FS Counterparty Energy2 - Fitch Ratings · We consider Credit Default Swaps for the counterparty as liquid sources of ... exact meaning of the spot is difficult ... GD t t T D t NPV

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www.fitchsolutions.com Jun 24, 2008

Quantitative ResearchSpecial Report Counterparty Risk Valuation for

Energy-CommoditiesSwapsImpact of Volatilities and Correlation

SummaryIt is commonly accepted that Commodit ies futures and forward prices, in principle,agree under some simplifying assumptions. One of the most relevant assumptions is the absence of counterparty risk. Indeed, due to margining, futures have pract ically nocounterparty risk. Forwards, instead, may bear the full risk of default for thecounterparty when t raded with brokers or outside clearinghouses, or when embeddedin other contracts such as swaps. In this paper, we focus on energy commodities and on Oil in part icular. We use a hybrid commodit ies-credit model to asses impact ofcounterparty risk in pricing formulas, both in the gross effect of default probabilit iesand on the subt ler effects of credit spread volat ility, commodit ies volat ilit y and credit -commodit ies correlat ion. We illust rate our general approach with a case study basedon an oil swap. This case study shows that an accurate valuat ion of counterparty riskdepends on volat ilit ies and correlat ion. Thus, counterparty risk cannot be accuratelyaccounted for through a predefined multiplier.

AMS Classification Codes: 60H10, 60J60, 60J75, 62H20, 91B70.

JEL Classification Codes: C15, C63, C65, G12, G13.

Keywords: Counterparty Risk, Credit Valuat ion adjustment , Commodit ies, Swaps, Oilmodels, Convenience Yield models, Stochastic Intensity models.

Analysts

Damiano BrigoFitch Solutions and Department ofMathematics, Imperial [email protected]

Kyriakos Chourdakis, CCFEAUniversity of Essex

Imane Bakkar

Related ResearchCount erparty Risk for Cred itDefaul t Swaps: Impact of spreadvolat i l i t y and defaul t correlat ion,28-May-2008A Stochastic Processes Toolkit forRisk Management, 17- Nov-2007

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2 Counterparty Risk Valuation for Energy-Commodities Swaps Jun 24, 2008

1. Introduction In this paper, we consider counterparty risk for commodit ies payoffs in presence of correlat ion between the default event and the underlying commodity, while taking into account volat ilit ies for both credit and commodit ies. We focus on Oil but much of our reasoning can be adapted to other commodities with similar characteristics (storability, liquidity, and similar seasonality).

Past work on pricing counterparty risk for dif ferent asset classes is in Sorensen & Bollier (1994), Brigo & Maset t i (2006) and Brigo & Pallavicini (2007, 2008) for interest rate swaps and exot ics underlyings. Leung & Kwok (2005) and Brigo & Chourdakis (2008) worked on counterparty risk for credit (CDS) underlyings.

Here we analyze in detail counterparty-risky (or default -risky) Oil forward and swaps cont racts. In general, the reason to int roduce counterparty risk when evaluat ing a contract is linked to the fact that many financial contracts are traded over the counter, so that the credit quality of the counterparty can be relevant . This is part icularly appropriate when thinking of the dif ferent defaults experienced by some important companies during the last years, especially in the energy sector.

Earlier works in counterparty risk for commodit ies include for example Cannabaro, Picoult, & Wilde (2005), who analyze this not ion more from a capital adequacy/ risk management point of view. In part icular, their approach is not dynamical and does not consider explicit ly credit spread volat ilit y and especially correlat ion between the underlying commodity and credit spread. In our approach, wrong way risk is modeled through said correlat ion. Most ly, however, the dif ference is in the purpose. We are valuing counterparty risk more from a pricing than a risk management perspect ive. Thus, we use a fully arbitrage free and fine-tuned risk neutral approach. This is why all our processes are calibrated to liquid market informat ion on both forward curves and volat ilit ies. Correlat ions are harder to est imate but we analyze their impact by let t ing them range across a set of possible values. A limitat ion of our approach is that we refer to a single counterparty.

In general, we are looking at the problem from the viewpoint of a safe (default -free) inst itut ion entering a f inancial cont ract with another counterparty having a posit ive probabilit y of default ing before the f inal maturity. We formalize the general and reasonable fact that the value of a generic claim subj ect to counterparty risk is always smaller than the value of a similar claim having a null default probabilit y, expressing the discrepancy in precise quantitative terms.

