Counterparty Risk - wrong way risk and liquidity issues€¦ · Counterparty Wrong-Way Risk Liquidity and Counterparty Risks Counterparty Risk-wrong way risk and liquidity issues

Counterparty Wrong-Way RiskLiquidity and Counterparty Risks

Counterparty Risk

-wrong way risk and liquidity issues

Antonio [email protected]

-

www.iasonltd.com

2011

Iason 2011 - All rights reserved


Index

1 Counterparty Wrong-Way RiskContract Exposure and Default ProbabilitiesCVA and VaR with wrong-way risk

2 Liquidity and Counterparty RisksLiquidity Risk Pricing in OTC Derivatives



Contract Exposure and Default ProbabilitiesCVA and VaR with wrong-way risk

Index






Definition

Counterparty risk is the risk that a party to an OTC derivativecontract may fail to perform on its contractual obligations, causinglosses to the other party.

Losses are usually quantified in terms of the replacement cost of thedefaulted derivatives .

Counterparty risk can be:1 One-Way: One party faces the exposures depending on the (ever

positive) value of the position it holds against the other party;2 Two-Way: Both parties may face exposures depending on the value

of the positions they hold against each other.

The feature distinguishing counterparty risk from lending risk isuncertainty of exposure at any future date:

1 Loan: exposure at any future date is the outstanding balance, whichis certain (not taking into account prepayments);

2 Derivative: exposure at any future date is the replacement cost,which is determined by the market value at that date and is,therefore, uncertain.




Index






Counterparty Risk Exposure of a Contract

We start by assuming that no netting or margin agreement is inplace.

The market value of contract i with a counterparty is known only forthe current date t = 0. For any future date t, the value Vi (t) israndom.

If the counterparty defaults at time τcpt before the contract’smaturity, the economic loss is equal to the replacement cost of thecontract:

if Vi (τcpt) > 0, we do not receive anything from defaultingcounterparty, but have to pay Vi (taucpt) to another counterparty toreplace the contract;if Vi (τcpt) < 0, we receive |Vi (τcpt)| from another counterparty, buthave to give this amount to the defaulting counterparty.

Combining these two scenarios, we can specify contract-levelexposure Ei (t) at time t as:

Ei (t) = max[Vi (t), 0]




Counterparty Risk Exposure of a Contract

At a future time T > t, the exposure is uncertain:




Theoretical Approaches to Default Modelling

The second building block of the Counterparty Risk measurement isthe prediction of the default of the counterparty.

Jointly to the contract and portfolio level exposure, defaultdetermines the counterparty risk fully.

In theoretical literature two approaches to model single defaults:

Structural Models: Default occurs as soon as the firm value crossesa given barrier (from above)Reduced (Intensity-based) Models: The default time is modelledas the first jump time of a given jump process (typically a Poissonprocess), occurring with an intensity λ(t), also called hazard rate.This is the probability of a default occurring at an infinitesimal timeafter t given that it did not occur before, and can be a stochasticprocess itself.




Index






Wrong/Right-Way Risk

Wrong/right-way risk arises from dependence between credit quality of acounterparty and exposure to that counterparty.

The risk is wrong (right) way when the exposure tends to increase(decrease) when counterparty credit quality worsens.

Wrong/right-way risk can be general (dependence caused by systematicrisk factors) or specific (dependence caused by counterparty-specific riskfactors).

Some examples:

1 We sell credit protection on X to Y: general right-way2 We enter into oil receiver swap with oil producer: general wrong-way3 We buy a put option on X stock from Y: general wrong-way4 We buy a put option on X stock from X: specific wrong-way

Specific wrong-way risk should be avoided.

We analyse the impact of wrong-way risk for a swap portfolio on:

The CVA adjustment for the risk-free value of a portfolio of swaps, toaccount for the expected losses given the default of the counterparty;The counterparty credit VaR.




Theoretical Framework to Include Wrong-Way Risk

We assume that the default probability of the counterparty is stochasticover the reference period.

Default is a jump whose probability of occurrence is determined by anintensity λ(t), which is a stochastic process.

Roughly speaking, the intensity indicates the annual probability of default,so that if λ(t) = 2%, there is a 2% probability that our counterparty willgo defaulted in next year. In our framework, the intensity varies over time,so that the defalt probability is not constant.

