CVA+FFVA+MV+Perspective.4

CVA and FVACVA and FVA from the Model Validation Perspectivefrom the Model Validation Perspective

Marcus EvansMarcus Evans

7th Annual Pricing Model Validation Workshop

Presented by

Raphael Albrecht

London, September 11‐12, 2014

Raphael AlbrechtIndependent Validation Unit

Barclays

1

Disclaimer

• All of the opinions expressed in this presentation are solely those of the speaker and should under no circumstances be taken to represent any spea e a d s ou d u de o c cu sta ces be ta e to ep ese t a ybank, regulatory agency or other institution, financial or otherwise.

• Any views regarding practice of model validation are personal opinions of th k d d t il l ith ti l i t lthe speaker and do not necessarily comply with any particular internal or regulatory guidance – please consider those as idealised statements of personal opinion

• All models discussed here are non‐proprietary #, all examples are hypothetical

# In the sense that they documentation is available in the public domain, see list of references at the end. This is notwithstanding any copyrights pertaining to those to any publications quoted

2Raphael Albrecht on CVA and FVA ‐ the Model Validation Perspective

Model Risk in Pricing Models

• Model Risk is the potential for adverse consequences (financial or other) from decisions based on incorrect or misused model ( a c a o ot e ) o dec s o s based o co ect o sused odeoutputs

• Can arise from fundamental model flaws leading to inaccurate outputs, i i l t ti i t/i i terrors in implementation or incorrect/inappropriate use

• Model Validation can help mitigating model risk in pricing models by identifying and documenting Model Assumptions and Limitationsy g g p


FVA in the Presence of Counterparty Credit Risk – EconomicsBenchmark Case: risk‐free exposure = fully collateralised tradef p f yMissing /excess collateral comes at a funding cost/benefit

Simple example: counterparty exposure = vanilla IR Swap – cases of increasing complexity w.r.t. default risk:

1. trading desk enters into a swap with an external counterparty that “cannot” default

– Funding Swap: just a simple funding contract between trading desk and CVA Trading Desk, whereby CVA Trading Desk agrees to provide the trading desk with borrowing/lending at the Bank's funding rate in exchange for the prevailing risk‐free rate (the value of this contract is called the “FVA")

2. trading desk enters into a swap with a counterparty that can default this has two impacts:g p p y p

– firstly, a new contract (“Contingent CDS = CCDS”) between trading desk and CVA Trading Desk, whereby CVA Trading Desk provides the trading desk with insurance against counterparty default (the value of this contract is called the "CVA“)

– secondly, the funding contract must be modified to be cancelled in the event of counterparty default (the value of thissecondly, the funding contract must be modified to be cancelled in the event of counterparty default (the value of this contract is still called the “FVA", but the funding swap is now an extinguisher in the event of counterparty default, since no funding requirement after the counterparty defaults)

3. trading desk enters into a swap with a counterparty that can default and has a collateral agreement this has three impacts:

i. the funding contract is now applicable only to the excess over and above the collateral value (since the collateral value can be used as a substitute for funding) (still called “FVA" and still extinguished on counterparty default)

ii. the counterparty default contract is also now applicable only to the excess over and above the collateral value (since the collateral value can be used to offset any loss from counterparty default) (still called "CVA" and still extinguished y p y ) ( gon counterparty default)

iii. an additional amount to cover any benefit [or cost] of the collateral carry that accrues to CVA Trading Desk because the counterparty is providing collateral (called the "LVA", which also extinguishes in the event of counterparty default since collateral will be sold/settled in this event)

Note that this is all couched in terms of what service or contract the CVA Trading Desk desk must supply to the trading desk in order to leave the trading desk free from counterparty default risk and funding spread risk (and actually also collateral spreadrisk). 4Raphael Albrecht on CVA and FVA ‐ the Model Validation Perspective

Cash Flows between CVA Trading Desk and Derivatives Desk on uncollateralised Exposure

Recap: CVA is the upfront cost to be paid by a trading desk to receive insurance against counterparty default, FVA is the upfront a trading desk would have to pay/or would receive in return for borrowing and lending at the risk‐free rate ri until maturity of the portfolio, or counterparty default (whichever happens earlier).

