Multi-currency CDS

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Multi Currency Credit Default Swaps Quanto effects and FX devaluation jumps

Prof. Damiano Brigo Dept. of Mathematics

Mathematical Finance Group & Stochastic Analysis Group Imperial College London

Joint work with Nicola Pede and Andrea Petrelli

Quant Summit Europe

London, 13 April 2016

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Introduction: Credit Default Swaps and Technical Setting

1 Introduction: Credit Default Swaps and Technical Setting Default Risk: reduced form / intensity credit risk models CDS

2 Financial Motivation: CDS in multiple currencies The Italy CDS example

3 Mathematical framework for multi-currency CDS Multi-currency pricing

FX rate dynamics and symmetries Quanto survival probabilties Default intensity model

Instantaneous correlation FX/Credit Spreads Pricing equations

Instantaneous correlation not enough for Default-FX contagion Adding jump-to-default FX contagion to explain the currency basis Direct Default-FX contagion and impact on CDS spreads

A numerical study of Italy’s CDS in EUR and USD

4 Conclusion

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Introduction: Credit Default Swaps and Technical Setting Default Risk: reduced form / intensity credit risk models

Default: Reduced-form intensity modelling & Cox process

Let us consider a probability space(Ω,G,Q,(Gt))satisfying the usual hypothesis and a process(λt,t>0), the intensity, defined on this space. LetΛ(s) =Rs

0λudu.

Default is a jump process(Dt,t>0)with the property thatλtis theGt−intensity ofD. We only focus on the first jump time ofDand we call itτ, the default time.

Let us consider

the filtration generated byλand “default-free processes" (eg short ratert),(Ft); the filtration generated byD,Ht=σ((τ < u),u6t)(“default monitoring") separable filtration assumption: Total filtrationGt= (Ft)∨(Ht);

a r.v.ξ∼exp(1)“jump to default risk", that is independent of(Ft). Let us define

τ:=Λ⁻¹(ξ) assumingλ >0. From the default time definition we get that

Q(τ∈[t,t+dt)|τ > t,Ft) =λtdt, (λtdt is a“local default probability");

Q(τ > T|Ft) =E



Q ZT

0

λsds < ξ

FT

!

|Ft



=E h

e⁻^R^T⁰^λ^s^ds|Ft

i

(λ also“credit spread").

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Introduction: Credit Default Swaps and Technical Setting CDS

Credit Defaul Swaps

A CDS is a contract between two partiesAandBwritten with respect to a set of securities issued by the reference entityCwhere

before default of the reference entity or until a final maturity, one party [protection buyer] pays the other [protection seller] a protection premium; this can be paid upfront, running or both;

upon default of the reference entity, if this happens before the final maturity, the protection seller pays the buyer aloss given defaulton the reference entity’s securities.

A B

Premium:S

LGD

For a running CDS with premium spreadS^c, the premium and the protection leg cash flows discounted back at time 0 and not yet present valued are given respectively by

Π^Premium=S^c XN i=0

1τ>T_iαiD^ccy(0,Ti) +accrual-term, Π^Protection=LGD1τ6TND^ccy(0,τ)

where accrual-term =S^c(τ−Tcoupon beforeτ)D^ccy(0,τ)1_τ<T_Nand

(T0, . . . ,TN)quarterly spaced payment times,αiyear fraction betweenTi−1&Ti; D^ccy(t,T)stochastic discount factor for currencyccyat timetfor maturityT;

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Financial Motivation: CDS in multiple currencies

Multi-currency CDS

Running CDS is often quoted via the spreadS^cthat matches the two legs (“par spread”).

CDSs on a given entity can be traded in different currencies.

We will consider the following currencies:

for each CDS we denote the currency in which premium leg and protection leg are settled as thecontractual currency;

for each reference entity we denote the contractual currency corresponding to the most liquid CDS in the market as theliquid currency.

A B A B

S(ccy1)

LGD(ccy1)

S(ccy2)

LGD(ccy2)

One would prefer to buy protection against the default of Italy in USD rather than EUR.

