Bounding Wrong-Way Risk in CVA Calculation

Advances in Financial Mathematics Conference

January 10, 2014

Paul Glasserman and Linan Yang Columbia University

CVA – The Price of Counterparty Risk

• Counterparty risk – still one of the main risks of and to the financial system

• CVA = Credit Valuation Adjustment – Adjustment made to the price of a derivative (or portfolio of

derivatives) to reflect counterparty credit risk – Measures the market price of the counterparty’s option to default

• CVA calculations are among the most computationally demanding tasks

faced by banks • CVA capital charge for counterparty risk is among the most significant

additions in Basel III

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Wrong-Way Risk

• CVA depends on joint distribution between default time and exposure at default

• Wrong-way risk refers an adverse joint distribution • Straightforward examples

– A company sells a put option on its own stock – A bank sells CDS protection on another similar bank – A bank enters into a currency swap paying a foreign currency

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Default time Exposure at default

Wrong-Way Risk, Continued

• More generally, joint modeling of credit and market risk is difficult – Independence is often assumed (e.g., Basel standardized formula) – Market and credit models may live in different systems and may not

communicate

• What we do – Estimate worst-case CVA given marginal models for market and credit

risk – Develop family of estimates that “interpolate” from independent case

to worst case that can be calibrated to observed data

• In contrast, simple copula models do not achieve worst case and do “interpolate” well

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Problem Formulation – Single Counterparty

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Scalar Case

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CVA is a Vector Version of This Problem

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Extremal Distributions in the Vector Case

• Rather little is known in general – see the recent book of Rüschendorf (2013) for special cases

• For distributions with finite support, extremal distributions are easy to find computationally

• Use finite number of simulated paths (simulated separately from market and credit models) and then investigate what happens as number of paths increases

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Simulation Formulation

• Generate paths (separately for exposures and default times) • We are trying to match exposure paths with default times

Implied default time probabilities or simulated frequencies

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Solution

CVA contributions Joint probabilities

Pij

Finding the worst case is a linear programming problem:

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Comments on the Worst-Case CVA

• Linear program can be solved very quickly, even with a large number of paths

• Reduces to classical co-monotonic solution in the scalar case

• We can use simulated default probabilities or market-implied probabilities for the qj

• If these default-time probabilities are multiples of 1/N, then the LP solution has

only 0-1 probabilities: each exposure path gets assigned to exactly one default time

• Chapter 8 of Glasserman and Yao (1994) has other applications to simulation, connections with Monge sequences, greedily solvable transportation problems, and antimatroids

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Convergence to Theoretical Upper Bound

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Convergence to Theoretical Upper Bound

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Convergence to Theoretical Upper Bound

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Bilateral Formulation

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Bilateral Formulation

• Finding the worst case, constraining three marginals:

• We can constrain the joint distribution of the two default times, if we know it

• We could incorporate downgrade triggers – we just need to know the contribution Cijk for every combination of paths for the market and the two parties

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Adding Counterparty-Specific Information

CVA contributions Joint probabilities

Pij

Some of the default risk has nothing to do with exposure (the worst case is less bad):

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Adding Pricing Constraints

• Suppose we want to enforce conditions of the form

for some payoffs Z • Finite sample version is a linear constraint: Let

We require

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Duality and Deltas

• As a byproduct of the LP solution, we get dual variables on the constraints • Dual variables on the column-sum constraints give sensitivities to default

probabilities

Pij

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Contrast with Gaussian Copula

• Rosen-Saunders (2012) method: Sort exposure paths based on a scalar attribute (e.g., path average or path max) and then link to default time with GC

• Perfect correlation does not yield the worst (or best) case

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• The worst case looks something like this… …lower correlation like this

• When we back away from the worst case using GC, we spread out the probability • But points that are close in the grid may have very different CVA contributions, we

get a sharp drop in CVA • Need a better way to find the full range of CVA from independent to worst case

Contrast with Gaussian Copula

Default Time

Path Index

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Constraining the Worst Case

• Constrain deviations from a reference model Fij using a relative entropy constraint

• We use the independent case as the reference model, Fij =qj/N • Interpret wrong-way risk as a type of model risk in the sense of Hansen and