We consider Credit Default Swaps for the counterparty as liquid sources of market default probabilit ies. Different models can be used to calibrate CDS data and obtain default probabilities: here we resort to Brigo & Alfonsi (2005) stochastic intensity model, whose j ump extension with analyt ical formulas for CDS opt ions is illust rated in Brigo & El-Bachir (2008).

As a model for oil, we adopt a two-factor model shaping both the short -term deviat ion in prices and the equilibrium price level, as in Smith & Schwartz (2000). This model can be shown to be equivalent to a more classical convenience yield model like in Gibson & Schwartz (1990), and a stochast ic volat ilit y extension of a similar approach is considered in Geman (2000). What is modeled is the oil spot price, under the implicit assumpt ion that such a spot price process exists. This is not t rue for elect ricity, for example, and even for markets like crude oil where spot prices are quoted daily, the

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Counterparty Risk Valuation for Energy-Commodities Swaps Jun 24, 2008 3

exact meaning of the spot is diff icult to single out . Nonetheless, we assume, alongwith most of the industry and with Carmona & Ludkowski (2004), that there is a t radedspot asset.

In the paper we find that counterparty risk has a relevant impact on the products pricesand that , in turn, correlat ion between oil and credit spreads of the counterparty has arelevant impact on the adj ustment due to counterparty risk. Similarly, oil and creditspread volat ilit ies have reasonable impacts on the adj ustment . The impact pat terns donot involve the peculiar behaviour one observes in the case of credit underlyings,observed in Brigo & Chourdakis (2008). Nonetheless, the impact is quant itat ivelyrelevant, and we illustrate this with a case study based on an oil swap.

The paper is organized as follows: Sect ion 2 lays down the general framework for thevaluat ion of counterparty risk. In sect ion 3, we present the CIR++ specif icat ion, whichserves as the credit model, and in sect ion 4, we out line the two-factor Smith andSchwartz commodity model. Sect ions 5 and 6 illust rate the counterparty adjustmentsfor forwards and swaps respect ively. An example, based on a swap cont ract withbetween a bank and an airline company is presented in section 7.

2. General Valuation of Counterparty RiskWe denote by the default time of the counterparty and we assume the investor who is considering a t ransact ion with the counterparty to be default -free. We place ourselvesin a probabilit y space ( , , t , ). The filt rat ion ( t)t models the f low of informat ionof the whole market, including credit and defaults. is the risk neutral measure. This space is endowed also with a right -cont inuous and complete sub-filtration trepresent ing all the observable market quant it ies but the default event (hence

t t t t where t = ({ u} : u t) is the right -cont inuous filt rat iongenerated by the default event ). We set t := t , the risk neut ral expectat ionleading to prices.

Let us call the f inal maturity of the payoff we need to evaluate. If > there is nodefault of the counterparty during the life of the product and the counterparty has noproblems in repaying the investors. On the contrary, if the counterparty cannotfulfil it s obligat ions and the following happens. At , the Net Present Value (NPV) ofthe residual payoff unt il maturity is computed: If this NPV is negat ive (respect ivelyposit ive) for the investor (defaulted counterparty), it is completely paid (received) bythe investor (counterparty) itself . If the NPV is posit ive (negat ive) for the investor(counterparty), only a recovery fraction REC of the NPV is exchanged.

Let us call D t, (somet imes abbreviated into D t ) the discounted payoff of ageneric claim at t under counterparty risk. This is the sum of all cash flows from t to ,each discounted back at t, and under counterparty risk. This is a stochast ic payoff ,whose price would be given by risk neut ral expectat ion. We denote by t, theanalogous quant ity when counterparty risk is absent , or when the counterparty isdefault free. All payoffs are seen from the point of view of the "investor" (i.e. thecompany facing counterparty risk). Then we have NPV( ) = { t, } and

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EC

,

, , R NPV NPV

DT

t T

t t T

t D t

1

1(2.1)

being D u,v the stochastic discount factor at time u for maturity v. This last expression is the general price of the payoff under counterparty risk. Indeed, if there is no earlycounterparty default this expression reduces to risk neut ral valuat ion of the payoff(f irst term in the right hand side); in case of early default , the payments due beforedefault occurs are received (second term), and then if the residual net present value isposit ive only a recovery of it is received (third term), whereas if it is negat ive it is paidin full (fourth term).

Calling t the discounted payoff for an equivalent claim with a default -freecounterparty, i.e. t = t, , it is possible to prove the following.