We will always take the liquid currency economy as the reference pricing measure.

When CDSs are traded in contractual currencies different from the liquid one, a joint model for the reference entity’s credit worthiness and the FX rate is needed to price the basis between the par-spreads.

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Financial Motivation: CDS in multiple currencies The Italy CDS example

A look at the market

Italy’s case

1 Italy’s CDS in EUR:

contractual currency: EUR liquid currency: USD

2 Italy’s CDS in USD:

contractual currency: USD liquid currency: USD

May 2011 Sep 2011 Jan 2012 May 2012 Sep 2012 Jan 2013 May 2013 Sep 2013 0

100 200 300 400 500 600

Spreads (bps)

S^5YEUR

S^5YUSD

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Mathematical framework for multi-currency CDS

1 Introduction: Credit Default Swaps and Technical Setting Default Risk: reduced form / intensity credit risk models CDS

2 Financial Motivation: CDS in multiple currencies The Italy CDS example

3 Mathematical framework for multi-currency CDS Multi-currency pricing

FX rate dynamics and symmetries Quanto survival probabilties Default intensity model

Instantaneous correlation FX/Credit Spreads Pricing equations

Instantaneous correlation not enough for Default-FX contagion Adding jump-to-default FX contagion to explain the currency basis Direct Default-FX contagion and impact on CDS spreads

A numerical study of Italy’s CDS in EUR and USD

4 Conclusion

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Mathematical framework for multi-currency CDS Multi-currency pricing

Mathematical framework for multi-currency pricing

We will consider the case where the contractual and liquid currencies are different.

Consider

a risk neutral measureQwith numeraire(B_t,t>0)with dB_t=r(t)Btdt, B0=1 is associated to the liquid currency:

a risk-neutral measureQˆwith numeraire(Bˆt,t>0)where dˆBt=ˆr(t)Bˆtdt, Bˆ0=1 is associated to the contractual currency;

an exchange rate(Z_t,t>0)between the currencies of the two economies (Zis the price of one unit of the contractual currency in the liquid currency);

interest rates are deterministic functions of time, although we will keep notation general in view of generalizations

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Mathematical framework for multi-currency CDS Multi-currency pricing

Mathematical framework for multi-currency pricing

Recall:Bˆcontractual currency,Bliquid currency.

The link between the two measure is provided by the Radon-Nikodym derivative LT:= ^{d ˆ}_d^Q

Q|_G_T. This can be calculated using a change of numeraire argument and a generic functionφTrepresenting a payoff denominated in the liquid currency. The price of the liquid currency payoffφis thenEt

hB_t B_TφT

i

, satisfying the usual 1

ZtEt

Bt

BT

φ_T

=Eˆt

"

Bˆt

BˆT

φT

ZT

#

⇒Et

Bt

BT

φ_T

=Eˆt

"

ˆBtZt

BˆTZT

φ_T

#

=Et

"

BˆtZt

BˆTZT

φ_TdQˆ dQ _G

T

#

“First discount liquid, then price liquid, then change to contractual=

=first change to contractual, then discount contractual, then price contractual"

The liquid-ccy expectation can be rewritten also as Et

Bt

BT

φT

=Et

"

BtBˆTZT

B_TBˆ_tZ_t BˆtZt

Bˆ_TZ_TφT

# .

From there, we can deduce the Radon-Nikodym derivative to be Lt= ZtBˆt

Z0Bt

, L0=1,

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Mathematical framework for multi-currency CDS FX rate dynamics and symmetries

FX rates no-arbitrage dynamics

The fact that(L_t,t>0)is aG-martingale inQcan be used to deduce no-arbitrage constraints. For example, ifrandˆrare deterministic functions of time and if(Zt,t>0)is a GBM

dZ_t=µ^ZZtdt+σZtdW_t, Z0=z, we can deduce the dynamics of the Radon-Nikodym derivativeLas

dLt=d Bˆt

Bt

Zt

Z0

!

