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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
2
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
16
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
22
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]
27
Example With Normal Marginals
• Weight function is exp(θx2y)
• Can’t get this through a Gaussian copula
28
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
29
Example Continued
2-year KRW foreign exchange forward
θ
CVA (% of Independent)
Worst Case
Independent
30 θ
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
31
CVA with Wrong Way Risk
At Each θ, Estimate Correlation Between Exchange Rate and Credit Spread – Match Empirical Value
32
Independent
“Correlation” Mapping
CVA with Wrong Way Risk
Choose θ to Match Correlation
33
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
34
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
36