The Expected positive exposure (EPE) of a swap is computed assumingthat all Euribor/Libor rates have a terminal Lognormal distribution. It ispossible to determine the distribution of the swap rates from thedistributions of the single Euribor/Libor rates (an analytical approximationis used in our framework)

We correlate the default intensity with the swap rates, and we deriveanalytical approximation for the expected losses (EPE× PD). We testedthis approximation against Montecarlo simulations and we found it veryaccurate.




Credit Value Adjustment

The Credit Valuation Adjustment (CVA) of an OTC derivativesportfolio with a given counterparty is the risk-neutral expectation ofthe discounted loss of value of the portfolio, due to default by thecounterparty

CVA =

Tn∑

k=1

Sprd · P(t, tk) · EPE(tk) ·∆tk

Sprd ≈ PD× LGD is the CDS spread dealing in the market for thecounterparty’s debt.

CVA can be computed analytically only at the contract level forseveral simple cases.

Calculating discounted EPE at the counterparty level requiressimulation.

The market value of a portfolio of derivatives with a riskycounterparty is given by the risk-free market value minus therelevant CVA, as defined above.




A Practical Example: Market Data

We show an example, assuming the following market data for interestrates:

Time Eonia Fwd Sread Fwd Libor

0 0.75% 0.65% 1.40%0.5 0.75% 0.64% 1.39%1 1.75% 0.64% 2.39%1.5 2.00% 0.63% 2.63%2 2.25% 0.63% 2.88%2.5 2.37% 0.62% 2.99%3 2.50% 0.61% 3.11%3.5 2.65% 0.61% 3.26%4 2.75% 0.60% 3.35%4.5 2.87% 0.60% 3.47%5 3.00% 0.59% 3.59%5.5 3.10% 0.59% 3.69%6 3.20% 0.58% 3.78%6.5 3.30% 0.58% 3.88%7 3.40% 0.57% 3.97%7.5 3.50% 0.57% 4.07%8 3.60% 0.56% 4.16%8.5 3.67% 0.56% 4.23%9 3.75% 0.55% 4.30%9.5 3.82% 0.55% 4.37%10 3.90% 0.54% 4.44%

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

3.50%

4.00%

4.50%

5.00%

0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9

Euribor Fw d

Eonia Fw d




A Practical Example: Market Data

Market data for caps&floors and swaptions volatilities are:

Caps&FloorsExpiry Volatility0.5 30.00%1 40.00%1.5 45.00%2 40.00%2.5 35.00%3 32.00%3.5 31.00%4 30.00%4.5 29.50%5 29.00%5.5 28.50%6 28.00%6.5 27.50%7 27.00%7.5 26.50%8 26.00%8.5 25.50%9 25.50%9.5 25.50%10 25.50%

SwaptionsExpiry Tenor Volatility0.5 9.5 27.95%1 9 28.00%1.5 8.5 27.69%2 8 27.09%2.5 7.5 26.61%3 7 26.32%3.5 6.5 26.16%4 6 26.02%4.5 5.5 25.90%5 5 25.79%5.5 4.5 25.68%6 4 25.57%6.5 3.5 25.46%7 3 25.37%7.5 2.5 25.28%8 2 25.22%8.5 1.5 25.21%9 1 25.34%9.5 0.5 25.50%10 0




A Practical Example: Default Probabilities

We assume that default is a jump oc-curring with an intensity λ followinga CIR process:

dλt = κ(θ − λt)dt + ν√λtdZt

Parameters are chosen to be:

λ0 3.0%κ 27.0%θ 3.0%ν 20.0%

The resulting PD are shown beside.

Years PD

1 2.94%2 5.72%3 8.34%4 10.80%5 13.13%6 15.35%7 17.48%8 19.53%9 21.52%10 23.44%

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

1 2 3 4 5 6 7 8 9 10

PD




A Practical Example: Constant Notional Portfolio

We analyse how the CVA is affected bydifferent levels of correlation between the(synthetic) swap rate of the portfolio andthe intensity of default. The portfolio ofswaps has a constant notional amountover next 10 years as shown below andit is a net receiver fixed rate.

