1. Presenter: Alexander Kaganov Supervisor: Paul Chow
Hardware Acceleration of Monte-Carlo Structural Financial Instrument Pricing Using a Gaussian Copula Model
Presenter: Alexander Kaganov
Supervisor: Paul Chow

2. Increasing Computational Requirements (1/3)
In recent years the financial industry has seen:
1. Increasing contract/model complexity
Every year new models are developed
Unavailability of closed-form solution
Necessitate Monte-Carlo pricing

3. Increasing Computational Requirements (2/3)
2. Increasing portfolio sizes
Increase in simple to price instruments
Increase in complex derivate security
CDO issuance has increased from $157 billion in 2004 to $507 billion in 2007 (>3x)¹
3xN instruments
3xY time (at least)
N instruments
Y time
¹ SIFMA

4. Increasing Computational Requirements (3/3)
3. Ever-present need to make real-time decisions
Market trends can change quickly
Instruments traded electronically
1 ms in Latency is Worth $100 M in Stock Trading Business Value (AMD Analyst Day-26 july 2007)

5. Trends in Financial Monte-Carlo Algorithms
1. Computationally intensive
Accuracy Time2
2. Highly repetitive
A large portion of the calculation time is spent in a small portion of the code
(~90% of the time is spent in ~10% of the code)
3. High degree of coarse and fine-grain parallelism
Fine-Grain
Coarse-Grain
Typical MC Financial simulation

6. Collateralized Debt Obligation (CDO)

7. CDO Problem: Solution:
Banks typically hold portfolios with highly volatile assets.
Solution:
Sell assets to an outside entity (SPV), which combines the different assets together into one collateral pool
Repackage the pool as CDO tranches.
Sell tranches as form of protection to investors in return for premium payments

8. CDO Structure/Pricing (1/2)
Each tranche has attachment and detachment points
Losses below attachment point → the tranche is unaffected
Losses above the detachment point → the tranche becomes inactive
Investor premium is paid based on the tranche width minus tranche losses
Investors
Borrowers
Super Senior: 12%-100%
Bonds
Senior: 6% -12%
Loans
Collateral Pool
(Credit Default Swap)
Mezzanine: 3% -6%
CDS
SPV
CDOs
Equity: 0% -3%
Sponsor (Bank)
Tranches

9. CDO Structure/Pricing (2/2)
Mezzanine Tranche:
Detachment (6%)
Paid premium based on the full investment
Investor Premium Payments
Investor Premium Payments
Losses 1/3 of the principal investment. Premium paid based on 2/3 of the original investment 4%
Tranche Losses
Attachment (3%)
CDO Tranche Value = Premium Leg – Default Leg
S =tranche thickness si= Premium
di= Discount factor Li= Tranche loses at time interval i

10. Li’s Gaussian Copula Model
Monte-Carlo (MC) pricing model
For each path:
One-Factor
Multi-Factor
1. Generate: or
2. Compare:
3. Record losses:

11. Implementation

12. Multi-Core Architecture
Three portions: Distributor, Pricing cores, and Collector.
Coarse Grain Parallelism: MC paths divided among OFGC cores
All cores have the same input data except for random generator seeds
Data transfer occurs in parallel to calculations
Double Buffering
Minimal required data transfer rate with an external processor of: 11.3MBytes/sec
1-Lane PCI express- 250 MBytes/sec
Data transfer latency can be hidden

13. Pricing Core Design Phase 1: Generate Yi
Phase 2: Compare Yi<Φ-1[P(τi<t)]. Record partial losses
Phase 3: Combine the partial sums, L(ti)’s.
Phase 4: Convert collateral pool losses to tranche losses
Phase 5: Accumulate tranche losses

14. Phase 1 One-Factor Multi-Factor
Four factors accumulated per clock cycle
Accumulation is performed in parallel to Phase 2 execution
One-Factor
Multi-Factor

15. Pricing Core Design Phase 1: Generate Yi
Phase 2: Compare Yi<Φ-1[P(τi<t)]. Record partial losses
Phase 3: Combine the partial sums, L(ti)’s.
Phase 4: Convert collateral pool losses to tranche losses
Phase 5: Accumulate tranche losses

16. Phase 2 Compare Yi<Φ-1[P(τi<t)]. Record Losses
Fine-grain parallelism: parallelize over time
8 replicas
More replicas → higher speedup (potentially)
However, large portions of the hardware become underutilized
Pipelined adder latency creates multiple partial sums