= Bˆt

BtZ0

(dZt+ˆrZtdt−rZtdt),

= Bˆt

BtZ0

(µ^ZZtdt+σZtdWt+ˆrZtdt−rZtdt), L0=1.

The martingale condition onLis given byEt[dL_t] =0, from which we can deduce the FX drift rate:

µ^Z=r(t) −ˆr(t)

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Mathematical framework for multi-currency CDS FX rate dynamics and symmetries

Symmetry for FX rates

What if we had started from^{d ˆ}_d^Q

Q|GTand from the reciprocal FX rate,(Xt,t>0)where X=¹/Z, instead?

If(Zt,t>0)(orX) are GBM, it doesn’t matter which RN derivative we start from. If, for example, we first start fromL, from which we calculate the no-arbitrage drift forZas above(r−ˆr), we subsequently deduceX’s dynamics though Ito’s formula for1/Z, and we finally use Girsanov’s theorem to moveX’s dynamics under the measureQˆ, we obtain

µ^X=ˆr(t) −r(t)

This is the same drift we would have obtained postulating a GBM forXand deriving its drift starting from^{d ˆ}_d^Q

Q|GT.

The same cannot be guaranteed to happen for other type of stochastic process forZ orX(e.g. stoch vol in FX, CEV...).

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Mathematical framework for multi-currency CDS Quanto survival probabilties

Quanto survival probabilities

We start from the price of defaultable zero-coupon bond:

Eˆt

"

Bˆt

BˆT

1τ>T

#

=Et

"

Bˆt

BˆT

1τ>T

d ˆQ dQ

#

= This can be written as

=B(t,T)

Zt Et[ZT1τ>T] (∗),

whereB(t,T) =D(t,T) =Bt/BTis the discount factor from timeTto timet6T. Remember we are assuming deterministic risk free discount rates.

We define thequanto-adjusted survival probabilityas pˆt(T) :=

Eˆt

h_ˆ

B_t Bˆ_T 1τ>T

i B(t,ˆ T) We will often use(Ut,t>0)

Ut:=ZtEˆt

"

Bˆt

BˆT

1τ>T

#

=by(∗)above=B(t,T)Et[ZT1τ>T] which is therefore aQ-price and as such has drift rUdt.

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Mathematical framework for multi-currency CDS Default intensity model

Default intensity modelling

Although for credit spreads the square root processes are better in terms of tractability and closed form solutions (B. et al [2, 3]), for lognormal consistency in this work we model the intensityλunder the liquid measureQas an exponential Ornstein-Uhlenbeck process

λt=e^Y^t

dY_t=a(b−Yt)dt+σ^YdW_t^Y Y0=y, which leads to the default timeτ=Λ⁻¹(ξ),ξ∼exp(1)and recall

λis instantaneous credit spread,λtdtlocal default probability in[t,t+dt); ξis jump to default risk.

Dependence between the credit component and the FX component is modeled though instantaneous correlation,ρ,

dZt=µ^ZZtdt+σ^ZZtdW_t^Z, Z0=z;

dhW^Y,W^Zit=ρdt

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Mathematical framework for multi-currency CDS Instantaneous correlation FX/Credit Spreads

Impact of correlation

Example: CDS on Italy, EUR=Contractual ccy, USD= liquid ccy.

Z is the amount of USD needed to get one unit of EUR.

Ifρnegative⇒intensity tends to grow when FX rateZdecreases.

Whenλ↑default of Italy becomes more likely and the amount of USD needed to get one EUR will tend to decrease (EUR devaluation), so that the EUR protection offered by the CDS will be worth less when benchmarked against the liquid USD CDS This results in a lower par spread for the EUR CDS.

Positive correlation will lead to a larger par spread.

More generally, ceteris paribus, we expect the par spread to increase with the correlation.