Proposition 2.1. (General counterparty-risk credit-valuation adjustment (CR-CVA)formula). At valuat ion t ime t, and on { > t } , t he price of our payof f undercounterparty risk is

1 ,DGDt t t t Tt t L D t NPV

Positive CR-CVA

(2.2)

where LGD = 1 - REC is t he Loss Given Default and t he recovery f ract ion REC is assumedto be deterministic. It is clear that the value of a defaultable claim is the value of the corresponding default -f ree claim minus an opt ion part , in t he specif ic a cal l opt ion(wit h zero strike) on t he residual NPV giving nonzero cont ribut ion only in scenarioswhere . Counterparty risk adds an optionality level to the original payoff.1

Not ice f inally that the previous formula can be approximated as follows. Take t = 0 forsimplicity and write, on a discretization time grid 0 1, , , bT T T T ,

1

1

1

1

0, 0, 0, ,

0, 0, ,

j

j j

j

j

bDGDb b bT Tj

bGDb bT Tj j T jT T

T T L D T

T L D T

1

1

approximated (positive) adjustment

(2.3)

where the approximat ion consists in postponing the default t ime to the f irst i

following . From this last expression, under independence between and , one canfactor the outer expectat ion inside the summat ion in products of default probabilit iest imes opt ion prices. This way we would not need a default model for the counterpartybut only survival probabilit ies and an opt ion model for the underling market of . Thisis what led to earlier results on swaps with counterparty risk in interest rate payoffs in

1 Refer to Brigo & Masetti (2006) for a proof.

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Brigo & Maset t i (2006). In this paper, we do not assume zero correlat ion, so that ingeneral we need to compute the counterparty risk without factoring the expectations.

3. Default Modeling AssumptionsIn this sect ion, we consider a reduced form model that is stochastic in the defaultintensity for the counterparty. We will later correlate the credit spread of this modelwith the underlying commodity model.

More in detail, we assume that the counterparty default intensity is , and we denote

the cumulated intensity by0

tt s ds . We assume intensities to be strict ly

positive, so that t t are invertible functions.

We assume determinist ic default -free instantaneous interest rate r (and hencedeterminist ic discount factors D s, t ,…), although our analysis would work well evenwith stochastic rates independent of oil and credit spreads.

We set ourselves in a Cox process setting, where

1 ,

with standard (unit-mean) exponential random variable.

3.1. CIR++ Stochastic Intensity ModelsFor the stochastic intensity model we set

; , 0 ,t y t t t (3.1)

where is a deterministic funct ion, depending on the parameter vector (whichincludes y0), that is integrable on closed intervals. The init ial condit ion y0 is one moreparameter at our disposal. We are free to select its value as long as

0 0 .0; y

We take y t o be a Cox Ingersoll Ross process (see for example Brigo & Mercurio (2001)or (2006)):

,ydy t y t dt v y t dZ t

where the parameter vector is 0, , ,v y , with 0, , , &v y positive

deterministic constants. As usual, yZ is a standard Brownian mot ion processes underthe risk neut ral measure, represent ing the stochastic shock in our dynamics. Weassume the origin to be inaccessible, i.e.

22 .v

We will often use the integrated quantities

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0 0 0, , and , , .

t t t

s st ds Y t y ds t s ds

3.2. CIR++ Model: CDS CalibrationSince we are assuming deterministic rates, the default t ime and interest ratequantities , , ,r D s t are trivially independent. It follows that the (receiver) CDSvaluation at time 0 becomes model independent and is given by the formula

1

GD, , ,

1

GD

0,CDS 0, ,L ;

0,

L 0,

b

a

b

a

ttTba b a b a b T

i i ii a

T

tT

P t t T d tS S

P T T

P t d t

(3.2)

This means that if we strip survival probabilit ies from CDS in a model independent wayat t ime 0, to calibrate the market CDS quotes we j ust need to make sure that thesurvival probabilit ies we st rip from CDS are correct ly reproduced by the CIR++ model.Since the survival probabilities in the CIR++ model are given by

exp ,tmodel

t e t Y t

(3.3)

we just need to make sure

exp ,market

t Y t t

from which

CIR 0, , 0 ;, ln ln

Y t

market market

t ye Pt

t t

(3.4)

(3.5)

(3.6)

where we choose the parameter in order to have a posit ive funct ion (i.e. anincreasing ) and PCIR is the closed form expression for bond prices in the t imehomogeneous CIR model with init ial condit ion y0 and parameters (see for exampleBrigo & Mercurio (2001, 2006)). Thus, if is selected according to this last formula, aswe will assume from now on, the model is easily and automat ically calibrated to themarket survival probabilities for the counterparty (possibly stripped from CDS data).

