Dr. Tony Webb, Director of Analytics, FINCAD lead a discussion on CVA best practices and current techniques at Random Walkers Roundtable on CVA organized by Maroon Analytics and hosted by 7city Learning. Random Walkers is a regular, open discussion on current topics in quantitative finance. With topics proposed by participants, each session is led by an expert in the field and is formatted to encourage active participation from all attendees. FINCAD's F3 is a fast & flexible CVA pricing solution. Watch a demo at http://bit.ly/Rdv7lf

1. CVA SENSITIVITIES WITHOUT BUMPING
Applications to CVA Allocation and Incremental CVA
PRESENTER:
Tony Webb, Director of Analytics
18th April 2012

2. Outline
• CVA as a Pricing Problem
• Overview of Sensitivity Methodology
• Performance of Sensitivity Methodology
• Application to CVA Allocation
• Application to Incremental CVA
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3. Typical CVA Process
Simulation of 1 Risk neutral for CVA/DVA;
market evolution1 Real world for PFE
2 Risk neutral for pricing
Market data
Trade data
paths
Pricing Netting &
Engine2 collateral rules
Exposure
paths
Default Expected
PFE profile
probabilities Exposure profile
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4. Unified CVA Approach
Default Netting &
Market data Trade data
probabilities collateral rules
CVA Pricing 1 Risk neutral simulation for
Engine1 unified pricing and
CVA/DVA calculation
CVA/DVA Sensitivities
(with WWR) (Analytically)
Incremental
Hedge factors CVA Allocation
CVA
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5. UNCOLLATERALIZED CVA
Where
is the random time of default
is the value at time t of cashflows remaining after time t
is the value of money market account at time t
is the default probability
is the recovery rate
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6. UNCOLLATERALIZED CVA
• This is equivalent to the value of a portfolio of N
synthetic contracts
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7. EXPOSURE OF A NETTING SET
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8. EXPOSURE OF A NETTING SET
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9. COLLATERALIZED CVA
• Again, this is equivalent to the value of a portfolio of
N synthetic contracts
• The kth synthetic contract has a value at time
of:
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10. Forming the Synthetic Contract
The value of the kth synthetic contract is the expected loss from a product,
or portfolio of products, due to counterparty default during the kth period.
The synthetic contract is formed as follows:
• form a product that represents the remainder of the underlying
portfolio after tk-1 by filtering out cash-flows prior to that time
• weight each cash flow in this product by a “Credit Index”; this
weighting has the effect of making each cash flow in the remainder of
the product after the tk-1 explicitly contingent on the occurrence of the
default event between tk-1 and tk. This weight also includes the LGD.
• form a new product which represents the positive part of this weighted
remainder, as the choice between it and a zero-value product.
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11. Outline
• CVA as a Pricing Problem
• Overview of Sensitivity Methodology
• Performance of Sensitivity Methodology
• Application to CVA Allocation
• Application to Incremental CVA
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12. ANALYTIC SENSITIVITY
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13. ANALYTIC SENSITIVITY
• First–order sensitivity of portfolio value to all
market data quotes and all parameters
• No finite difference (“bumping”)
• Chain Rule applied at each function composition
• Automatic tracking of all intermediate sensitivities
• Incorporated into software architecture design
• On-the-fly, no source code generation
• Works with generic pricing engine for arbitrary trades
• Sensitivities propagate through any calibration step
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14. Outline