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Mathematical framework for multi-currency CDS Pricing equations

The pricing equation

The dependency of survival probability value of the default event that is given by the additional processDt=1τ<tmust be explicitly considered.(Ut,t>0)can be represented as some functionf(t,Xt,Yt,Dt)and we have (Itô’s lemma with jumps)

dU_t=rfdt+∂_tfdt+∂_zf

µ^Zzdt+σ^ZzdW^Z_t +∂_yf

a(b−Y_t)dt+σ^YdW_t^Y

+1 2

σ^Zz2

∂zzfdt+1 2

σ^Y2

∂yyfdt+ρσ^Zσ^Yz∂zyfdt+∆fdDt,

where∆f=f(t,x,y, 1) −f(t,x,y, 0)is the jump to default of the survival probability. To write the pricing equation, it has to be noted thatDis not a martingale, so its compensator must be calculated. The process(M_t,t>0)given by

Mt=Dt− Zt

0

(1−Ds)λsds (1)

is aG-martingale inQ^.

Imposing that the drift ofdUto berUt=rfyields

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The pricing equation

∂tf+µ^Zz∂zf+a(b−Yt)∂yf+1 2

σ^Zz2

∂zzf

+1 2

σ^Y2

∂yyf+ρσ^Zσ^Yz∂zyf+e^y(1−d)∆f=0,

The PDE forfcan be solved it in two steps:

1 calculating firstu(t,x,y) :=f(t,x,y, 1);

2 usinguto solve forv(t,x,y) :=f(t,x,y, 0)

They only get coupled through the terme^y(1−d)∆fand that term only enters the v−equation. Final conditions for the two functions are respectively given by

v(T,z,y) =f(T,z,y, 0) =z;

u(T,z,y) =f(T,z,y, 1) =0 (see Jeanblanc et al [1] for a more general discussion).

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Two pricing equations

The PDE foruis given by

∂tu= −µ^Zz∂zu−a(b−y)∂yu−1 2

σ^Zz2

∂zzu

−1 2

σ^Y 2

∂yyu−ρσ^Zσ^Yz∂zyu u(T,z,y) =0.

and leads tou≡0. The PDE forvis then given by (∆f=v):

∂tv= −µ^Zz∂zv−a(b−y)∂yv−1 2

σ^Zz 2

∂zzv

−1 2

σ^Y2

∂yyv−ρσ^Zσ^Yz∂zyv+e^yv v(T,z,y) =z.

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The pricing equation

uandvcan be interpreted as pre-default and post-default values of a derivative with payoff functionφ:

f(t,x,y,d) =Et

φ(XT,YT,DT)|Xt=x,Yt=y,Dt=d

=1τ>tEt

φ(X_T,Y_T,D_T)|X_t=x,Y_t=y,D_t=0 +1τ6tEt

φ(XT,YT,DT)|Xt=x,Yt=y,Dt=1

=1τ>tv(t,x,y) +1τ6tu(t,x,y) We will solve numerically the PDEs by using anAlternating Directions Implicitscheme variant, accurate fourth order inxand second order int(see [6]).

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Mathematical framework for multi-currency CDS Instantaneous correlation not enough for Default-FX contagion

Limits of instantaneous correlation on Quanto CDS par spreads S

1.0 0.5 0.0 0.5 1.0

ρ

6 4 2 0 2 4 6

S(ρ,σY)−S(ρ=0,σY)

(b ps )

σ^Y=0.2

0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60

σ^Y

10 15 20 25 30 35

S(ρ=1,σY)−S(ρ=−1,σY)

(b ps )

z µ σ^Z a b y T

0.8 0.0 0.1 0.0001 204.0 -4.089 5.0

Italy CDS. Contractual currency: EUR. Liquid Currency: USD. We see that even the maximum excursion of correlation cannot explain the discrepancy between CDS spreads in two currencies, since

differences in the markets can reach much larger ranges than what we see here (case of Italy).

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Mathematical framework for multi-currency CDS Direct Default-FX contagion and impact on CDS spreads

Adding Jump to default: Direct Default-FX contagion

We can account for the devaluation factor directly in the dynamics of(Zt,t>0)by considering

dZ_t=µ¯^ZZtdt+σ^ZZtdW_t^Z+ γZt−dD_t, Z0=z,

Take again the example of Italy.