Once we have done this and calibrated CDS data through ( , ), we are left with theparameter , which can be used to calibrate further products. However, this will beinterest ing when single name opt ion data on the credit derivat ives market will becomemore liquid. Current ly the bid-ask spreads for single name CDS opt ions are large andsuggest to either consider these quotes with caut ion, or to t ry and deduce volat ilit yparameters from more liquid index opt ions. At the moment we content ourselves ofcalibrat ing only CDS's. To help specifying without further data we set some values ofthe parameters implying possibly reasonable values for the implied volat ilit y of

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hypothetical CDS options on the counterparty.

4. Commodity ModelWe consider crude oil as a first important case.

Suppose we have an airline company that buys a forward cont ract on oil from a bankwith a very high credit quality, so that we assume the bank to be default -free. Thebank wants to charge counterparty risk to the airline in defining the forward price, asthere is no collateral posted and no margining is occurring.

As a model for oil we adopt a two factor model shaping both the short term deviation in prices and the equilibrium price level, as in Smith & Schwartz (2000). This model canbe shown to be equivalent to a more classical convenience yield model like in Gibson &Schwartz (1990), and a stochastic volat ilit y extension of a similar approach isconsidered in Geman (2000). What is modeled is the oil spot price, under the implicitassumption that such a spot price process exists.

This is not t rue for elect ricity, for example, and even for markets like crude oil wherespot prices are quoted daily, the exact meaning of the spot is difficult to single out .Nonetheless, we assume, along with most of the indust ry, that there is a t raded spotasset.

If we denote by St the oil spot price at time t, the log-price process is written as

ln ,tS x t L t t

where, under the risk neutral measure,

,

,

, ,x x x

L L L x L x L

dx t k x t dt dZ

dL t dt dZ dZ dZ dt

(4.1)

(4.2)

and is a deterministic shif t we will use to calibrate quoted futures prices. Theprocess x represents the short term deviat ion, whereas L represents the backbone ofthe equilibrium price level in the long run.

For applicat ions it can be important to derive the t ransition density of the spotcommodity in this model. For the two factors we have a joint Gaussian transition,

2,

2

,

, ,

exp Cov ,, 1 exp 2

2

Cov , 1 exp

x x Lxx

LxLx s L s

x Lx L x L x

x

x s k t sx t s tk t s

L t t skL s t s

s t k t sk

where

This can be used for exact simulat ion between t imes s and u. As we know that the sumof two jointly Gaussian random variables is Gaussian, we have

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,

22

,

ln , , ,

, exp ,

, 1 exp 2 2 ,2

x s L s

x L

xx L x L

x

S t x t L t t m t s V s t

m t s x s k t s L s t s t

V t s k t s t s Cov s tk

from which, in particular, we see that

exp

, exp

, 2

x

L

x s k t s

L s

S t x s L s t s

t

V s t

Hence we can compute the forward price [S(T) x(t), L(t)] at t ime t of the commodity atmaturity T when counterparty risk is negligible and under deterministic interest rates,as

exp

, exp

, 2

x

L

t Tx k

L

F t T

t

t

T t

t

T

TV

(4.3)

In part icular, given the forward curve T FM(0,T ) f rom the market , the expression for

the shift M(T ) that makes the model consistent with said curve is

0 0ln 0, exp , 2 .M Mx LT F T x k T L T V T t

The short term/ equilibrium price model (x,L), when = 0, is equivalent to the moreclassical Gibson & Schwartz (1990) model, formulated as

2

,

ln 2 ,

,

,

t S S S

q q q

S q q S

d S r t q t dt dZ

dq t k q t dt dZ

dZ dZ dt

(4.4)

the relationships begin

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2

2 2 2 2,

2 2 2,

2 2 2, , ,

1

1ln

2

2

2

2

q

tq

x q

L S

x q q

L S q q q S S q q

x q

qL S S q S q q q S S q q

q

qx L S q S S q q q S S q q

q

x t q tk

L t S q tk

k k

rk

k kdZ dZ

dZ dZ dZ k kk

k kk

5. Forward vs. Future Prices and Counterparty RiskConsider now a forward cont ract . The prototypical forward cont ract agrees on thefollowing.