17. Pricing Core Design Phase 1: Generate Yi
Phase 2: Compare Yi<Φ-1[P(τi<t)]. Record partial losses
Phase 3: Combine the partial sums, L(ti)’s.
Phase 3: Combine the partial sums, L(ti)’s.
Phase 4: Convert collateral pool losses to tranche losses
Phase 4: Convert collateral pool losses to tranche losses
Phase 5: Accumulate tranche losses
Phase 5: Accumulate tranche losses

18. Results

19. One-Factor Utilization*
Single Precision
Double Precision
Hybrid
“Integer”
Flip-Flops
6530
9910 (+51.2%)
6721 (+2.9%)
4906
(-24.9%)
LUTs
7052
13325 (+89.0%)
7599 (+7.8%)
5224
(-25.9%)
BRAMs 15
31 (+106.7%)
(0%)
DSP48Es 29
40 (+37.9%)
30 (+3.4%)
7 (-75.9%)
Frequency
248.8
190.9
(-23.3%)
244.8
(-1.6%)
268.2
(+7.8%)
Average Error (%)
0.37
[1.07]
3.02E-5
[5.27E-5]
*Utilization for Virtex 5 SX50T chip

20. Performance: Benchmarks
Credit rating and number of instruments are based on Dow Jones CDX
Notionals obtained from Moody’s, range from $600,000 to $6.6 billion
α: uniformly distributed in [0, 1]
Recovery rate: Normally distributed, N (0.4,0.15)
# of Time Steps: Normally distributed, N (20,10)
# of Systemic Factors: varied from 1 to 16 #
Based on Data From
# of Assets
# of Time Steps
# of Default Curves 1
CDX.NA.HY
100 15 5 2
CDX.NA.IG
125 35 3
CDX.NA.IG.HVOL 30 19 4
CDX.NA.XO 22
CDX.EM 14 6
CDX.DIVERSIFIED 40 23 7
CDX.NA.HY.BB 37 13 8
CDX.NA.HY.B 46 26 9
Semi-homogenous
400 24

21. Processor vs. FPGA setup
3.4 GHz Intel Xeon Processor
3GB RAM
C++ program
100,000 Monte-Carlo paths
Virtex 5 SX50T speed grade -3
Connected to host through PCI express
100,000 Monte-Carlo paths

22. Performance: One-Factor Single Core Results (1/3)

23. Performance: One-Factor Single Core Results (2/3)

24. Performance: One-Factor Single Core Results (3/3)
Single Core Average Acceleration:
Double Precision: 10.8 X
Single Precision: 14.0 X
Single/Double Hybrid: 13.8 X
Integer: 15.8 X

25. Performance: One-Factor Multi-Core Results
Monte-Carlo paths independence allows for a linear speedup as more pricing cores are incorporated.
Double
Single
Single/Double Hybrid
Integer
Single Core
Acceleration
10.8X
14.0X
13.8X
15.8X
Maximum # of Instantiations 2 4 5
Multi-Core Acceleration
15.9X
47.2X
47.5X
64.5X

26. Performance: Multi-Factor Results (1/3)
SFRF is the number of cycles before Phase 1 generates a new Yi
TRF is the number of cycles before Phase 2 requires a new Yi
Consumer: Phase 2
Ratec: 1/TRF
Producer: Phase 1
Ratep: 1/SFRF

27. Performance: Multi-Factor Results (2/3)
Ratec<= Ratep
Ratec>Ratep

28. Performance: Multi-Factor Results (3/3)
Ratep>Ratec
Ratep= Ratec
Ratep<Ratec
Average
Double Precision
1 Core
13.3X
15.9X
12.0X
13.5X
2 Cores
18.7X
22.5X
16.9X
19.0X
Single Precision
16.7X
20.1X
15.1X
17.0X
4 Cores
52.8X
63.4X
47.7X
53.6X
Hybrid
16.5X
19.9X
14.9X
16.8X
50.3X
60.4X
45.4X
51.1X
Integer
17.7X
22.4X
18.6X
18.9X
5 Cores
67.0X
84.5X
70.1X
71.5X

29. Summary
Presented a hardware architecture for pricing Collateralized Debt Obligations using Li’s models
Demonstrating the flexibility of an FPGA in exploiting both coarse- and fine-grain parallelism
Established that either a single/double hybrid or “integer” representations could be used to balance resource utilization and accuracy
Achieved a maximal acceleration of over 64-fold and 71-fold, for the one-factor and multi-factor models, respectively

30. Future Work Expand to a multi-FPGA system
Attempt the algorithm on a different accelerator architectures
GPU
3. Incorporate into a financial software suite

31. Thank You
(Questions?)