Years Notional

1 100.002 100.003 100.004 100.005 100.006 100.007 100.008 100.009 100.0010 100.00

We compute the CVA for different levels ofcorrelation (nil=-0%, medium=-50%, high=-90%),and for different levels of the synthetic swap rate(at-the-money, in-the-money=ATM+1%,out-of-the-money=ATM-1%).

Corr Swap- OTM ATM ITMDef.Intens. 1.63% 2.63% 3.63%

0% 0.0362 0.2853 0.8621-50% 0.0788 0.4905 1.2640-90% 0.1130 0.6546 1.5856

Adjustment over the risk-free swap rate to includethe counterparty risk.


0% 1.6341% 2.6621% 3.7271%-50% 1.6389% 2.6853% 3.7724%-90% 1.6427% 2.7037% 3.8086%




A Practical Example: Constant Notional Portfolio

Expected Positive Exposure (OTM,ATM,ITM)

-

0.05000

0.10000

0.15000

0.20000

0.25000

0.30000

1 2 3 4 5 6 7 8 9

EPE

-

0.20000

0.40000

0.60000

0.80000

1.00000

1.20000

1.40000

1.60000

1.80000

2.00000

1 2 3 4 5 6 7 8 9

EPE

-

1.00000

2.00000

3.00000

4.00000

5.00000

6.00000

7.00000

8.00000

1 2 3 4 5 6 7 8 9

EPE

Expected Loss Given Default (OTM,ATM,ITM)

0

0.005

0.01

0.015

0.02

0.025

1 2 3 4 5 6 7 8 9

ELgd Zero Corr

ELgd Medium Corr

ELgd High Corr

0

0.02

0.04

0.06

0.08

0.1

0.12

1 2 3 4 5 6 7 8 9

ELgd Zero Corr

ELgd Medium Corr

ELgd High Corr

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1 2 3 4 5 6 7 8 9

ELgd Zero Corr

ELgd Medium Corr

ELgd High Corr




A Practical Example: Decreasing Notional Portfolio

The same analysis on how the CVA isaffected by different levels of correlationbetween the (synthetic) swap rate of theportfolio and the intensity of default, isperformed for a declining notional swapportfolio over next 10 years, as shownbelow. It is a net receiver fixed rate.

Years Notional

1 100.002 90.003 80.004 70.005 60.006 50.007 40.008 30.009 20.0010 10.00



0% 0.0168 0.1420 0.4020-50% 0.0345 0.2250 0.5526-90% 0.0486 0.2914 0.6731



0% 1.6333% 2.6579% 3.7089%-50% 1.6368% 2.6742% 3.7385%-90% 1.6395% 2.6872% 3.7622%




A Practical Example: Decreasing Notional Portfolio


-

0.50000

1.00000

1.50000

2.00000

2.50000

3.00000

3.50000

4.00000

4.50000

5.00000

1 2 3 4 5 6 7 8 9

EPE

-

0.20000

0.40000

0.60000

0.80000

1.00000

1.20000

1.40000

1.60000

1 2 3 4 5 6 7 8 9

EPE

-

0.02000

0.04000

0.06000

0.08000

0.10000

0.12000

0.14000

1 2 3 4 5 6 7 8 9

EPE


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

1 2 3 4 5 6 7 8 9

ELgd Zero Corr

ELgd Medium Corr

ELgd High Corr

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

1 2 3 4 5 6 7 8 9

ELgd Zero Corr

ELgd Medium Corr

ELgd High Corr

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

1 2 3 4 5 6 7 8 9

ELgd Zero Corr

ELgd Medium Corr

ELgd High Corr




A Practical Example: Increasing Notional Portfolio

Finally we operate the analysis on howthe CVA is affected by different levels ofcorrelation between the (synthetic) swaprate of the portfolio and the intensity ofdefault, for an increasing notional swapportfolio over next 10 years, as shownbelow. It is still a net receiver fixed rate.