1. Trade initiation t=0:

: Value of a derivatives portfolio to Bank (incl. credit and funding) CVA FVA

: Upfront fee from desk to CVA desk : = day1 for CVA desk

t̂ t t

t t t

V V UUU U

= +

= +−

2. Trade is alive at 0 < t <T :

0

0

0: Upfront fee from desk to CVA desk, : day1 for CVA desk

V : credit&funding-risk-free day1 valuation o

U U

0̂f the derivative, = day1 for deskV

trading desk is either required to hold funds against contingent liability or receives funding benefit on (use as collateral) t tV V− +− CVA desk (CRT) pays /receives funding spread on risk-free deri−

( )( ) :

( ) ( ) s ( ) s ( )

vative value, assumi

( s

ng

)

( )

= : ) (f i

f i t f t f ft t

r t

t t t t t

r t

r r V dt V dt V dt V dt− +

≥

− − −−=

3. Cpty defaults , while Bank alive: 0 < t=τC < T0 CVA desk to trader 0 trader to CVA desk≥ ≤

or unwinds by paying t- desk either receives from CVA desk

CVA d

o Cpty

- receives from Cptysket t

C t

V V

R V

+ −

+

−

×

4. Bank defaults , while Cpty alive: 0 < t=τB < T

receives from Cpty or pays ) to Cpty- desk either (t B tV R V −+ × −

p y- funding swap extinguishes

C t

receives from Cpty or pays ) to Cpty- f desk

unding swap extinguis ei

hether (

s t B tV R V


Joint Valuation of Credit and Funding

Combined value of the hedges (in a simple model assuming deterministic interest , funding rates and credit spreads independent from the exposure profiles)

( ) ( )Ut = (CVAt = Contingent Credit Default Swap CCDS) + (FVAt = swap on notional = exposure profile at t)

[ ]

( )

( )

CVA (1 ) ( ) ( , ) ( , ( ))

FFVA ( ) ( , ) ( , ( ))

F C

f C

T

C C r s ttT

F r tt

R u D t u V u S u du

s u D t u V u S u du

t

t

λ

λ

λ ++ +

+=

⎡ ⎤= − − ⎣ ⎦

−

∫∫

E

E [ ]:

( ) : risk-free value of the derivatives portfolio

s ( ) ( ) ( ) : funding sp

:

read of Ban

haza d

k

r

f Ct

f f i

c

t

t

trtr

Vλ

= −∫

rate for Cpty ( )

( , ) : : Survival Probability of the Cpty from to

: Recovery Rate of the Counterparty

Ds

Ct

C

u du

C

t s e t s

R

λ

λ

−∫=

FVA can be split into a funding benefit adjustment (FBA) and the funding cost adjustment (FCA)

FVA FCA( ) FBA( )

FCA ( ) ( , ) ( , ( )) 0

FBA

( )

( )

( ) ( ) ( ) ( ( )) 0

f C

T

F r tt

T

t t


s u D t u V

t

S u du

t

t u

λ+

+

−

+

⎡ ⎤− ≤⎣ ⎦

⎡ ⎤−

=

=

= ≥⎣ ⎦

∫∫

E

EFBA( ) ( ) ( , ) ( , ( )) 0f CF r tt

s u D t u V S u dut uλ+ ⎡ ⎤= ≥⎣ ⎦∫ E


Overlap between FBA and DVA

The FVA can be split into a funding benefit adjustment (FBA) and the funding cost adjustment (FCA)

FVA FCA( ) FBA( )

FCA ( ) ( ) ( ( )) 0

( )

( )T

t t


t

t +

+

⎡ ⎤− ≤⎣ ⎦

=

= ∫ E

(unilateral) Debit Value Adjustment (DVA) is the CVA calculated from the other parties’ side

FCA ( ) ( , ) ( , ( )) 0

FBA

( )

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

f C

f C

F r ttT

F r tt


s u D t u V S u du

t

t u

λ

λ

+

−+

⎡ ⎤ ≤⎣ ⎦

⎡ ⎤−

=

= ≥⎣ ⎦

∫∫

E

E

1. Replace Counterparty C by Bank B

2. Replace exposure by reverse exposure V → (‐V) i.e. as seen from Cpty

(1 ) ( )DV ( , ) ( , ( )) 0A( )f B

T

B B r ttR u D t u V u S u dut λλ −

+ ⎡ ⎤− − ≥⎣ ⎦= ∫ E

Lemma: In the absence of bond‐cds‐basis for Bank: DVA (t) = FBA (t) , for all t. Proof:

(1 ) ( )

(1 ) ( ) ( ) (

Assume: ( ) , then we get:

DVA( ) ( )) 0

f B B

T

s R t

R u D t u V u S

t

t u du

λ

λ −

−

⎡ ⎤ ≥⎣ ⎦

=

= ∫ E

DVA arises naturally in the context of Bilateral CVA ie when both Bank and Cpty can default:

(1 ) ( ) ( , ) ( ,DVA( ) ( )) 0

( ) ( , ) ( ,

(

( ))

FBA ) . . .

f B

f C

B B r ttT

f r tt

R u D t u V u St

t

u du

s u D t u V u S u d

q

u

e d

λ

λ

λ +

−+

⎡ ⎤− − ≥⎣ ⎦

⎡ ⎤− ⎣= ⎦

=

=

∫∫

E

E

DVA arises naturally in the context of Bilateral CVA, ie. when both Bank and Cpty can default:

BI MU MU

MU

CVA CVA ( ) DVA ( ) : Bilateral CVA

(1 ) ( ) ( , ) ( , ( )) : Modified Unilateral CVA

( )

CVA ( )f CB

T

r tt C

T

C

t t t

R u D t u V u S u dut λ λλ ++

+

+

⎡ ⎤− − ⎣ ⎦

=

= ∫ E

MU (1 ) ( ) ( , ) ( , ( )) : Modified UnilaDV erA ( ) tCf Br

T

B B ttR u D t u V u S u dt uλλλ −

+ + ⎡ ⎤−= − ⎣ ⎦∫ E al DVA


Extension to Collateral

• With the assumption that the collateral can be fully rehypothecated (i.e it can be re‐used as collateral), the extension of this model to the case when collateral is posted/received is straightforward.

– Non‐rehypothecable collateral should be priced as a 1 way CSA.

– Examples include assets for which a repo market does not exist or is not accessible to us, pledged collateral, collateral posted to a 3rd party custodian, etc.

• The total derivative value is given by V+U, where V is the counterparty‐risk‐free value– without any collateral posted nor received.

• The adjustment U is now split between

1. the credit valuation adjustment (CVA)

2. the funding value adjustment (FVA),

3. the collateral value adjustment (COLVA), sometimes referred to as the liquidity value adjustment(LVA).

ˆ ( ) ( )( )V V t U tt +=

( )

[ ]

CVA( ) (1 ) ( ) ( , ) ( , ( )) ( , ( ))

FVA( ) ( ) ( ) ( ( ) ( ( ))

( ) CVA( ) FVA( ) LVA( )

f C

T

C C r tt

T

t R u D t u V u S u u S u du

t D t V S S d

U t t t t

λλ φ

φ

++

⎡ ⎤= − − −⎣

= +

⎦

+

∫

∫

E

E [ ]

[ ]

FVA( ) ( ) ( , ) ( , ( ) ( , ( ))

LVA( ) ( ) ( , ) ( , ( ))

f C

f C

f r tt

T

r tt

t s u D t u V u S u u S u du

t s u D t u u S u du

λ

φ λ

φ

φ

+

+

=

=

− −

−

∫∫

E

E


Multi‐Currency Funding and Collateral

At each point in time we need to balance in‐ and outflows per currency, so it may be necessary to e.g. use collateral posted to us in EUR to reduce the need for USD unsecured funding. This leads to the following modifications to the FVA formula:modifications to the FVA formula:

• We get one FVA per funding currency where the funding spread sF,I is the funding spread for that currency and the amount to be funded depends on the amount of trade MTM and collateral we allocate to that currency.

• Whenever we allocate trade MTM or collateral in one currency to funding in another currency we need to put on an FX‐hedge in the form of a cross‐currency swap. The cross currency swap basis spreads lead to further valuation adjustments.

More specifically we need the following extinguishable cross currency swaps:More specifically we need the following extinguishable cross currency swaps:

• Trade currency to USD. By trade currency we mean the collateral currency for the trade if it were done on a fully collateralised basis on e.g. LCH. It follows that xccy swaps and other FX trades are considered to be USD trades as they are collareralised in USD.

• Counterparty collateral currency to USD.

• USD to each funding currency according to the funding currency allocation rule.