• CVA as a Pricing Problem
• Overview of Sensitivity Methodology
• Performance of Sensitivity Methodology
• Application to CVA Allocation
• Application to Incremental CVA
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15. SENSITIVITIES: ANALYTIC VS BUMPING
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16. PERFORMANCE: ANALYTIC VS BUMPING
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17. NETTING SET FOR CVA HEDGE TEST
Portfolio Swap Pay/Rec Maturity Notional Fixed Rate
# (Million) (Par Rate)
1 Pay 1Y 2 0.33%
2 Rec 2Y 1 0.49%
3 Rec 3Y 3 1.64%
4 Pay 4Y 5 2.93%
Netting 5 Pay 5Y 5 3.69%
Set
6 Rec 6Y 4 4.19%
7 Rec 7Y 5 4.54%
8 Rec 8Y 2 4.80%
9 Pay 9Y 3 5.00%
10 Rec 10Y 1 5.15%
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18. EXPECTED EXPOSURE
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19. CVA HEDGE (BY CDS)
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20. Outline
• CVA as a Pricing Problem
• Overview of Sensitivity Methodology
• Performance of Sensitivity Methodology
• Application to CVA Allocation
• Application to Incremental CVA
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21. CVA ALLOCATION
• Calculation of additive contributions of M
individual trades to CSA-Level CVA
• CVA =
• This is sufficient for CSA-level CVA to be
allocated to contributing desks or traders
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22. EULER ALLOCATION
• Homogeneous Function of degree
• Euler Allocation
If homogeneous function is piecewise
differentiable, then
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23. CVA ALLOCATION (NO COLLATERAL)
• CVA (No Threshold)
• is the stochastic discount factor
• CVA is a positive homogeneous function of degree 1
of portfolio weights
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24. CVA ALLOCATION (NO COLLATERAL)
• CVA Allocation
; so
• Marginal CVA:
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25. CVA ALLOCATION (NO COLLATERAL)
• CVA Allocation
; so
• Marginal CVA:
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26. TEST PORTFOLIOS
Portfolio Swap Pay/Rec Maturity Notional Fixed Rate
# (Million) (Par Rate)
1 Pay 1Y 2 0.33%
2 Rec 2Y 1 0.49%
3 Rec 3Y 3 1.64%
4 Pay 4Y 5 2.93%
5 Pay 5Y 5 3.69%
B A
6 Rec 6Y 4 4.19%
7 Rec 7Y 5 4.54%
8 Rec 8Y 2 4.80%
9 Pay 9Y 3 5.00%
10 Rec 10Y 1 5.15%
11 Pay 11Y 1 5.28%
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27. CVA ALLOCATION
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28. CVA ALLOCATION AS A SPREAD (BP)
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29. CVA ALLOCATION WITH NEW TRADE
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30. TEST PORTFOLIOS
Portfolio Swap Pay/Rec Maturity Notional Fixed Rate
# (Million) (Par Rate)
1 Pay 1Y 2 0.33%
2 Pay 2Y 1 0.49%
3 Pay 3Y 3 1.64%
4 Pay 4Y 5 2.93%
5 Pay 5Y 5 3.69%
B* A*
6 Pay 6Y 4 4.19%
7 Pay 7Y 5 4.54%
8 Pay 8Y 2 4.80%
9 Pay 9Y 3 5.00%
10 Pay 10Y 1 5.15%
11 Pay 11Y 1 5.28%
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31. CVA ALLOCATION WITH NEW TRADE
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32. CVA ALLOCATION (WITH COLLATERAL)
• CVA ( Threshold)
• CVA is a positive homogeneous function of degree
1 of the augmented set of arguments
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33. CVA ALLOCATION (WITH COLLATERAL)
• Step 1: Allocation includes Threshold
• Marginal CVA
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34. CVA ALLOCATION (WITH COLLATERAL)
• Step 1: Allocation includes Threshold
• Marginal CVA
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35. CVA ALLOCATION (WITH COLLATERAL)
• Step 2: Allocation on Threshold
• Proposed by Pykhtin + Rosen(2010)
• Expected Exposure Level
• Trade contributions occur when the portfolio value exceeds
the threshold
• Suitable for allocation of time-dependent Expected Exposure
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36. CVA ALLOCATION (WITH COLLATERAL)
• Alternative:
• Weight at the CVA Level
• Integrate into default probability
Each trade has a greater contribution to threshold allocation