Let t be the valuat ion t ime. At the future t ime T a party agrees to buy from a secondparty a commodity at the price K fixed today. This is expressed by saying that the first party has entered a payer forward rate agreement . The second party has agreed toenter a receiver forward rate agreement . The value of this cont ract to the f irst andsecond party respectively, at maturity, will be

,T TS K K S

i.e. the actual price of the commodity at maturity minus the pre-agreed price in thepayer case, and the opposite of this in the receiver case. Let us focus on the payercase. When this is discounted back at t with determinist ic interest rates, and riskneutral expectation is taken, this leads to the price being given by

,

,

, , .

T

t T

D t T S K

D t T S K

D t T F t T K

(5.1)

Note that the forward price is exact ly the value of the pre-agreed rate K that sets thecont ract price to zero, i.e. K = F(t, T). Let us maintain a general K in the forwardcontract under examination.

In the oil model above, the forward cont ract price is given by plugging Formula (4.3)

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into (5.1). Let us denote by Fwdp(t, T; K) such price (“ p” is for payer),

wdp

exp

F , ; , exp

, 2

x

L

x t k T t

L t

t T K D t T T t K

T

V T t

Whereas the opposite of this quantity is denoted by Fwdr(t, T; K) - (“ r” for receiver),

We may apply our counterparty risk framework to the forward cont ract , where nowt, = D(t, T)(ST - K), and NPV(t) = Fwdp(t, T; K). We obtain as price of the payer

forward under counterparty risk from Equation (2.2). We obtain

wdp wdp wdpF t,T;K = F , ; , F , ;Positive counterparty-risk adjustment

Dt t TGDt T K L D t T K1 (5.2)

Under the bucketing approximation given by Equation (2.3), we obtain

11

wdp wdp wdpF , ; F , ; , F , ; .j j

bD

j t jT Ti

GDt T K t T K L D t T T T K1 If

one assumes independence between the underlying commodity and the counterpartydefault, one may factor the above expectation obtaining

11

wdp wdp

wdp

F , ; F , ;

, F , ; .

D

b

j j t j ji

GD

t T K t T K

L T T D t T T T K

The last term is the price of an opt ion on a forward price, which is known in closedform in the Schwartz & Smith model, although we have to incorporate the shif t in ourformulation. We have

wdp, F , ;

, , ; , , , ; , , , ln, exp

, , 2 , ,

, , ; , ln,

, ,

t j j

j t t j t t j

j j

j t t

j

D t T T T K

M t T T x L M t T T x L V t T T KD t T

V t T T V t T T

M t T T x L KD t T K

V t T T

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where

2

2,

, , ; , exp , 2

, , exp 2 1 exp 22

exp 2 1 exp

j t t t x t j

xj x j x j

x

x LL j x j x L x j

x

M t T T x L x k T t L T V T T

V t T T k T T k T tk

T t k T T k T tk

where is the cumulative distribution function of the standard Gaussian.

So we have the adj ustment as a st ream of opt ions on forwards weighted by defaultprobabilities.

If we do not assume independence, then we need to subst itute for the intensity model.Through iterated conditioning we obtain easily

1

1

wdp wdp

wdp

F , ; F , ;

exp exp, .

F , ; ; ,

D

b j j

j tj

j j j

GD

t T K t T K

T TL D t T

T T K x T L T

If in particular we select K = F(t, T), then Fwdp(t, T; K) will be zero.

This price can be computed by j oint simulat ion of , x and L. We may correlate thecredit spread to the commodity by correlat ing the shock Zy in the default intensity tothe shocks Zx, ZL in the commodity. If we assume

, ,,x y x y L y L ydZ dZ dt dZ dZ dt

Then the instantaneous correlation of interest is

, ,

2 2,

corr ,2

x x y L L yt t

x L x L x L

d dS

This is the correlat ion one may t ry to infer from the market , through historicalest imat ion or implying it from liquid market quotes. In general the only parametersthat have not been calibrated previously are x,y and L,y. If we make, for example, theassumption that the two are the same,

, , 1 ,:x y L y

then we get the model correlat ion parameters as a funct ion of the already calibratedparameters and of the market correlation as

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2 2,

1

2corr , x L x L x L

t tx L

d dS

6. Swaps and Counterparty RiskConsider now a swap contract. The prototypical swap contract is actually a portfolio of forward contracts with different maturities, and agrees on the following.