Now, if Italy defaults, a negativeγwould pushZdown with a jump.

The amount of USD needed to buy 1 EUR jumps down, we have a intantaneous devaluation jump for the EUR, which makes sense in a scenario of default of Italy.

As a consequence the CDS offering protection in EUR will be worth much less when benchmarked with that in USD, so that we expect the par spread to go down with a negativeγ.

A less negative or positiveγwould instead give us a larger par spread.

We expect the par spread to increase withγ.

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Adding Jump to default: Direct Default-FX contagion

Re-doing calculations as above but accounting for the new jump term inZ: dU_t=rfdt+∂_tfdt+∂_zf

µ¯Zzdt+σ^ZzdW_t⁽²⁾+γ^ZzdD_t +∂yf

a(b−Yt)dt+σ^YdW⁽¹⁾_t +1

2

σ^Zz2

∂zzfdt+1 2

σ^Y2

∂yyfdt+ρσ^Zσ^Yz∂zyfdt+∆fdDt−∂zf∆Zt. The pricing equation associated to that is

∂tv= −(r−ˆr)z∂zv−a(b−y)∂yv−1 2

σ^Zz2

∂zzv

−1 2

σ^Y2

∂yyv−ρσ^Zσ^Yz∂zyv+e^y(v−γ^Zz∂zv) v(T,z,y) =z,

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Hazard rate’s dyamics in Q ˆ

So far we always worked in the benchmark liquid currency measureQ^.

However there is an important approximation where we can benefit from deriving the contractual currency measureQˆ dynamics for the intensity. By Girsanov’s theorem:

d ˆMt=dM_t−dhM,Lit

Lt

=dM_t−dhD,γ^ZDit

=dM_t− (1−Dt)γ^Zλtdt

=dDt− (1−Dt)(1+γ^Z)λtdt from which the intensity of the default process inQˆ is given by

ˆλ_t= (1+γ^Z)λ_t

This is important because there are cases when a CDS par-spread can be suitably approximated byS≈(1−R)λ(e.g. deterministic constant hazard rate models). In such cases, dividing both sides by LGD=1-R, which is fixed at the same level in both currencies by the auction (Doctor et al. (2010) [5]), an approximated relation can be written for the par-spreads of the contractual and liquid CDSs as

Sˆ= (1+γ^Z)S

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FX Symmetry

Jump diffusion

Requiring that(Lt,t>0)is aG-martingale inQ^{leads to}

¯

µ^Z=r(t) −ˆr(t) −λ_tγ^Z1τ>t

The dynamics inQˆ^of(Xt,t>0)can be calculated starting from the dynamics of (Zt,t>0)

dX_t= (ˆr−r)Xtdt−σ^ZXtd ˆW_t^Z+Xt−γ^Xd ˆMt, X0= 1 z, where

γ^X= − γ^Z 1+γ^Z That is a dynamics such that ^{d ˆ}_d^Q

Q|Gt is aG-martingale inQˆ^.

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Mathematical framework for multi-currency CDS A numerical study of Italy’s CDS in EUR and USD

Jump to default effect on quanto spreads

−1.0 −0.5 0.0 0.5 1.0

−6

−4

−2 0 2 4 6

∆S (bps)

γ =0

−1.0 −0.5 0.0 0.5 1.0

−62.0

−61.5

−61.0

−60.5

−60.0

−59.5

−59.0

−58.5

−58.0

−57.5 γ =0.6

−1.0 −0.5 0.0 0.5 1.0

ρ

−81.0

−80.5

−80.0

−79.5

−79.0

−78.5

∆S (bps)

γ =0.8

−1.0 −0.5 0.0 0.5 1.0

ρ

−106

−104

−102

−100

−98

−96

−94

γ =1

∆S:=S(ρ,γ) −S(0, 0)

z µ σ^Z a b y σ^Y T

0.8 0.0 0.1 0.0001 -210.0 -4.089 0.2 5.0

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Jump to default effect on default probabilities