Years Notional

1 10.002 20.003 30.004 40.005 50.006 60.007 70.008 80.009 90.0010 100.00



0% 0.0569 0.1825 0.5539-50% 0.0817 0.3266 0.8450-90% 0.0888 0.4419 1.0778



0% 1.6355% 2.6691% 3.7486%-50% 1.6422% 2.6999% 3.8109%-90% 1.6475% 2.7246% 3.8607%




A Practical Example: Increasing Notional Portfolio


-

0.50000

1.00000

1.50000

2.00000

2.50000

3.00000

3.50000

1 2 3 4 5 6 7 8 9

EPE

-

0.20000

0.40000

0.60000

0.80000

1.00000

1.20000

1 2 3 4 5 6 7 8 9

EPE

-

0.05000

0.10000

0.15000

0.20000

0.25000

1 2 3 4 5 6 7 8 9

EPE


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

1 2 3 4 5 6 7 8 9

ELgd Zero Corr

ELgd Medium Corr

ELgd High Corr

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

1 2 3 4 5 6 7 8 9

ELgd Zero Corr

ELgd Medium Corr

ELgd High Corr

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

1 2 3 4 5 6 7 8 9

ELgd Zero Corr

ELgd Medium Corr

ELgd High Corr




Basel II Regulation: the IMM

IMM approach is the most suited to properly take into account themarket risks related to a given counterparty’s portfolio

The building blocks of the IMM approach:

definition of a set of statistics for internal and regulatory purposesidentification of market factors and generation of future scenariospricing algorithms to price the contracts included in the booksaggregation rules to evaluate the effects of the risk mitigationagreementsa framework modelling the credit risk of the counterparties, thecorrelations amongst them and the correlation of counterparties withmarket risks to measure the “full” counterparty riskcalculation of the counterparty exposure measures, i.e. its risk profile,credit value adjustment, economic and regulatory capital




Exposure Measures in the IMM Method

Starting from exposure for each counterparty we can define otherstatistics and risk measures:

Expected exposure (EE )

EEi (tk) = E

[

max[Vi (tk), 0]

]

Expected positive exposure (EPE)

EPEi (tk) =1

tk − t0

k∑

j=1

EEi (tj)(tj − tj−1)

Effective Maturity M

M = min

[

ΣTk=1Df (tk) · EPEi (tk)

Σtk≤1Yk=1 Df (tk) · EPE (tk)

, 5

]




Regulatory Capital

The Regulatory Capital (RC ) is computed by means of the followingquantities

1 EPE2 The α factor (ratio of the EC calculated with full simulation, to the

EC calculated with a constant exposure equal to EPE )3 The Effective EPE (EEPE), which takes into account the roll-off risk

EEPE(tk) = max[EEPE(tk−1),EPE(tk)]

and it is actually the counterparty exposure measure which the RC

is computed upon

The α factor is calculated on a given time interval, but keptconstant otherwise. The period between two calculations depends onthe granularity and the time evolution of the portfolio




Regulatory Capital

For regulatory purposes, the capital can be determined as follows:

MCR = α× EEPE× RW × 8%

dove RW = 12.5× K and

K = LGD N

(

N−1(PDi ) + ri N−1(0.999)

√

1− r2i

)

− LGD × PDi

PDi = default probability for counterparty i

LGD = loss given default

ri = systemic risk load factor, also indicated by the Regulation equalto:

0.12×(1−e(−50×PD))/(1−e−50)+0.24×[1−(1−e(−50×PD))/(1−e−50)]

α has to be estimated according to an internal model approved by theSurveillance Authority (with a 1.20 minimum), otherwise it has to be setequal to 1.40.




Regulatory Capital and Wrong-Way Risk

We aim at introducing the wrong-way risk also in the counterparty creditVaR calculation.

The idea is still to have a “loan equivalent” of the exposure, so that wecan adjust the EPE or the EEPE measure to input into the regulatoryformula.

A possible approach to apply to the framework outlined above to computethe VaR is:

Compute the stressed (99.9% c.l.) PD according to the supervisoryformulae;Deduce which is the level of the default intensity consistent with thestressed PD;Compute the conditioned mean and variance of the risk factorsdetermining the exposure of the derivative contracts;Compute the new EPE with the conditioned mean and variance;Calculate the wrong-way-adjusted Economic Capital with the newEPE or the corresponding EEPE.




Counterparty Credit VaR: an Example

We analyze a swap portfolio expiring in 1 year, with monthly interest rateexchanges. The portfolio is net receiver fixed rate.