Let t be the valuation time. At the future times Ti in Ta+1, Ta+2, , Tb, a party agrees to buy from a second party a commodity at the price K fixed today, on a notional i. This is expressed by saying that the f irst party has entered a payer swap agreement . Thesecond party has agreed to enter a receiver swap. The value of the payer commodityswap (CS) contract to the first party, at time t, will be

,1

1

1

PCS , ,

, ,

, ; .

i

b

a b t i i Ti

b

i i ii

b

i ii

WD

t K D t T S K

D t T F t T K

F p t T K

Since the last formula is known in our oil model, in terms of the processes x(t) and L(t),we easily obtain a formula for the commodity swap by summation.

If we look for the value of K that sets the cont ract price to zero, i.e. the so calledforward swap commodity price Sa,b(t), we have

,

, ,.

,

bi i ii a

a b bi ii a

D t T F t TS t

D t T

Using this rate we can also express the payer commodity swap price at a general st rikeK as

, ,+1

PCS , ,b

a b a b i ii a

t K S t K D t T

whereas the receiver commodity swap would be

, ,+1

RCS , , .b

a b a b i ii a

t K K S t D t T

These formulas provide the value of these contracts when a clearing house or margining agreements are in place. However, swaps are often t raded outside such contexts andas such they embed counterparty risk.

Our general formula (2.2), for a payer CS, when including counterparty risk, would read

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in the swap case:

, , ,

,1

wdp

PCS ; PCS ; 1 , PCS ;

PCS ; 1 , F , ;

Positive counterparty-risk adjustmentb

b

Da b a b t a bt T

b

a b t i it Ti a

GD

GD

t K t K L D t K

t K L D t T K

(6.1)

Since the forward formula is known in our model, we can proceed similarly to theforward case to value the counterparty risk adj ustment for the swap case throughsimulation. The receiver case is completely analogous.

7. A Case Study and Conclusions

As a case study we consider an oil swap. An airline needs to buy oil in the future and is concerned about possible changes in the oil price. To hedge this price movement theairline asks a bank to enter a swap where the bank pays periodically to the airline a(f loat ing) amount indexed at a relevant oil futures price at the coupon date. Inexchange for this, the airline pays periodically an amount K that is f ixed in thebeginning.

In the following, we are taking an example of a bank with current ly high credit spreadsas receiver, and one internat ional airline as the payer of the swap. We will look at thecounterparty risk adj ustment from the point of view of each of the two part iesseparately, by calibrating the credit model adequately in each case.

The Oil model has been calibrated to the At-The-Money Futures opt ion’ s impliedvolatility shown in Figure 1.

The shif t has been calibrated in order to fit the forward curve ext racted from WestTexas Intermediate Futures in Figure 2.

FIXED-FLOATING (for hedge purposes)

Pays Floating price indexed on WTIF FuturesBank Airline

Pays a Fixed price K

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Figure 1 Calibration: ATM Volatility Curve.

Figure 2 Calibration: Forward Curve.

We reformulated the commodity model by set t ing the parameter L in the long term

equilibrium price to zero, as it can be included implicit ly in the determinist ic shif t .The resulting Oil model parameters are shown in Table 1.

The oil swap we consider has a final maturity of 5 years, monthly payments and strike Kgiven by K = 126 USD, that is the st rike set t ing to zero the value of the 5-year default

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free oil swap. i is equal to one (barrel).

Table 1 Calibration Parameters

kx x L x.L

0.7170 0.3522 0.19 -0.0392

7.1.Counterparty Risk from the Payer Perspective (the Airline computescounterparty risk)

First , we use the CDS spreads for the bank, which are given in Table 2 along with theyield curve is given in Table 3.

Table 2 CDS Spreads Term Structure for the Bankmaturity (years) 0.5 1 2 3 4 5

spread (bps) 345 332 287 256 232 217

Table 3 Zero Coupon Continuously Compound Spot Interest Ratesmaturity (years) 3/12 6/12 2 5 10 30

yield (percent) 2.68 2.92 3.40 4.27 4.87 5.376

In the following we assume the bank credit quality to be characterized by a CIR++stochast ic intensity model that , as spread levels, is consistent with Table 2 through theshift , while allowing for credit spread volat ilit y through the CIR dynamics. We usethe base CIR parameter set given in Table 4. Later, we change the spread volat ilit yparameter v by reducing it through mult iplicat ive factors smaller than one, andrecalibrate the model shif t to maintain consistency with Table 2. This way weinvestigate the impact of the spread volatility on the counterparty adjustment.