We can link the ratio of quanto-adjusted and single-currency default probabilities and the factorγ. For

smallT the two are linked by

1+γ= 1−pˆt(T) 1−pt(T)

−1.0 −0.8 −0.6 −0.4 −0.2 0.0

γ^Z 0.0

0.2 0.4 0.6 0.8 1.0 1.2

1−ˆp(T) 1−p(T) T =1 T =3 T =5 T =10

Figure:Low spread (≈100bps)

−1.0 −0.8 −0.6 −0.4 −0.2 0.0

γ^Z 0.0

0.2 0.4 0.6 0.8 1.0

1−ˆp(T) 1−p(T) T =1 T =3 T =5 T =10

Figure:High spreads, (≈700bps).

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CDS spreads on Italy in 2011-2013

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CDS spreads on Italy in 2011-2013

Linking parameters to market data

Let us recall the change in intensity of default induced by the JTD in the Radon-Nikodym derivative:

ˆλt= (1+γ^Z)λt

This relation can be inverted and used as a way to estimateγfrom market data.

−0.40 −0.35 −0.30 −0.25 −0.20 −0.15 −0.10 −0.05

SEUR^1Y−S^1YUSD S^1YUSD

−0.5

−0.4

−0.3

−0.2

−0.1 0.0 0.1

γ

Figure:Relative basis spread for 1Y maturity CDSs.

−0.40 −0.35 −0.30 −0.25 −0.20 −0.15 −0.10 −0.05

SEUR^5Y−S^5YUSD S^5YUSD

−0.5

−0.4

−0.3

−0.2

−0.1 0.0 0.1

γ

Figure:Relative basis spread for 5Y maturity CDSs.

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Conclusion

Conclusions

Throughout this presentation we showed

a model that can consistently accounts both for instantaneous correlation between FX and hazard rate and for a devaluation effect;

that jump-to-default effects in FX rates are needed to account for observed quanto basis in the market;

how the introduction of jump to default in the RN derivative affects the intensity of the default event in different measures;

that introducing jump to default effects in the FX rate dynamics does not break the

“symmetry”;

Numerical example and a practical recipe to estimate devaluation effect on FX from CDS data.

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Conclusion

References I

[1] Bielecki, T. R., Jeanblanc M. and M. Rutkowski.

Pde approach to valuation and hedging of credit derivatives.

Quantitative Finance, 5, 2005.

[2] Brigo, D., Alfonsi, A.,

Credit Default Swap Calibration and Derivatives Pricing with the SSRD Stochastic Intensity Model,Finance and Stochastic(2005), Vol. 9, N. 1.

[3] D. Brigo, and El-Bachir, N. (2010).

An exact formula for default swaptions’ pricing in the SSRJD stochastic intensity model.Mathematical Finance, Volume 20, Issue 3, July 2010, Pages: 365- [4] D. Brigo, N. Pede, and A. Petrelli (2015).

Multi Currency Credit Default Swaps

SSRN,http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2703605. [5] A. Elizalde, S. Doctor, and H. Singh (2010).

Trading credit in different currencies via quanto CDS. Technical report, J.P. Morgan, October 2010. Retrieved on April 10, 2016 on

http://www.wilmott.com/messageview.cfm?catid=11&threadid=97137

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Conclusion

References II

[6] Düring, B., Fournié, M. and A. Rigal (2013).

High-order ADI schemes for convection-diffusion equations with mixed derivative terms. In: Spectral and High Order Methods for Partial Differential Equations - ICOSAHOM’12, M. Azaiez et al. (eds.), pp. 217-226, Lecture Notes in Computational Science and Engineering 95, Springer, Berlin, Heidelberg, 2013

[7] Pykhtin, M. and Sokol, A. (2013).

Exposure under systemic impact. Risk Magazine.

[8] P. Ehlers and P. Schönbucher.

The Influence of FX Risk on Credit Spreads.

Working Paper, ETH, 2006.

[9] D. Lando.

Credit Risk Modeling – Theory and Applications Princeton Series in Finance, 2004.