The counterparty has a PD = 2.94% in next year, and this is produced bya jump process whose intensity is a stocahstic process with parametersseen above. We test 3 versions: constant (P1), moderately decreasing(P2) and incresing (P3) notional.

According to the supervisory formula, the stressed PD at a 99.9% c.l. is22.34%.

Market rates and volatilities:

Months Eonia 1M Libor Libor Vol

1 0.79% 0.99% 34.00%2 0.82% 1.02% 34.50%3 0.85% 1.05% 35.00%4 0.87% 1.07% 35.50%5 0.90% 1.10% 36.00%6 0.92% 1.12% 36.50%7 0.95% 1.15% 37.00%8 0.97% 1.17% 37.50%9 1.00% 1.20% 38.00%10 1.02% 1.22% 38.50%11 1.05% 1.25% 39.00%12 1.07% 1.27% 39.50%

Months P1 P2 P3

1 100 100 452 100 95 503 100 90 554 100 85 605 100 80 656 100 75 707 100 70 758 100 65 809 100 60 8510 100 55 9011 100 50 9512 100 45 100





Portfolio P1 (Constant Notional).Tables below show the effective EPE fordifferent levels of correlation and averagefixed rate of the portfolio, and the ratio(α) with the 0-correlation case.Beside the figures show the EPE and theeffective EPE for a correlation of −10%,for the three fixed rate values indicatedin the tables.

OTM ATM ITMCorr Swap-Def.Intens. 0.85% 1.10% 1.35%

0% 0.0045 0.0426 0.2120-5% 0.0067 0.0610 0.2578-10% 0.0096 0.0881 0.3024


0% 1.00 1.00 1.00-5% 1.47 1.43 1.22-10% 2.12 2.07 1.43

EPE and EEPE (OTM,ATM,ITM)

-

0.05000

0.10000

0.15000

0.20000

0.25000

0.30000

0.35000

0.08 0.17 0.25 0.33 0.42 0.50 0.58 0.67 0.75 0.83 0.92

EPE

EPE WV

EEPE WV

EEPE

-

0.01000

0.02000

0.03000

0.04000

0.05000

0.06000

0.07000

0.08000

0.09000

0.10000

0.08 0.17 0.25 0.33 0.42 0.50 0.58 0.67 0.75 0.83 0.92

EPE

EPE WV

EEPE WV

EEPE

-

0.00200

0.00400

0.00600

0.00800

0.01000

0.01200

0.01400

0.08 0.17 0.25 0.33 0.42 0.50 0.58 0.67 0.75 0.83 0.92

EPE

EPE WV

EEPE WV

EEPE





Portfolio P2 (Decreasing Notional).Tables below show the effective EPE fordifferent levels of correlation and averagefixed rate of the portfolio, and the ratio(α) with the 0-correlation case.Beside the figures show the EPE and theeffective EPE for a correlation of −10%,for the three fixed rate values indicatedin the tables.


0% 0.0024 0.0282 0.1533-5% 0.0036 0.0428 0.1849-10% 0.0054 0.0642 0.2155


0% 1.00 1.00 1.00-5% 1.50 1.52 1.21-10% 2.23 2.27 1.41


-

0.05000

0.10000

0.15000

0.20000

0.25000

0.08 0.17 0.25 0.33 0.42 0.50 0.58 0.67 0.75 0.83 0.92

EPE

EPE WV

EEPE WV

EEPE

-

0.01000

0.02000

0.03000

0.04000

0.05000

0.06000

0.07000

0.08 0.17 0.25 0.33 0.42 0.50 0.58 0.67 0.75 0.83 0.92

EPE

EPE WV

EEPE WV

EEPE

-

0.00100

0.00200

0.00300

0.00400

0.00500

0.00600

0.00700

0.00800

0.08 0.17 0.25 0.33 0.42 0.50 0.58 0.67 0.75 0.83 0.92

EPE

EPE WV

EEPE WV

EEPE





Portfolio P3 (Increasing Notional).Tables below show the effective EPE fordifferent levels of correlation and averagefixed rate of the portfolio, and the ratio(α) with the 0-correlation case.Beside the figures show the EPE and theeffective EPE for a correlation of −10%,for the three fixed rate values indicatedin the tables.