Approach to Pricing CVA & FVA via Hedging and Replication

One possible way of approaching the pricing is by setting up a self‐financing replication portfolio containing the following assets such that movements in the underlying assets (i.e., market risks) and jumps at the g y g ( , ) j pcounterparty default (i.e., counterparty credit risk) are all hedged:

• Underlying assets (e.g. shares) Si denominated in currency i. These are financed via separate repo‐accounts denominated in the same currency as the asset.

• Foreign risk‐free zero coupon bonds Pi with corresponding FX rates Xi. These are financed via separate domestic repo‐accounts.

• A domestic risky counterparty zero coupon zero recovery bond , financed with a domestic repo account. The currency of the counterparty risky bond has no impact on the final result, not would using CDS rather y p y y p , gthan bonds.

• Collateral received or paid in each currency (assumed re‐hypothecable for now).

To run this hedging strategy and fund the collateral the issuer can issue or repurchase own risky bonds PF,idenominated in currency i and all FX risks introduced by foreign currency funding are hedged


Hypothetical Funding Strategy

The hypothetical funding strategy corresponding to the current FVA calculation looks as follows:

• Financing constraint: The total value of the derivative plus the value of the different bonds must equal the collateral posted minus the collateral received.

• Posted collateral is funded in the currency in the collateral currency and received collateral is invested inPosted collateral is funded in the currency in the collateral currency and received collateral is invested in the collateral currency.

• Each trade is assigned a trade currency (see below for how this is done), and each trade currency is assigned a funding currency. One strategy would be to fund all trades in the same currency regardless of trade currency Alternatively one could for example say that all trades are funded in their own tradetrade currency. Alternatively one could for example say that all trades are funded in their own trade currency.

• The adjustments themselves (FC CVA and MC FVA) are funded in a single currency.


Trade and Collateral Currencies

First, we need to decide what trade currency to assign to a trade. The rule used is that single currency trades have the trade currency equal to the cashflow currency (e.g. a EUR vanilla swap is a EUR trade), whilst FX and other multi‐currency trades are considered to be USD trades The key to understanding the logic behind the rules is to consider thecurrency trades are considered to be USD trades. The key to understanding the logic behind the rules is to consider the trade on a collateralised basis (e.g. LCH). In this case the trade currency is simply the currency of the collateral account for the trade. On most clearing houses single currency trades in majors are collateralised in the cashflow currency and we make the simplification that this applies to all currencies. Furthermore, cross currency swaps and other FX trades are collateralised in USD which is why we consider them to be USD tradesare collateralised in USD which is why we consider them to be USD trades.

If information about the eligible collateral currencies is not available one can for example assume that collateral is posted according to the following rule referred to as the ``proportional LCH rule":posted according to the following rule, referred to as the proportional LCH rule :

where V and ф are the total trade and collateral values (in domestic currency) respectively and V is the trade mark‐to‐

i iVVφφ =

where V and ф are the total trade and collateral values (in domestic currency) respectively and Vi is the trade mark‐to‐market of the currency trades. This rule behaves in the following way:

• No CSA: zero collateral in all currencies i

• Two‐way CSA: фi =Vi, i.e. LCH collateral so the counterparty collateral satisfies the funding needs for each y фi i, p y gcurrency in isolation without any need for internal cross‐currency swapping.

• One‐way CSA: like No‐CSA when not posting and like a two‐way CSA when posting.

Note that this assumption plus the assumption that collateral is kept in its own currency implies that no cross‐currency basis adjustments need to be applied to the collateral due to FX hedging.


Multi‐Currency FVA ‐ Details

As in the single currency case the total derivative value is given by where V is the counterparty default risk‐free value (i.e., discounting at some reference rate), FC CVA the funding‐consistent CVA, MC FVA the Multi‐Currency FVA and LVA the liquidity valuation adjustment.

ˆ ,V V FCCVA MCFVA LVA= + + +

The FC CVA is given by

,FCCVA( , , ) (1 ) ( ) ( , ) ( ( , ( ), ( )) ( , ( ), ( )))

F A C

T

C C r ttt s x R u D t u V u S u X u u S u X u duλλ φ +

+ ⎡ ⎤= − − −⎣ ⎦∫ E

which is a unilateral CVA discounted at the effective adjustment funding rate (see Table 1 for its definition).

