XVAs Series: CVA - London Quantslondonquants.com/wp-content/uploads/2015/01/CVAandRels.pdfUnilateral CVA and Simpliﬁcations To get the standard CVA formula that is usually implemented,

XVAs Series: CVA

An Introduction

Marıa R. Nogueiras ([email protected])

March 2016

London Quants

XVAs

Before the crisis: the price of derivatives was computed evaluating the expected returnand risk of the underlying asset (at a trade level)

After the crisis: “New” Risks that not negiglible anymore are taken into account inpricing, risk managemant and regulation (at a counterparty, netting set, and/or portfoliolevel)

Counterparty Credit risk (CVA)

Own Credit risk (DVA)

Funding Cost/Benefit (FVA)

Capital Costs (KVA)

Initial Margin on CCPs, Initial Margin on new CSA (MVA)

. . .

. . . and Friends Date: 24 de marzo de 2016 2

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Counterparty Risk

The counterparty credit risk is defined as the risk that the counterparty to a tran-saction could default before the final settlement of the transactions cash flows. Aneconomic loss would occur if the transactions or portfolio of transactions with thecounterparty has a positive economic value at the time of default. Basel II.

If we have an open position with a counterparty C, with value V ; if the counterpartydefaults we may loose the exposure E:

E = max(V, 0)

In fact, we may loose a percentage of the position value, the Loss Given Default,

LGD

Similarly, we recuperate a percentage of the portfolio value, the recovery rate R,

LGD = 1−R


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In Risk contexts we are used to answer the question (concept of Credit VAR):

How much can I loose of this portfolio, within one year, at a some confidence level,due to default risk and exposure?

LGD × EAD × PD

where EAD denotes the exposure at default and the PD is the probability of default.

Under Basel II, the risk of counterparty default and credit migration risk were ad-dressed but mark-to-market losses due to credit valuation adjustments (CVA) werenot. During the financial crisis, however, roughly two-thirds of losses attributed tocounterparty credit risk were due to CVA losses and only about one-third were dueto actual defaults. Basel Committee on Banking Supervision, BIS (2011).

Counterparty Risk was measured before the crisis, but not for (risk-neutral) pricing. Theperception of counterparty risk (probability of default) was low.


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From [B-11-slides]


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CVA Definition

Credit Valuation Adjustment or CVA is

The market price of credit/counterparty risk on a financial instrument that is marked-to-market

The price of a CDS on the counterparty with a contingent nominal equal to theexposure.

The reduction in price we ask to our counterparty C for the fact that C may default

. . .

Market convection is:

CV A = V (default free)− V (risky)


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Credit Risk versus Counterparty Risk

Both terms are referred to the default risk.

In general, we talk about “credit risk” when considering loans, bonds, . . . (or creditderivatives as CDSs, CDOs, etc)

For a loan, for instance, the exposure/value is (almost) certain/deterministic and it isalways positive

E = max(V, 0) = V

And the uncertainty about the instrument is mainly due to credit risk. When modelling,prices can expressed as functions of default probabilitities and LGD (and discountedcashflows).

“Counterparty risk” is in this context a more general concept

The exposure is uncertain and not always positive; in fact it is a nonlinear function ofthe derivative/position value.


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In the general, when talking about counterparty risk and CVA:

The exposure is an option on future (netting set/position with my counterparty)prices: Non linearity

Even the most simple derivative will need a model specification to compute CVA(non static replication): Model dependent

If fact, not only the (various, different asset classes) underlyings’s volatilities arerequired, but also the probability of default of the counterparty, and correlationbetween underlyings and probability of default: Hybrid models required


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Unilateral CVA

Assuming that the bank B can not default, and the counterparty C is risky, we canderive the following expression:

CV A(t) = E

[LgdC 1τC≤TD(t, τC) (V (τC))

+|Ft

]

The model is formulated under risk neutral measure, with numeraire the bank account

dβt = rtβtdt

with rt the risk free rate at time t, and

D(t, τ) = e∫ τt rsds: stochastic discount factor between t and τ .

LgdC: Loss given default of the counterparty

τC: Time of default of the counterparty


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V (t): is the risk-free value of the portfolio for the Bank (Assumption on the closeout!)