0% 0.0052 0.0403 0.1677-5% 0.0076 0.0555 0.2029-10% 0.0107 0.0755 0.2371


0% 1.00 1.00 1.00-5% 1.44 1.38 1.21-10% 2.05 1.87 1.41


-

0.05000

0.10000

0.15000

0.20000

0.25000

0.08 0.17 0.25 0.33 0.42 0.50 0.58 0.67 0.75 0.83 0.92

EPE

EPE WV

EEPE WV

EEPE

-

0.01000

0.02000

0.03000

0.04000

0.05000

0.06000

0.07000

0.08000

0.08 0.17 0.25 0.33 0.42 0.50 0.58 0.67 0.75 0.83 0.92

EPE

EPE WV

EEPE WV

EEPE

-

0.00200

0.00400

0.00600

0.00800

0.01000

0.01200

0.01400

0.01600

0.08 0.17 0.25 0.33 0.42 0.50 0.58 0.67 0.75 0.83 0.92

EPE

EPE WV

EEPE WV

EEPE




Counterparty Credit VaR: Some Conclusions

The experiments we have presented above seem to show that, at least asfar as the wrong-way risk is concerned, the regulatory coefficient α is inmany cases too low. This is even more evident if consider the fact that wechose very small value for the correlation.

Actually, the stressed PD’s in a Merton (Gaussian copula) approach, suchas the regulatory one, are determined by the stressed level of a commonfactor affecting all the debtors and producing a correlation amongstdefaults.

The correlation with respect to this factor does not need to be the sameas the correlation between the single total PD and market risk factors (inour examples, the swap rates) and it is likely lower than the correlationwith the specific factors, although not always this is the case.

We have not explored in the analysis this extension, but it would bestraightforward to set the default intensity of each conuterparty as:

λD = λ

i + pi × λC

where λC is an intensity process common to all counterparties and λi isspecific to each counterparty.




Counterparty Credit VaR: Some Conclusions

The α (for the part due to the wrong-way) is also a function of thefeatures of the portfolio of contracts with a counterparty, in terms ofaverage fixed rate received (or paid) and evolution of the aggregatednotional over time. The main finding is that one α good for all occasionsis not a wise choice.

When considering also the correlation amongst the defaults of all thecounterparties, some diversification effects may be expected, so that the α

can actually be lower than 1.40 which is the standard level set byregulation if the bank is not able to compute a full deployed VaR.

To introduce a rich structure of correlation amongst counterparties’defaults, the regulatory formula has to be enhanced so as to include morefactors. Extensions of the Merton’s (Gaussian copula) approach areavailable and some of them can be effectively solved in very good analyticapproximations.

It is important to check which is the real contribution of the correlationsto the Economic Capital for management purposes. For regulatorypurposes, the proposed α = 1.40 seems to be very favorable to banks tosave allocated capital.



Liquidity Risk Pricing in OTC Derivatives

Index






Collateral and Margin Agreements

Collateral agreement is a contract between two counterparties thatrequires one or both counterparties to post collateral (typically cash orhigh quality bonds) under certain conditions.

Margin agreement is a legally binding collateral agreement with specificrules for posting collateral, which include:

1 Minimum transfer amount: defines the minimum amount ofcollateral that can be exchanged. If the exposure entails a collateralposting below the minimum, amount, no collateral is provided;

2 A threshold, defined for one (unilateral agreement) or both (bilateralagreement) counterparties. If the difference between the net portfoliovalue and already posted collateral exceeds the threshold, thecounterparty must provide collateral sufficient to cover this excess(subject to minimum transfer amount);

3 Frequency: defines the periodicity of the exposure calculation and ofthe determination of the collateral to post.

The terms of the rules depend mainly on the credit qualities of thecounterparties involved.




Index






Pricing OTC Derivatives with CSA

In a very general fashion, the price at time 0 of a derivatives contractwhich is not subject to counterparty risk is:

V0 = EQ

[

e−

∫T0 rsdsVT

]

where

VT is the terminal pay-off of the contract;rt is the (possibly time dependent) risk-free interest rate.