Table 4 CIR Parameters for the Base Case Bank Credit Spread Volatility

y v

0.0560 0.6331 0.0293 0.5945

The graphs in Figures 3 and 5 illust rate some of our results for the CR-CVA. Thecounterparty risk is expressed as a percentage of a 5Y maturing swap fixed leg value,which is 6852.35 USD. First we observe the effect of varying the commodity volat ilit ywhile keeping the credit intensity volat ilit y f ixed at vBank = 59%. 2 The commodity

2 The CDS implied volat il it y associated to these parameters is 26%. Brigo (2005, 2006), under the CDSmarket model, shows that implied volatilities for CDS options can easily exceed 50%

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volat ilit y was varied by applying mult iplicat ive factors to the two factors instantaneousvolatilities x and l.

As an indicat ion of implied volat ilit y levels, the term st ructure of the commodity-implied volatility, when we apply the multiplicative factor 2 is given in Figure 4.

Figure 3 Commodity Swap CR-CVA Results Overview: Commodity Volatility Effect

Figure 4 Model Implied Volatility without (right scale) and with (left scale)Multiplicative Factors

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Figure 5 Commodity Swap CR-CVA Results Overview: Credit volatility effect

Secondly, we observe the effect of varying the intensity volat ilit y while keeping thecommodity spot volatility fixed at S = 32.82% as implied by Table 1.

The same results are presented in a dif ferent way in Tables 5 and 6. In these tables,we give the absolute value of the adjustment in USD. We also express it as an adjusted St rike price that the payer might choose to pay to its counterparty by taking intoaccount the estimated adjustment:

Table 5 Effect of Credit Spread Volatility on the CR-CVA

Intensity Volatility vR 0.0295 0.295 0.59

CR-CVA (USD) 63.49 25.17 21.58-68.9Adjusted Strike 124.84 125.54 125.6



CR-CVA (USD) 73.3 65.23 63.420Adjusted Strike 124.66 124.8 124.84

CR-CVA (USD) 74.62 76.63 77.36+13.8Adjusted Strike 124.63 124.59 124.58



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Table 6 Effect of Oil Volatility on the CR-CVA

Commodity SpotVolatility S 0.0330 0.1642 0.3285 0.6570

CR-CVA (USD) 1.17 11.05 21.58 57.11-68.9Adjusted Strike 125.98 125.79 125.60 124.95



CR-CVA (USD) 1.98 32.4 63.42 164.270Adjusted Strike 125.96 125.41 124.84 122.98

CR-CVA (USD) 2.15 38.85 77.36 200.08+13.8Adjusted Strike 125.96 125.28 124.58 122.33



7.2. Counterparty Risk from the Receiver Perspective (the Bank computescounterparty risk)

Now we place ourselves from the point of view of the bank, and we use the CDS spreads for the airline, which are given in Table 7.

Table 7 CDS Spreads Term Structure for the Airline

maturity (years) 0.5 1 2 3 4 5

spread (bps) 76 82 104 122 139 154

We use the same discount curve as in Table 3.

Here, the airline credit quality is represented by a CIR++ stochast ic intensity modelthat, as spreads levels, is consistent with Table 7 through the shift , while allowing for credit spread volat ilit y through the CIR dynamics. We use the base CIR parameter setgiven in Table 8. Later, we reduce the spread volat ilit y parameter v via mult iplicat ivefactors smaller than one, and recalibrate the shif t to maintain each t ime the modelconsistent with Table 7. This way we invest igate again the impact of the spreadvolatility on the counterparty adjustment.

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Table 8 CIR Parameters for the Base Case Airline Credit Spread Volatility

y v

0.0000 0.5341 0.0328 0.2105

As before, we observe the effect of varying the commodity volat ilit y and of the airlinecredit intensity volat ilit y, start ing from S = 32.82% as from Table 1 and vAirline = 21%.We apply the same mult iplicat ive factors as before and the results are summarized inthe graphs in Figures 6 and 7.

The same results are presented more in detail in Tables 9 and 10.