Ft: is the sigma-algebra with the information up to time t (market risk and counter-party credit)

Idea of the proof

Lets omit the notation C, and let us introduce

R = 1− Lgd

π(s1, s2): cash flows from s1 to s2 discounted at time s1; they may be contingentbut they are independent of the default of the counterparty

By definition

V (s) = E[π(s, T )|Ft], and π(s, t) +D(s, t)π(t, u) = π(s, u)

Let us introduce the corresponding (counterparty) risky quantities: π and V


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From the point of view of B we have the general risky payoff

π(t, T ) = 1τ≥T π(t, T )+1τ<T [π(t, τ)+

D(t, τ)(Rec(V (τ))+ − (−V (τ))+

)]

Taking the risk neutral expectations,

V (t) = E [π(t, τ)1τ<T |Ft] + E [π(t, T )1τ≥T |Ft] +

E[R D(t, τ)(V (τ))+1τ<T |Ft

]− E

[D(t, τ)(−V (τ))+1τ<T |Ft

]

By using that (V (t))+ − (−V (t))+ = V (t), V (t) = E[π(t, T )|Ft],

V (t) = V (t)− E[(1−R)D(t, τ)(V (τ))+1τ<T |Ft]


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If we denote by Gt the filtration to time t just containing the market state variable andnot the credit worthiness of the counterparty we can write

CV A(t) =

∫ T

t

fC(t, u)E[(1−R)D(t, u)(V (u))+|Gt, τ = u]du

with fC(t, u) the probability density function of variable time to default τ = τC.

fC(t, u) = λC(u)e−

∫ ut λC(s)ds

with λC the counterparty hazard rate

Notice that the expected value is conditioned to the default of the counterparty!


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Unilateral CVA and Simplifications

To get the standard CVA formula that is usually implemented, we have to make furtherassumptions.

The recovery is a deterministic quantity

CV A(t) = (1−R)

∫ T

t

fτ(u) E[D(t, u)V (u))+|Gt, τ = u] du


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Bucketing time t = t0 < t1 < . . . < tn = T and approximatting the integral

CV A(t) = (1−R)

n−1∑

i=0

P(τ ∈ [ti, ti+1]) E[D(t, ti+1)(V (ti+1)+|Gt, τ ∈ [ti, ti+1])]

or, also

CV A(t) = (1−R)

n−1∑

i=0

(GC(t, ti)−GC(t, ti+1) E[D(t, ti+1)(V (ti+1)+|Gt, τ ∈ [ti, ti+1])]

with GC(t, s) the probability of being solvent at time s (contidioned to be solvent attime t).


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Credit risk is independent of any market risk factors

CV A(t) = (1−R)n−1∑

i=0

P(τ ∈ [ti, ti+1]) E[D(t, ti+1)(V (ti+1)+|Gt]

CV A(t) = (1−R)

n−1∑

i=0

(GC(t, ti)−GC(t, ti+1) E[D(t, ti+1)(V (ti+1)+|Gt]

Making explicit the hazard rates and the risk neutral rate r:

CV A(t) = (1−R)

n−1∑

i=0

(e−∫ tit λ(s)ds − e−

∫ ti+1t λ(s)ds)E[e−

∫ ti+1t r(s)ds(V (ti+1)

+|Gt)]

Warning Big Assumption: WWR, credit derivatives, etc


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It λ = λC is the counterparty hazard rate

GC(t, s) = e−∫ st λ(s)ds

With this notation, the default probability at time s, FC(t, s) is

FC(t, s) = 1−GC(t, s) = 1− e−∫ st λ(s)ds

And, consistently with the previous definitions, the probability density function of timeto default τC (for time s conditined to survival at t) is

fC(t, s) =∂FC(t, s)

∂s= λ(s)e−

∫ st λ(u)du


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Unilateral DVA

If we consider same assumptions from the counterparty point of view, i.e., the Bank isunrisky and the counterparty is risky, we will arrive to the symmetrical quantity calledDebit Value Adjustment.

More precisely, if we denote by VX, and VX the risk neutral and risky prices from thepoint of view of X we have,

VB(t) = −VC(t), VB(t) = −VC(t)

VC(t) = −(VB(t)− CV AB(t)) = VC(t) + CV AB(t)

whereCV AB(t) = E[(1−RC)D(t, τC)(VB(τC))

+1τC<T |Ft]

Let us introduce the quantity

DV AC(t) = E[(1−RC)D(t, τC)(−VC(τC))+1τC<T |Ft] = CV AB(t)

ThereforeVC(t) = VC(t) +DV AC(t).


