Efficient method for KVA pricing under mixed measures

1. KVA Mind your P’s and Q’s!
Shashi Jain, ING Bank
12th Fixed Income Conference, Berlin
Joint work with:
Patrik Karlsson
Quantitative Analytics
Financial Markets NL - ING Bank
Drona Kandhai
Head of Quantitative Analytics
Financial Markets NL - ING Bank

2. The views expressed in the presentation are personal views of the speakers and do not
necessarily reflect the views or policies of their employers.
Disclaimer
2

3. “The rise of KVA: How 10 banks are pricing the capital crunch”, RISK.net, September 2015.
Motivation
3
What method do you use to calculate KVA?
Approximation methods 2
Constant return on equity on all trades 4
Nested Monte Carlo simulations 0
Others 3
Do not calculate 1
Number of respondents 10

4. 1. Need for KVA and associated numerical challenges
2. An efficient method for KVA pricing
3. Tapping in on technology
4. Study on impact of approximations
5. Hedging KVA
Agenda
4

5. • Regulatory Capital is a buffer to absorb losses
• Contributes significant to the banks creditworthiness
• Minimize capital
• Maximize business (e.g., trades) and profits
• Can be seen as another type of funding like FVA
• Is a cost since shareholders requires returns on their investments
Regulatory Capital
5

6. 1. We cannot charge clients, looking back at expiry, the capital consumption
2. We can however, calculate the expected capital consumption cost
3. We charge the expected capital consumption cost
4. And we hedge the expected capital consumption cost.
KVA = 3) + 4)
Regulatory Capital
6

7. 1. Counterparty Credit Risk (CCR) Capital
2. CVA Capital
3. Market risk (MR) Capital
Capital Valuation Adjustment (KVA)
7

8. 1. KVA (risk-neutral) at inception
2. Capital
3. Effective EPE (EEPE), 1Y running maximum
4. Expected exposure (historical)
8
Capital Valuation Adjustment (KVA): CCR component

9. CCR is an interesting capital, especially from Quants perspective
Difficulties:
1. The ℙ-in-ℚ problem
2. Conditional pathwise future exposures
3. Non-linearity due to the max function
Capital Valuation Adjustment (KVA)
9

10. Monte Carlo methods for KVA

11. Most numerical effort is in producing pathwise future exposures
1. Nested Monte Carlo
• ℙ-in-ℚ simulation
2. Regression
• Least Squares Monte Carlo (LSMC)
• Two simulations, ℙ and ℚ
• Stochastic Grid Bundling Method (SGBM)
• One simulation in ℚ
• Closed form function: ℙ → ℚ
3. Use the un-conditional EE's instead of future EE's and that ℙ ∽ ℚ
• Reducing the complexity to that of CVA/FVA.
• Similar work in Elouerkhaoui (2016), without ℙ ∽ ℚ assumption.
Methods
11

12. 1. At each time 𝑇𝑘, the paths 𝑆 are clustered into non-overlapping bundles ℬ
2. Law of iterated approximation
3. The inner expectation is approximated using least squares
4. The outer conditional expectation can be calculated in closed form
Remark: Efficiently calculate the floored portfolio values in different measures for each time
Stochastic Grid Bundling Method (SGBM)
12

13. SGBM-1
13
1. Simulate risk-neutral paths 𝑆

14. SGBM-2
14
1. Simulate risk-neutral paths 𝑆
2. Calculate Exposures 𝑉+

15. SGBM-3
15
1. Simulate risk-neutral paths 𝑆
2. Calculate Exposures 𝑉+
3. Bundle at time 𝑡 𝑘

16. SGBM-4
16
1. Simulate risk-neutral paths 𝑆
2. Calculate Exposures 𝑉+
3. Bundle at time 𝑡 𝑘
4. For each bundle ℬ𝑖

17. SGBM-5
17
1. Simulate risk-neutral paths 𝑆
2. Calculate Exposures 𝑉+
3. Bundle at time 𝑡 𝑘
4. For each bundle ℬ𝑖
i. Regression

18. SGBM-6
18
1. Simulate risk-neutral paths 𝑆
2. Calculate Exposures 𝑉+
3. Bundle at time 𝑡 𝑘
4. For each bundle ℬ𝑖
i. Regression
ii. Conditional EE in ℙ

19. SGBM-7
19
1. Simulate risk-neutral paths 𝑆
2. Calculate Exposures 𝑉+
3. Bundle at time 𝑡 𝑘
4. For each bundle ℬ𝑖
i. Regression
ii. Conditional EE in ℙ
5. Capital
Remark: known moments (GBM, LMM, HW etc.)  We do not need to simulate the ℙ paths!