The LVA (aka COLVA – Collateral Value Adjustment)

t∫

The simplest case is when all the trades are denominated and funded in a single currency ''i'' and all collateral is also posted/received in this currency. In this case the MC FVA is given by

[ ],,( , , ) ( ) ( , ) ( ) ( , ( ), ( ))

F A C

T

i r i iti

LVA t s x s u D t u X u u S u X u duφ λ φ+= −∑∫ E

is also posted/received in this currency. In this case the MC FVA is given by

,, , ,MC FVA( , , ) ( ) ( , ) ( )( ( , ( ), ( )) ( , ( ), ( )))F A C

T

F i r i f i f itt s x s u D t u X u V u S u X u u S u X u duλ φ+ ⎡ ⎤= −⎣ ⎦∫ E


MC FVA ‐Multi currency with cross‐currency funding In the general case where there are trades that are not funded in their own currency and/or collateral that is allocated to a currency different from the collateral currency the MC FVA becomes more complex. The reason for this is that each currency conversion needs to be hedged (using a xccy swap) resulting in extra adjustment terms:

MC FVA( )t S X =

,, , ,

MC FVA( , , )

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

( ) ( ) ( )( ( ( ) ( )) ( ( ) ( )))

F A C

T

F i r i f i f iti

T

t S X

s u D t u X u V u S u X u u S u X u du


λ φ

φ

+ ⎡ ⎤− −⎣ ⎦

⎡ ⎤− −⎣ ⎦

∑∫

∑∫

E

E,

,

, , ,

,

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

( ) ( , ) ( )( (

F A C

F A C

B i r i f i f iti

T

B i r i iti


s u D t u X u V u

λ

λ

φ+

+

⎡ ⎤⎣ ⎦

+

∑∫

∑∫

E

E[ ], ( ), ( )) ( , ( ), ( )))iS u X u u S u X u duφ−

line one: The adjustment is the cost of funding the currency i shortfall or gain (similar to a single currency FVA).

line two: Cost of hedging the FX‐risk from conversion from the domestic currency to the funding currency i

line three: Cost of hedging the FX‐risk i from the conversion from trade/collateral currency i to domestic currency.

V and ф are the total trade and collateral values in base currency (USD)

Vi and фi are the total trade and collateral values in trade currency i

Vf,i and фf,i are the total trade and collateral values allocated to funding currency i

sF,I Instantaneous currency i funding spread of Bank.

s B,I Instantaneous basis spread for currency i

I t t b i d f ir F,A = Instantaneous basis spread for currency i

Dx(t,u) Discount factor from t to u using rate x (which may itself be dependent on u)14Raphael Albrecht on CVA and FVA ‐ the Model

Validation Perspective

Notation Index


CVA+FVA Model Validation: Theoretical Consistency – Model Assumptions & Limitationsp

• What is the Model used for (Pricing/Reserving/Risk)?

• Is CVA approach (unilateral/modified/bilateral) consistent with FVA?• Is CVA approach (unilateral/modified/bilateral) consistent with FVA?

• Is there any double‐counting? (candidate: DVA vs. FBA)

• A&L in risk factor dynamicsy– How many risk factors are captured? Any missing? Stochastic or deterministic?

– What are the underlying volatility and correlation assumptions?

• Are discount factors applied specifically and consistently?(“risk‐free”/credit‐risky, funding/collateral ‐ aware)

• Are CSA provisions adequately reflected?Are CSA provisions adequately reflected? – Netting & clean‐up

– Rating‐dependent collateral thresholds or ATEs

– Independent Amounts

– For Multicurrency CSAs: is actual collateral considered or are there any ad‐hoc assumptions in place about distribution of collateral by currencyassumptions in place about distribution of collateral by currency


CVA+FVA Model Validation: Indentify Implementation – related Assumptions & Limitationsy p p

Particular Points in CVA+FVA:

• Implementation: MC/Analytic/Hybrid?• Implementation: MC/Analytic/Hybrid?– Limits on Implementation Testing

• Exposure Calculation: – Risk Factor Simulation

• Calibration: Risk‐neutral or historic, or none at all?

T d R l ti– Trade Revaluation• FO Pricer or bespoke?

• Default Time Intensityy– Consistent with standard credit curve build?

• LGDs– Trader marks or model‐based?

• Bespoke Rating Transition Model?