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DVA is the increase in value I need to pay to enter a deal with a counterparty becauseI am default risky.

DVA introduces symmetry in prices, (theoretically) “the law of one price” again.

Controversial quantity

Paradox: My DVA increases (and therefore my portfolio value) when my creditworthiness decreases

Impossible to hedge: Selling protection on myself? We can not sell our own CDSs; wecould trade on our own debt, or hedging with proxies, as CDSs indexes (but problemswhen actual defaults).

Basel III does not consider DVA when computing CVA charges, but FASB aproved it.

Double counting with FVA?


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WWR

Important (and common) assumption: Independency between Counterparty default andmarket value of the portfolio (exposure); it does not take into account the possiblecorrelation between them.

If fact, we may have cases (specific wrong way risk):

with Wrong Way Risk: my exposure is higher when my counterparty is more likely todefaultExample: I have a put option on my counterparty

with Right Way Risk: my exposure is higher when my counterparty is less likely todefaultExample: I have a call option on my counterparty


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We may consider also systemic wrong way risk:

Example:My portfolio exposure increases if the interest rates decrease; in general, in crisis periodsthe probability of defaul of any counterparty increases, and we have low interest rates.

Example:During the crisis, wrong way risk were especially severe for monolines, and was notrecognized in most credit models

In general, default rates are highly correlated between them; therefore a credit derivativesare likely exposed to WWR.

WWR is, in general, very difficult to model!

Common practice is assuming independence, and analysing separately cases of WWR.


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Bilateral CVA

Now assuming that both the bank and the counterparty are risky, we can compute bothCVA and DVA from the banks point of view

CV AB(t) = E[(1−RC)D(t, τC)(VB(τC))+1τC<τB<T |Ft]

DV AB(t) = E[(1−RB)D(t, τB)(−VB(τB))+1τB<τC<T |Ft]

And finallyVB(t) = VB(t)− CV AB(t) +DV AB(t)


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With simplifications: deterministic R, bucketing, and independence between default andthe rest of market risk factors, we obtain

CV A(t) = (1−R)

n−1∑

i=0

P(τC ∈ [ti, ti+1], ti+1 < τB) E[D(t, ti+1)(V (ti+1)+|Gt)]

Modelling the dependence between default times τC and τB is difficult (1st to defaultmodelling), not only because of the models themselves but also because a correlationparameter between defaults needs to be specified.

Standard assumption is that they are independt, under which

P(τC ∈ [ti, ti+1], ti+1 < τB) = (GC(t, ti)−GC(t, ti+1))GB(t, ti+1))


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Risk Mitigants: Netting and Collateral

Netting:This is the agreement to net all positions towards a counterparty in the eventof the counterparty default.

CCR calculations are typically computed at a counterparty and a netting set level.

Collateral:It is a guarantee that is deposited in a collateral account in favour of theinvestor party facing the exposure. If the depositing counterparty defaults, thus notbeing able to fulfill payments associated to the above mentioned exposure, collateralcan be used by the investor to offset its loss

Adding the collateral to the CVA formula:

CV AB(t) = E[(1−RC)D(t,mın(τC))(VB(τC)−M(τC))+1τC<τB<T |Ft]

with M(t) the value of the collateral portfolio at time t.


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Ideal CSA contract: Instantaneaous posting, no thresholds, no minimum transferamounts, cash, etc. The collateral would inmediatly reduce the CVA to zero.

Realistic CSA contract: The collateral is posted with certain frequency; there is a“grace period” that consideres the number of days to realise a counterparty hasdefaulted; we may have desagreements about the collateral to post, the post may beno cash etc. There is still GAP RISK and a remaining CVA.

Even under daily collateralization there can be large mark to market swings that makecollateral rather ineffective. This is called GAP RISK and is one of the reasons whyCentral Clearing Counterparties (CCPs) and the new standard CSA have an initialmargin as well.


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Gap Risk and MPoR

The GAP RISK is the possibility of a market crash between the time the counterpartydefaulted and the time when close-out of the positions is completed.

Margin period of risk (MPoR) is the length of time between the default event and thetime when positions with the counterparty are closed out

If we consider a marging period of risk of δ

M(t) = f(V (t− δ), X1, X2, . . .)

where Xi are characteristics of the collateral contract, as threshold, minimum transferamount etc

In the case of 0 thresholds, for instance

CV AB(t) = E[(1−RC)D(t,mın(τC))(VB(τC)− VB(τC − δ))+1τC<τB<T |Ft]

Very recent MPoR model in [APS-16-paper].