When counterpaty risk is considered, then we have to include the so calledCVA (the expected losses we suffer when on default of the counterparty)the and DVA (the expected losses the counterparty suffers on our default):

VCCP0 = E

Q

[

e−

∫T0 rsdsVT

]

− CVA+DVA

The terminal value of the contract is still discounted ad the risk-free ratert , but then the price is adjusted with the net effect due to the lossesupon default of the two counterparties involved in the trade.





Assume now we have a CSA agreement operating between the twocounterparties. The CSA provides for a daily margining mechanism of thefull variation of the NPV (nowadays a very common form of the CSA).The party that owns a positive balance on the collateral account(corresponding to a positive NPV of the contract) pays the rate ct to theother party.

The pricing of the contract can be now be operated by excluding thedefault risk (there is still a very small residual risk between two dailymargining).

It can be shown that the pricing formula is very similar to the standardcase we have seen above, but with the collateral rate ct replacing therisk-free rate rt .:

VCSA0 = E

Q

[

e−

∫T0 csdsVT

]

(1)





This result is very convenient, since we have a well defined rate that hasto be paid on the collateral balance (set within the contract), whereas therisk-free rate is very difficult to determine in the current marketenvironment (it used to be the Libor in the interbank market).

Usually the daily margined CSA agreements set the remuneration of thecollateral at the EONIA for contracts in euro (or some equivalent OIS ratefor other currencies). EONIA (OIS) rates can be considered the bestapproximation of a risk-free rate.

Nevertheless there is still one assumption that is made when deriving thepricing formula with the CSA:

The rate at which the bank can lend money is the same of the one itcan borrow money.

This assumption can be easily relaxed when we price contracts whoseNPV can be always either positive or negative (e.g.: a long or a shortposition on an option contract).

When the NPV of the contract can switch from positive to negativeand/or from negative to positive (e.g.: forward and swap contracts) thenrelaxing the assumption is trickier.





Assume we have a contract whose value during the life of the contract atany time 0 < t < T , Vt can be positive or negative.

We also assume that the bank can invest cash at a risk-free rate equal tothe collateral rate rt = ct , but it has a funding spread ft when borrowingmoney over a short period, so that the total funding cost is rt + ft .

When considering the funding spread in the pricing of a collateralizedderivatives contract, it can be shown that the valuation equation can bewritten as:

VCSA0 = E

Q

[

e−

∫T0 cu−fu1{Vu<0}duVT

]

= EQ

[

e−

∫T0 cuduVT

]

+ LVA (2)

where

LVA = EQ

[∫ T

0

e−

∫s0 cudu min(Vs(0), 0)fsds

]

(3)





Equation (3) can be computed numerically with a good degree ofapproximation: it accounts for the funding cost that the bank has to paywhen financing the cash injection in the collateral account, expressed hasa spread ft over the risk-free rate rt = ct , which on the contrary is the ratethe bank can invest at. We name this quantity Liquidity Value Adjustment

(LVA)

As an example, when we price a (K -fix rate receiver) swap contractstarting in ti and ending in T = tN , the minimum value of the contract ismin(Vt , 0) = −min(−Vt , 0) = −max(Si,n − K , 0).

Assume we divide the period between the evaluation time t0 = 0 and theexpiry in N intervals. The LVA can be written as:

LVASwp = −N∑

i=1

Pay(ti ; ti ,T ,K )fti∆ti (4)

where Pay is the value of a payer swaption struck at K , expiring in ti andwritten on a swap starting in ti and maturing in T .




A Practical Example

We show an example, assuming the following market data for interestrates:

Time Eonia Fwd Spread Fwd Libor

0 0.75% 0.65% 1.40%0.5 0.75% 0.64% 1.39%1 1.75% 0.64% 2.39%1.5 2.00% 0.63% 2.63%2 2.25% 0.63% 2.88%2.5 2.37% 0.62% 2.99%3 2.50% 0.61% 3.11%3.5 2.65% 0.61% 3.26%4 2.75% 0.60% 3.35%4.5 2.87% 0.60% 3.47%5 3.00% 0.59% 3.59%5.5 3.10% 0.59% 3.69%6 3.20% 0.58% 3.78%6.5 3.30% 0.58% 3.88%7 3.40% 0.57% 3.97%7.5 3.50% 0.57% 4.07%8 3.60% 0.56% 4.16%8.5 3.67% 0.56% 4.23%9 3.75% 0.55% 4.30%9.5 3.82% 0.55% 4.37%10 3.90% 0.54% 4.44%

