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Market Expectations Trough Derivative Instruments

Stefano Caprioli

2nd LessonEuribor Crisis through EONIA

Discounting

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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.

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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

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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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Two Curves and Fx Framework: Academic ApproachIn 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->TRemember also that, having a single curve, xfd(0)=1Note 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 FrameworkConsidering 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:

1

2

)()()(exp),,,,(

),,,,(,,,,

1

121211

T

t

XfXffd

XffdffQt

duuuuTtQA

TtQATTtFTTTFETd

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+ ].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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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.

1

0211

1

0

)()(),,(),()(),(n

i

TTii

n

ii

ii dKNdNTtLTtPtTt

Using Black Model we have (for Cap):

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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

iiifiififff TTtFTTTtPStATtS

111 ),,(),(),(),(),(

Where the Annuity is:

m

jjjfjff SSStPStA

11 ),(),(),(

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If we choice Af(t,S) as numeraire we have:

ffff dWSTtvSTtSSTtdS ),,(),,(),,(

Where vf is the Volatility and dWf is a Brownian Motion. We have:

m

jffd

m

jjjfjffdjfdjjdjd StAtxSSStPtxStXSSStP

1 111 ),()(),(),()(),(),(),(

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:

),(

),()(),(

:

),(),(),()(

StA

StAtxStY

or

StAStYStAtx

d

ffdfd

dfdffd

Yfd(t,S) is a martingale under “Discounting Curve Annuity”, so we have:

Yf

YYfdfd

dWdWdt

dWStvStYStdY

),(),(),(

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Again we make a change of numeraire in order to obtain forecasting Swap Rate dynamic under discounting swap measure:

),,(),,(),,(

),,(),,(),,(),,(

STtvSTtvSTt

dWSTtSvdtSTtSTtSSTtdS

Yff

ffffff

We thus obtain the following expressions:

0

),,(),,(exp),,,,(

),,,,,(),,(,,0T

t YfYffd

YffdffQd

STvSTvvvSTtQA

vvSTtQASTtSSTTSESd

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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CONCLUSIONSWe 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 PremiumsOIS 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