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Close-out value

We have assumed that the value of the residual deal computed at the closeout time wasthe risk-neutral value.

However, there is a debate on that:

Risk neutral residual may generate a discountinuity in the mark to market prices

We may have considered a replacement closeout, where the remaining deal is pricedby taking into account the credit quality of the surviving party and of the party thatreplaces the defaulted one.

Obviously, this second approach is more complicated

Still a debate: What is the risk neutral rate? (when no collateral)


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Particular cases: Analytical Exposures

Equity Forward

Let us consider an Equity forward which pays at time T the strike K and receives S(T ).Assmuning

E

[D(t, Tj) (V (Tj))

+|Ft

]= Et

[D(t, Tj)

(S(Tj)−Ke−r(T−Tj)

)+|Ft

]

which is the prime of a call option on S maturing at Tj with strike Ke−r(T−Tj).


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IRS

Let us consider a payer IRS, swaping fixed coupons K by floating coupons at timesTa+1, . . . , Tb, and let ∆i = Ti − Ti−1

Et

[D(t, Tj) (V (Tj, T ))

+|Ft

]=

Et

D(t, Tj)

(TS(Tj;Tj, Tb)−K)

b∑

j

D(Tj, Tk)∆k

+

|Ft

which is the time t swaption prime of a swaption with maturity Tj, underlying Tb − Tj

and strike K.


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General case: Implementation

Even with the simplified (unilateral) formula, but we still need to compute the expectedexposure at a netting set level.

Except from some particular, mainly theoretical, but very useful, exercises (see [SS-15-paper]), the expected exposure is computed by simulation:

Monte Carlo simulation for generating different states of the world are calculated(risk factor simulation),

Pricing of the instruments at scenario level.

Simplifications in some of the pricers are needed (to avoid nested Monte Carlosimulations, for instance).

Aggregation rules are performed (taking into account netting, collateral, etc).


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Incremental CVA

We are insterested in computing the impact of adding a new instrument to the nettingset.

It can only be calculated on a differential basis, that is through two calculations, onewith the original portfolio and one with the new portfolio.

From the computational efficiency point of view:

We should be computing MC scenarios and intruments prices (all of them, both theportfolio and the new instrument) once;

and then developing the aggregations twice: with the new instrument and without.


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Allocated CVA

For simplicity, we omit the time-dependency and the discount factors in the notation.

The objective is to compute CV Aiatributed quantities for i = 1, . . . , n portfolio instru-

ments

CV A =∑

i

CV Aiatributed

Let Vi be the trade i value. The LGD and default probabilities are counterpartydependent and not trade-dependent. Therefore our problem is to compute EEi

allocated

such that

EE =∑

i

EEiallocated

However, in general, expected exposure

EE = EE(V1, . . . , Vn) = E[(∑

i

Vi)+]

is not additive in the trades.


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Let us define auxiliary functions and variables:

Vi = αiui

f(α1, . . . , αn) = EE(α1u1, . . . , αnun) = E[(∑

i

αiui)+]

Since f is homogeneous order 1, Eulers theorem gives

EE(α1u1, . . . , αnun) =∑

i

∂EE

∂αi

(α1u1, . . . , αnun)

In case αi = 1

EE(V1, . . . , Vn) =∑

i

∂EE

∂αi

(V1, . . . , Vn)

Finally

EEiallocated(V1, . . . , Vn) =

∂EE

∂αi

(V1, . . . , Vn)

One way to compute the above partial derivative is to change the size of a transactionby a small value and calculate the marginal EE using a finite difference.


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Alternatively, it can be also computed via a conditional expectation ([PR-10-paper]):Using that

E

((∑

i

xi)+

)= E

((∑

i

xi)1(∑

i xi)+>0

)

∂EE

∂αi

(α1u1, . . . , αnun) = E

(ui1(

∑i αiui)+>0

)

And for αi = 1∂EE

∂αi

(V1, . . . , Vn) = E

(Vi1(

∑i Vi)+>0

)

The intuition behind the above formula is that the future values of the trade in questionare added only if the netting set has positive value at the equivalent point.


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Further discussion: Risk or Price when modelling

CVA and Counterparty Risk Management

Focused on risk management and capital.