Figure 6 Commodity Swap CR-CVA Results Overview: Credit volatility effect

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Figure 7 Commodity Swap CR-CVA Results Overview: Commodity volatility effect

Table 9 Effect of Credit Spread Volatility on the CR-CVA

Intensity Volatility vR 0.0295 0.295 0.59




CR-CVA (USD) 27.99 29.37 29.160Adjusted Strike 126.51 126.54 126.53




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Table 10 Effect of Oil Volatility on the CR-CVA

Commodity Spot Volatility S 0.0330 0.1642 0.3285 0.6570




CR-CVA (USD) 0.07 15.42 29.16 48.590Adjusted Strike 126.00 126.28 126.53 126.89


CR-CVA (USD) 0.05 11.86 23.42 39.09+27.6Adjusted Strike 126.00 126.22 126.43 126.72CR-CVA (USD) 0.03 7.4 16.31 27.16+68.9Adjusted Strike 126.00 126.13 126.30 126.50

7.3. ConclusionsThe pat terns we observe in the counterparty-risk credit valuat ion adj ustment (CR-CVA)are natural. Start ing with the receiver case, for a fixed credit spread volat ilit y, thereceiver CR-CVA increases in oil volat ilit y and decreases in correlat ion. Given theembedded oil opt ion, the increase with respect to oil volat ilit y is natural (as is in thepayer case). As concerns correlat ion, as this increases, the oil tends to move in linewith credit spreads. This means that higher credit spreads will lead to higher oil values, and the opt ion will end up less in the money as the oil spot goes up. The oppositeappears in the payer case. Patterns in credit spread volatility are similarly explained.

The size of the CVA hence depends on the precise value of the volatility and correlation dynamic parameters that cannot be explained via rough multipliers.

References[1] Brigo, D. (2005). Market Models for CDS Opt ions and Callable Floaters, Risk,

January issue. Also in: Derivat ives Trading and Opt ion Pricing, Dunbar N. (Editor),Risk Books, 2005.

[2] Brigo, D. (2006). Constant Maturity Credit Default Swap Valuat ion with MarketModels, Risk, June issue.

[3] Brigo, D., and Alfonsi, A. (2005) Credit Default Swaps Calibrat ion and Derivat ivesPricing with the SSRD Stochast ic Intensity Model, Finance and Stochast ic, Vol. 9,N. 1.

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[4] Brigo, D., and El{Bachir, N. (2008). An exact formula for default swapt ions pricingin the SSRJD stochastic intensity model. To appear in Mathematical Finance.

[5] Brigo, D., and Maset t i, M. (2006) Risk Neut ral Pricing of Counterparty Risk. InCounterpart y Credit Risk Model ing: Risk Management , Pricing and Regulat ion, ed.Pykhtin, M., Risk Books, London.

[6] Brigo, D., Mercurio, F. (2001) Int erest Rat e Models: Theory and Pract ice - wit hSmile, Inflation and Credit, Second Edition, 2006, Springer Verlag.

[7] Brigo, D., and Pallavicini, A. (2007). Counterparty Risk under Correlat ion betweenDefault and Interest Rates. In: Miller, J., Edelman, D., and Appleby, J. (Editors),Numercial Methods for Finance, Chapman Hall.

[8] Brigo, D., and Pallavicini, A. (2008). Counterparty risk and Cont ingent CDS withstochastic intensity hybrid models. Risk Magazine, February issue.

[9] Cannabaro, E., Picoult , E., and Wilde, T. (2005). Counterparty Risk. Energy Risk,May issue.

[10] Carmona, R., and Ludkovski, M. (2004). Spot Convenience Yield Models for EnergyMarkets. AMS Mathemat ics of Finance, G. Yin & Y. Zhang eds., vol. 351 ofContemporary Mathematics, pp. 6580, 2004.

[11] Cherubini, U. (2005) Counterparty Risk in Derivat ives and Collateral Policies: TheReplicat ing Port folio Approach. In: Proceedings of t he Count erpart y Credit Risk2005 C.R.E.D.I.T. conference, Venice, Sept 22-23, Vol 1.

[12] Collin-Dufresne, P., Goldstein, R., and Hugonnier, J. (2004). A general formula forpricing defaultable securities. Econometrica 72(5), 1377-1407.

[13] Geman, H. (2000). Scarcity and price volat ilit y in oil markets. EDF t radingtechnical report.

[14] R. Gibson and E. S. Schwartz (1990). Stochast ic convenience yield and the pricingof oil contingent claims, Journal of Finance XLV (3), 959-976.

[15] Leung, S.Y., and Kwok, Y. K. (2005). Credit Default Swap Valuat ion withCounterparty Risk. The Kyoto Economic Review 74 (1), 25-45.

[16] Schwartz, E., and Smith, J. (2000). Short-Term Variations and Long-Term Dynamics in Commodity Prices. Management Science, Vol. 46, No. 7, July 2000 pp. 893-911

[17] Sorensen, E.H., and Bollier, T. F. (1994) Pricing Swap Default Risk. FinancialAnalysts Journal, 50, 23-33.

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