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

3.50%

4.00%

4.50%

5.00%

0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9

Euribor Fw d

Eonia Fw d




A Practical Example

Market data for caps&floors and swaptions volatilities are:

Caps&FloorsExpiry Volatility0.5 30.00%1 40.00%1.5 45.00%2 40.00%2.5 35.00%3 32.00%3.5 31.00%4 30.00%4.5 29.50%5 29.00%5.5 28.50%6 28.00%6.5 27.50%7 27.00%7.5 26.50%8 26.00%8.5 25.50%9 25.50%9.5 25.50%10 25.50%

SwaptionsExpiry Tenor Volatility0.5 9.5 27.95%1 9 28.00%1.5 8.5 27.69%2 8 27.09%2.5 7.5 26.61%3 7 26.32%3.5 6.5 26.16%4 6 26.02%4.5 5.5 25.90%5 5 25.79%5.5 4.5 25.68%6 4 25.57%6.5 3.5 25.46%7 3 25.37%7.5 2.5 25.28%8 2 25.22%8.5 1.5 25.21%9 1 25.34%9.5 0.5 25.50%10 0




A Practical Example: Collateralized Swap

We price under a CSA agreement areceiver swap whereby we we pay theLibor fixing semi-annually (set at theprevious payment date) and we re-ceive the fixed rate annually. Withmarket data shown above, the fairrate can be easily calculated (we areusing the new market standard ap-proach to employ the EONIA/OIScurve for discounting and the 6M Li-bor curve to project forward rates).We assume also that we have to paya funding spread of 15bps over theEONIA/OIS curve. This is applied tothe ENE plotted beside.

LVA -0.0512%Fair Swap rate 3.3020%Swap Rate + Coll. Fund 3.3079%Difference 0.0059%

ENE

(7.0000)

(6.0000)

(5.0000)

(4.0000)

(3.0000)

(2.0000)

(1.0000)

-

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5




A Practical Example: Collateralized Swap

We may be interested in calculat-ing the impact of the liquidity of acollateralized swap with respect to amore conservative measure than theENE, similarly to what happens inthe counterparty risk management.We choose the Potential Future Ex-posure, which is the expected nega-tive NPV of the swap at a given levelof confidence, set in this example atthe 99% and computed with marketvolatilities.The funding spread is still 15bps overthe EONIA/OIS curve. The PotentialFuture Exposure (blue line), and theENE (purple line, same as before) forcomparison, are plotted beside.

LVA -0.2156%Fair Swap rate 3.3020%Swap Rate + Coll. Fund 3.3264%Difference 0.0244%

(30.0000)

(25.0000)

(20.0000)

(15.0000)

(10.0000)

(5.0000)

-

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5

PFE

ENE




About Iason

Iason is a company created by market practitioners, financial quants and programmerswith valuable experience achieved in dealing rooms of financial institutions.Iason offers a unique blend of skills and expertise in the understanding of financialmarkets, in the pricing of complex financial instruments and in the measuring and themanagement of banking risks. The company’s structure is very flexible and grants afully bespoke service to our Clients.Iason believes that the ability to develop new quantitative finance approaches throughresearch as well as to apply those approaches in practice, is critical to innovation in riskmanagement and derivatives pricing. It brings into all the areas of the risk managementa new and fresh approach based on the balance between rigour and efficiency Iason’speople aimed at when working in the dealing rooms.Besides tailor made services, Iason offers software applications to calculate and monitorcredit VaR and conterparty VaR, fund transfer pricing and loan pricing, liquidity-at-risk.

c©Iason - 2011

This is a Iason’s creation.

The ideas and the model frameworks described in this presentation are the fruit of the intellectual efforts and of the skills of the peopleworking in Iason. You may not reproduce or transmit any part of this document in any form or by any means, electronic or mechanical,including photocopying and recording, for any purpose without the express written permission of Iason ltd.


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