CVA+FVA Model Validation: Implementation Testing – General Principles

• To verify the model is implemented in agreement with documentation• Standard procedure: comparison to independent implementation (replica) p p p p ( p )• applied to all key pricing elements individually

– Are different pricer versions used for various sub‐tasks? • Example: Pricing on PDE calibration on analytic approximation• Example: Pricing on PDE, calibration on analytic approximation

– If implementation is analytical, agreement should be to within numerical accuracy

– Otherwise it should be 100% clear that diffs are “small” and “random” (i.e. no systematic bias) in worst case argue by showing regressionsystematic bias), in worst case argue by showing regression

• Benchmarking: if model is too complex to replicate (ex: general PDE solver) or if natural benchmark models are readily available and were previously reviewedreviewed

• Example: Assuming FVA and CVA share exposure calculation and funding is deterministic (i.p. independent of exposure profile)

V if FVA d CVA l l ti i t t– Verify FVA and CVA calculations are consistent

– Verify FVA output from given exposure profile and input discount factors and recovery rates


CVA+FVA Model Validation: Model and Calibration Analysis

• To verify the model behaviour is as expected from theory• Impact on model and calibration when varying important inputImpact on model and calibration when varying important input

parameters– Input parameters are market data (e.g., funding , credit curves, vols, etc.) or

model parameters (ex: IR‐CR correlation)model parameters (ex: IR CR correlation)

• Stress tests – to see if and when the model breaks down/fails to calibrate when wide ranges of parameters are used

Note that it is not the primary intention of MV testing to determine precisely– Note that it is not the primary intention of MV testing to determine precisely when the model will brake (this is generally impossible!)

– only if we see that it brakes we would like to investigate why this is the case and determine if in that particular case this can be expected or notand determine if in that particular case this can be expected or not

• ex. when a no‐arbitrage bound is violated

– depending on the specific model/calibration in question it may not be possible/necessary to perform all of the above testspossible/necessary to perform all of the above tests

– Test trade / portfolio: should be realistic, but not too complex, such that expected behaviour can be figured out


CVA+FVA Model Validation: Testing related to Limitations and Approximations

• To investigate the impact of model‐ and implementation‐related assumptions and limitations on pricing and risk for representative trades assu pt o s a d tat o s o p c g a d s o ep ese tat e t ades

Examples:

• Impact of counter‐intuitive model assumptions (ex. negative hazard rates) on pricing and sensitivities

• Convergence of PV and sensitivities with number of MC runs, number of grid points in a latticegrid points in a lattice

• Impact of using a simplified analytic pricer for calibration on pricing accuracy

• Impact of any simplifications in scenario‐based re‐valuation within the expected exposure calculation


CVA+FVA Model Validation: Additional Testing

Fixed slot for any testing not fitting into any of the standard categories above

Possible tests would include:Possible tests would include:

• Thought experiments – Example: ad‐hoc scenarios to show the effect of any missing correlations

• CR – IR – Exposure – LGD

• Comparison to alternative models (usually ad‐hoc) l f l h d– Example: Impact of IR Vol on CVA+FVA with deterministic Rates

• Consistency among various payoffs– Example: Bilateral CVA ( no CDS‐bond‐basis) single currency FVAExample: Bilateral CVA ( no CDS bond basis) single currency FVA


References

CVA and FVA modelling discussed here is based on

[1] C. Burgard and M. Kjaer. Partial differential equation representations of derivatives with counterparty risk and funding costs. The Journal of Credit Risk, Vol. 7, No. 3, 1‐19, 2011.

[2] C. Burgard, M. Kjaer. In the balance, Risk, November, 72‐75, 2011.[ ] g , j , , , ,

[3] C. Burgard, M. Kjaer. Funding Costs, Funding Strategies, Risk, 82‐87, Dec 2013.

[4] C. Burgard, Kjaer, M., The FVA Debate, http://ssrn.com/abstract=2157634

[5] C B rgard Kjaer M CVA and FVA ith f nding a are close o ts Working Paper[5] C. Burgard, Kjaer, M., CVA and FVA with funding aware close outs, Working Paper, http://ssrn.com/abstract=2157631

S h // / f / d f / f f l i d lSee http://www.cvasource.com/references/cvadvafva/ for references to alternative models

Author: [email protected]


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CVA+FFVA+MV+Perspective.4