at times when higher default probability than with lower
probability
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37. CVA ALLOCATION (WITH COLLATERAL)
• Numerator of
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38. CVA ALLOCATION (WITH COLLATERAL)
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39. CVA ALLOCATION (WITH COLLATERAL)
• Numerator of
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40. CVA ALLOCATION (WITH COLLATERAL)
• Denominator of
• Weight
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41. CVA ALLOCATION (WITH COLLATERAL)
• Complete allocation scheme:
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42. CVA ALLOCATION (WITH COLLATERAL)
• Complete allocation scheme:
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43. TEST PORTFOLIO
Portfolio Swap Pay/Rec Maturity Notional Fixed Rate
# (Million) (Par Rate)
1 Pay 1Y 2 0.33%
2 Pay 2Y 1 0.49%
3 Pay 3Y 3 1.64%
4 Pay 4Y 5 2.93%
5 Pay 5Y 5 3.69%
C
6 Pay 6Y 4 4.19%
7 Pay 7Y 5 4.54%
8 Pay 8Y 2 4.80%
9 Pay 9Y 3 5.00%
10 Pay 10Y 1 5.15%
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44. EXPECTED EXPOSURE (NO THRESHOLD)
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45. CVA ALLOCATION(THRESHOLD 10 MILLION)
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46. EXPECTED EXPOSURE (THRESHOLD 2.5 MILLION)
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47. CVA ALLOCATION(THRESHOLD 2.5 MILLION)
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48. EXPECTED EXPOSURE (THRESHOLD 1.5 MILLION)
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49. CVA ALLOCATION(THRESHOLD 1.5 MILLION)
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50. EXPECTED EXPOSURE (THRESHOLD 0.5 MILLION)
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51. CVA ALLOCATION (THRESHOLD 0.5 MILLION)
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52. Outline
• CVA as a Pricing Problem
• Overview of Sensitivity Methodology
• Performance of Sensitivity Methodology
• Application to CVA Allocation
• Application to Incremental CVA
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53. INCREMENTAL CVA
• The increase (decrease) in CSA-level CVA caused
by adding a new trade to the netting set
• Fast computation needed for prospective trades
so that CVA can be priced into the deal
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54. MARGINAL CVA
• First Order Approximation
If function is differentiable,
then for small
• CVA is a differentiable function of notionals
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55. INCREMENTAL CVA
• CVA First-order Approximation
• Advantages
• Easy to get using analytic sensitivities
• Time saving (compared to recalculating CVA)
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56. INCREMENTAL CVA
• CVA First-order Approximation
• Advantages
• Easy to get using analytic sensitivities
• Time saving (compared to recalculating CVA)
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57. TEST PORTFOLIO
Portfolio Swap Pay/Rec Maturit Notional Fixed Rate
# y (Million) (Par Rate)
1 Pay 1Y 20 0.33%
2 Rec 2Y 10 0.49%
3 Rec 3Y 30 1.64%
4 Pay 4Y 50 2.93%
5 Pay 5Y 50 3.69%
D
6 Rec 6Y 40 4.19%
7 Pay 7Y 50 4.54%
8 Pay 8Y 20 4.80%
9 Rec 9Y 30 5.00%
10 Pay 10Y 10 5.15%
11 to 20 Same as Same as 0 Same as
1 to 10 1 to 10 1 to 10
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58. INCREMENTAL CVA ( 1Y SWAP)
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59. INCREMENTAL CVA ( 5Y SWAP)
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60. INCREMENTAL CVA ( 10Y SWAP)
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61. SUMMARY
• CVA is equivalent to the value of a portfolio of
synthetic contracts
• This portfolio can be valued using the same engine
used for real contracts
• Analytic sensitivities are fast and accurate
• Applications of analytic CVA sensitivities:
• CVA hedging
• CVA allocation
• Pre-trade estimation of incremental CVA
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62. QUESTIONS?
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