 1987 stock
market crash, led to significant reforms and improvements in
clearing and settlement that paid off in subsequent periods of
financial stress. Given the growing interdependencies among
clearing houses, along with the new mandates for central clearing,
now is a good time to reflect on the lessons of the recent crisis and
consider whether further improvements are possible.”
2011-US Federal Reserve Chairman Ben Bernanke
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