Model under real-world distributions.

Mostly driven by regulations

CVA and Counterparty Risk Pricing

Focused on pricing and hedging.

Model under risk-neutral distributions.

Mostly driven by business


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However, the debate about which measure/calibration to use is still open.See [HSW-14-paper] for a joint measure model !?

Compared to VAR, for instance, in CCR (and CVA) we consider the whole life of thenetting set, not just short term movements. In general

In the short term: the drift componente is dominated by the volatility component

In the long term: the drift term becomes dominant

Therefore we are interested not only in measuring well the volatility but also the drift.


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. . . and the Regulation

For a review on the regulation (Basel III and FRTB) see [G-16-slides].

Basell III: Current, 2 approaches

• Standarized: CVA capital charge as a function of EAD

• Advanced:◦ Approval for CCR and Market Risk IMM required◦ If full revaluation VAR: Formula for CVA similar to the unilateral CVA models(next slide)

◦ Only credit sensitivity considered (exposure remains constant in scenarios)◦ Exposure calculated from CRR engines, in general historical calibration, IMMapproval and backtesting required


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Basel III regulation asks explicitly for market implied probability of default and LGD inthe CVA ((for CVA VAR computations) as:

CV A = LGDmkt

∑

i

PDi−1,i

(EEi−1Di−1 + EEiDi

2

)

where

PDi−1,i = max

(0, exp

(−si−1ti−1

LGDmkt

)− exp

(−

siti

LGDmkt

))

is the approximation of the probability of default in the interval [Ti−1, Ti], and

LGDmkt the loss given default based on market expectations and not historicalestimates.

sj CDS spread for time Tj

EEj expected exposure for time Tj

Dj discount factor for time Tj


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What is coming: new methodology allows for 3 approaches

Basic CVA

FRTB-CVA

• Standarized Approach: SA-CVA• Internal Model approach: IMA-VA

Important: for FRTB sensitivities are needed!

Sensitivities for SA-CVA with finite differences; but in general the industry is movingtowards AAD, LS-regression based methods, etc

More details can be found in [G-16-slides] . . . or in another London Quants and Friendstalk?


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Some References

[APS-16-paper] Rethinking Margin Period of Risk. Andersen, L. and Pykhtin, M. andSokol, A. (2016)

[BMP-13-paper] Counterparty Credit Risk, Collateral and Funding: With PricingCases For All Asset Classes. Brigo, D. and Morini, M. and Pallavicini, A. (2013)

[B-11-slides] Counterparty Risk FAQ: Credit VaR, PFE, CVA, DVA, Closeout, Net-ting, Collateral, Re-hypothecation, WWR, Basel, Funding, CCDS and Margin Len-ding. Brigo, D. (2011)

[B-11-slides] Nonlinear valuation under credit gap risk, collateral margins, fundingcosts and multiple curves. Brigo, D. (2015)

[BM-05-paper] A Formula for Interest Rate Swaps Valuation under CounterpartyRisk in presence of Netting Agreements. Brigo, D. and Masetti, M. (2005)

[G-16-book] XVA Credit, Funding and Capital Valuation Adjustments, Green, A.(2016)


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[G-16-slides] FRTB, FRTB-CVA and implications for capital valuation adjustment(KVA), Green, A. (2016)

[G-12-book] Counterparty Credit Risk and Credit Value Adjustment: A ContinuingChallenge for Global Financial Markets, Gregory, J. (2012)

[G-15-book] The XVAs chagenllenge: Counterparty Credit Risk, Funding, Collateraland Capital, Gregory, J. (2015)

[HSW-14-paper] Modeling the Short Rate: The Real and Risk-Neutral Worlds,Hull,Sokol, White (2014)

[M-15-slides] XVAs: Funding, Credit, Debit and Capital in pricing. Morini, M. (2015)

[PR-10-paper] Pricing Counterparty Risk at the Trade Level and CVA Allocations,Pykhtin, M.and Rosen, D. (2010)

[SS-15-paper] Potential Future Exposure (PFE), Credit Value Adjustment (CVA) andWrong Way Risk (WWR) Analytic Solutions Syrkin, M. and Shirazi, A. (2015)


Documents

XVAs Series: CVA - London Quantslondonquants.com/wp-content/uploads/2015/01/CVAandRels.pdfUnilateral CVA and Simpliﬁcations To get the standard CVA formula that is usually implemented,