20. SGBM – Bundling Intuition
20
Increasing the number of bundles and paths  “nested” Monte Carlo effect

21. SGBM – Bundling Example (2 assets)
21

22. SGBM – Bundling Example (2 assets)
22
𝛼1
𝛼2

23. SGBM – Bundling Example (2 assets)
23
𝛼2
𝛼3
𝛼1

24. • In the limiting case, number of bundles, and paths
• the distribution would be similar to a point source
• Allow us to mimic nested Monte Carlo
• Efficiently price ℙ-in-ℚ
• The moments are known in both ℙ and ℚ (e.g., GBM, LMM, HW, Heston...)
• We do not need to simulate the historical paths
• Allow for high-dimensional (realistic) problems
SGBM – Summary
24

25. Approximation – ”Frozen” EEPEs.
25
We can reducing complexity to that of CVA/FVA
Additionally an assumption on a choice of single measure :

26. Hybrid parallel processing: GPU+CPU

27. KVA is computationally expensive, much more than CVA/FVA.
1. GPU infrastructure for CVA/FVA
• Parallel exposure calculations
2. Employment of simultaneous GPU/CPU processing.
GPU-CPU Calculation
27
2. Parallel over
portfolios and
time grid
3. Exposures
5. KVA
GPU
CPU
1. Data
4. Regression

28. SGBM-7
28
1. Simulate risk-neutral paths 𝑆
2. Calculate Exposures 𝑉+
3. Bundle at time 𝑡 𝑘
4. For each bundle ℬ𝑖
i. Regression
ii. Conditional EE in ℙ
5. Capital
Remark: known moments (GBM, LMM, HW etc.)  We do not need to simulate the ℙ paths!

29. Results

30. 1. Bermudan swaptions
i. Exposures, PFE, CVA
2. KVA
i. Nested Monte Carlo
ii. Regression
iii. “Frozen” EEPEs
• Assuming one measure, ℙ or ℚ
3. Hedging KVA
Numerical Examples
30

31. CVA variance reduction (vs. LSMC) for 5Y and 10Y Bermudan swaptions (HW1F)
Example – CVA on Bermudan Swaption
31 Karlsson, P., Jain, S. and Oosterlee, C.W. (2016). Counterparty Credit Exposures for Interest Rate Derivatives using the Stochastic Grid Bundling Method. Applied
Mathematical Finance. Forthcoming.

32. FX Portfolio (no collateral, initial margin)
• Three market setups
• Case 1: ℙ ∽ ℚ, calibrated to market data 19 Jan 2016.
• Case 2: Bump risk-neutral volatility 10%
Example – Setup
32
Case ℙ vol ℚ vol
1
10.08% 10.08%
2
10.08% 20.08%

33. Case Nested
MC
SGBM Frozen ℙ Frozen ℚ
1
16.15 16.14
(0.03%)
15.26
(~6%)
15.25
(~6%)
2
26.04 26.97
(~4%)
15.26
(~43%)
29.28
(~13%)
Example – KVA(0)
33

34. Case I Results
34

35. 35
Case II Results

36. Hedging KVA

37. 1. Charge KVA at inception
2. Delta hedge
3. Proceeds invested in the money-market bank account:
4. After one time step:
1. Bank account grows at risk-free rate
2. KVA changes due to run-off and market changes
3. Pay out Capital costs for this time interval
4. Hedge value changes.
5. All this should add up to 0 (or hedge error).
Example – Discrete-Time Hedging
37

38. The PnL of the total capital cost
1. Cruder approximation  larger potential bleeding.
2. If we do not hedge, the volatility can eat up the entire KVA charge.
Example – Discrete-Time Hedging
38
Case Hedged KVA Price
(NMC)
SGBM Frozen ℙ Frozen ℚ
KVA (t0) + H. Err KVA (t0) + H. Err KVA (t0) + H. Err
1
Yes
16.15
16.14
16.14
15.26
16.14
15.25
16.14
No 16.15 16.14 16.15
2
Yes
26.04 26.97
26.02
15.26
25.97
29.28
26.03
No 26.05 26.00 26.06

39. 39
Hedge results

40. Conclusions

41. • Frozen EEPE
• Okay if there is not too much convexity in the future EE and ℙ and ℚ are similar.
• Biased hedging errors (bleeding).
• Regression: SGBM
• Able to capture (most) convexity of the future EE
• Candidate for the ℙ-in-ℚ problem
• Volatility for the naked total cost position is significant larger than the hedged one
• If we do not hedge
• We will bleed
• Overcharge KVA to absorb the volatility of capital cost and the market risk components
• If we hedge
• Reducing the total capital cost volatility
• Acknowledge KVA on the Balance Sheet?
Conclusions
41

42. Jain, S., Karlsson, P. and Kandhai, D. (2016). KVA, Mind your P’s and Q’s!. Submitted for publication.
Available at SSRN: http://ssrn.com/abstract=2792956.
Jain, S. and Oosterlee, C.W. (2015). The stochastic grid bundling method: Efficient pricing of Bermudan
options and their Greeks. Applied Mathematics and Computation 269: 412-431.
http://dx.doi.org/10.1016/j.amc.2015.07.085
Feng, Q., Jain, S., Karlsson, P., Kandhai, D. and Oosterlee, C.W. (2016). Efficient computation of
exposure profiles on real-world and risk-neutral scenarios for Bermudan swaptions. The Journal of
Computational Finance 20: 139-172. http://dx.doi.org/10.21314/JCF.2017.337.
Karlsson, P., Jain, S. and Oosterlee, C.W. (2016). Counterparty Credit Exposures for Interest Rate
Derivatives using the Stochastic Grid Bundling Method. Applied Mathematical Finance. Forthcoming.
http://dx.doi.org/10.1080/1350486X.2016.1226144.
Karlsson, P., Jain, S. and Oosterlee, C.W. (2016). Fast and accurate exercise policies for Bermudan
swaptions in the LIBOR market model. International Journal of Financial Engineering 3: 1650005.
http://dx.doi.org/10.1142/S2424786316500055.
References
42

43. Thank you!
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