Sargent (2007) and Glasserman and Xu (2012) • Similar formulation used by Bosc and Galichon (2010) on extremal dependence

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Penalty Formulation

• Penalize deviations from a reference model Fij using a relative entropy penalty

• We use the independent case as the reference model, Fij =qj/N • With θ = 0, get the independent case; with θ = infinity, get the worst case

• No longer linear, but very simple to implement…

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Implementation: IPFP Algorithm

Initial guess (gives more weight to entries with high CVA contributions)

Rescale to match row marginals

Rescale to match column marginals

Iterate

• Converges to optimal solution with penalty 1/θ (Ireland-Kullback 1968) • Compare with copula: weight depends on CVA contribution, not on location in grid

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Comments On This Method

• Repeating this procedure for multiple θ values, we get the full range of possible “wrong-way” distributions from the independent case to the worst case

• (By taking negative θ values, we get the full range of “right-way” distributions)

• The cumulative rescalings give us dual variable for the original problem – these are credit Deltas for CVA

• Extends to bilateral CVA – need to rescale 3-dimensional joint distribution iteratively to match marginals for exposure paths and two default time distributions

• In principle, extends to portfolio CVA with multiple counterparties, but direct optimization may be faster than IPFP with many counterparties

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• Fix θ>0

• Proof uses duality result of Bhattacharya (2006) for upper bound and a uniqueness result of Rüschendorf and Thomsen (1993) for I-projections

Limit and Convergence of Penalty Problem

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Interpreting θ

• We can “apply” IPFP with standard normal marginals and weight exp(θxy) • Not hard to see that this produces a bivariate normal with correlation

• Thus,

• So, we can parameterize the full range using a pseudo-correlation parameter ρ in [-1,1]

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Example With Normal Marginals

• Weight function is exp(θx2y)

• Can’t get this through a Gaussian copula

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Choosing θ: An Example

• Portfolio of a single trade – 2 year Korean won (KRW) foreign exchange forward – Party A pays KRW, counterparty B pays USD – Counterparty B is Korean Bank

• Exposure of party A depends on foreign exchange rate of KRW – Simulate paths of positive exposure

• Credit curve of counterparty B – Use sovereign credit spread to approximate

• Wrong way risk is from the correlation between the foreign exchange rate and credit quality of counterparty B

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Example Continued

2-year KRW foreign exchange forward

θ

CVA (% of Independent)

Worst Case

Independent

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How to Choose θ?

• Choose θ to match some observed measure of dependence

• For example, we observe correlation between credit spread and exchange rate in the real world

• Simulation at each θ yields a correlation parameter as well, so we can choose the value that matches the empirical correlation

• Note that we are not saying that correlation determines the full joint distribution – if it did, this approach would be pointless

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CVA with Wrong Way Risk

At Each θ, Estimate Correlation Between Exchange Rate and Credit Spread – Match Empirical Value

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Independent

“Correlation” Mapping

CVA with Wrong Way Risk

Choose θ to Match Correlation

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Simulation upper bound at ρ=0.99

Independent

Correlation Mapping

How Do Credit Spread and Exposure Volatility Affect Worst-Case Wrong-Way Risk?

• 1 year horizon, lognormal exposure, flat CDS term structure • Look at ratio of Worst Case CVA/Independent CVA • Ratio increases with volatility, decreases with spread

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0 0.1 0.2 0.3 0.4 0.5 0.6

Ratio

Exposure Volatility

Worst-Case Wrong-Way Risk Ratio

250 bps

500 bps

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0 100 200 300 400 500 600

Rati

o

CDS Spread

Worst-Case Wrong-Way Risk Ratio

20% vol

40% vol

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Summary

• Practical method to find worst-case wrong-way risk in CVA – No assumptions required on joint distribution of exposure and default time – Some additional information can be added through constraints

• Extends to bilateral CVA and portfolio CVA with multiple counterparties

• By penalizing deviations from independence, we can sweep out the full range of

possible CVA from independent case to worst case

• This nonlinear problem can be solved easily through iterative matrix rescaling

• Choose penalty parameter to match empirical correlation (or other feature)

• Worst-case impact depends on exposure volatility and credit spread 35

Thank You

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