Collateralized Debt Obligation Pricing on the Cell/B.E. -- A preliminary Result

Transcription

1 Collateralized Debt Obligation Pricing on the Cell/B.E. -- A preliminary Result Lurng-Kuo Liu Virat Agarwal

2 Outline Objectivee Collateralized Debt Obligation Basics CDO on the Cell/B.E. A preliminary result Conclusion 2 CDO Pricing on Cell/Lurng-Kuo Liu/Virat Agarwal

3 Objective IBM TJ Watson Research Center Objective Demonstrate the competitive edge of the Cell/B.E. on CDO pricing using Monte Carlo simulation with Gaussian Copula No intention to develop new models for CDO pricing Why CDO? The fastest growing sector of the asset-backed securities market. According to SIFMA, global CDO issuance increased to $488.6 billion in 2006, nearly twice the $249.3 billion issued in CDO is challenging to price. Monte Carlo simulation has been the most popular method for CDO valuation. Monte Carlo simulation can be very resource intensive for large CDOs. Seems to be the good fit for the Cell/B.E. 3 CDO Pricing on Cell/Lurng-Kuo Liu/Virat Agarwal

4 CDO Basics A Collateralized Debt Obligation (CDO) is an asset-backed security backed by a diversified pool of defaultable instruments like loans, junk bonds, mortgages, etc. If the portfolio contains only credit default swaps (CDS), it is called a synthetic CDO. It is structured as multiple tranches and sold to investors. Each tranche has different priority to claim on the principal. Separate out the risks by prioritize the receipt of principal among the investors. Originating Bank Assets sold to the SPV Cash SPV Principal & interest Funding Cash Senior 30-70% Mezzanine 5-30% Equity 0-5% Loss Detachment point - d Attachment point - a 4 CDO Pricing on Cell/Lurng-Kuo Liu/Virat Agarwal

5 Distribution of Losses Loss given default amount of the i th reference obligation: L = (1 R ) N i where N i is the notional amount and R i is the recover rate. The accumulated portfolio loss is i i 1{ τ i t } L n ( t) = L i 1{ τ i t} i=1= 1 where is a default indicator Cumulative loss on a given trance d a Senior Mezzanine Equity Portfolio loss L a, d ( t) = ( L( t) a) + ( L( t) d) + where + ( x ) max( x,0) 5 CDO Pricing on Cell/Lurng-Kuo Liu/Virat Agarwal

6 CDO Pricing Losses due to defaults (the issuer fails to satisfy the terms of the obligation) are the main source of risk as payoffs. Estimate the present value of tranche losses due to defaults default t leg (floating leg) T r( u) du 0 DL = E e dla, d ( t) 0 Calculate the present value of the premium payments weighted by the outstanding capital premium leg (fixed leg) w Ti r( u) du 0 PL = sa, d E δi e min{max[ d L ( ti ),0], d a } i= 1 The fair price of the CDO tranche is defined to be spread such that the expected value of both legs is equal. t T r( u) du 0 E e dla, d ( t) 0 * s = a, d w Ti r( u) du 0 E δie min{max[ d L( ti ),0], d a} i= 1 6 CDO Pricing on Cell/Lurng-Kuo Liu/Virat Agarwal

7 Modeling Default Times Marginal Distributions Default time τ for a single firm is modeled as the first jump in a Cox process. t λ (u) u du 0 p( τ > t) = E e t t = E 0 p ( τ ) 1 e λ ( u) du Default intensity or hazard rate of a given firm determines its default time. 7 CDO Pricing on Cell/Lurng-Kuo Liu/Virat Agarwal

8 Modeling Default Times Joint Distributions The primary driver of loss distributions is default codependence correlation sensitivity. The higher the correlation, the more likely extreme loss events (multiple l defaults) become and therefore increases the spread of a senior tranche. Need to model the join distribution of the default times (τ i,, τ m ) of the obligations in the portfolio Gaussian copula is one of the first to be used for modeling the dependence structure in a credit portfolio p( τ t,..., τ t ) =Φ [ ( ( )),..., ( ( ))] 1 1 Φ F t Φ F t 1 1 N N Σ 1 1 N N 8 CDO Pricing on Cell/Lurng-Kuo Liu/Virat Agarwal

9 Monte Carlo Simulation with Gaussian Copula Draw a sample Z=(Z 1,,Z N ) from an N-dimensional Gaussian distribution, with correlation matrix R Generate independent uniform random numbers Convert them into normal random numbers (W) by using e.g. Box-Muller transformation Perform Cholesky decomposition on the correlation matrix R=C.C T Generate correlated normal random numbers with X=CW Convert this sample to a correlated N-dimensional uniform vector U=(U 1, U N ) = Φ(X) Turn each of these uniforms into a default time samples, by inversion: τ i = F -1 i (U i ) Sort the N-dimensional vector of default time in ascending order and select the default times that happen before maturity date. Use the random default times to generate the cash flow for the fixed leg and floating leg Discount these cash flow to get their present values Repeat the process for m times for the m-path Monte Carlo estimation * For simplicity, calibration process is not included in this work. 9 CDO Pricing on Cell/Lurng-Kuo Liu/Virat Agarwal

10 Introducing Cell/B.E. v1.0 Cell/B.E. is an accelerator extension to 64b Power Built on a Power ecosystem Used best know system practices for processor design Sets a new performance standard Exploits parallelism while achieving high frequency Supercomputer attributes with extreme floating point capabilities Sustains high memory bandwidth with smart DMA First Generation Cell/B.E. controllers Designed for natural human interaction Photo-realistic effects Predictable real-time response Virtualized resources for concurrent activities Designed for flexibility Wide variety of application domains Highly abstracted to highly exploitable programming models Reconfigurable I/O interfaces Virtual trusted computing environment for security Cell/B.E. is the chip powering the Sony PS3 (Shipped in volume the US in Nov 06) 10 CDO Pricing on Cell/Lurng-Kuo Liu/Virat Agarwal 90 nm 241M transistors 235mm2 9 cores, 10 threads >200 GFlops (SP) >20 GFlops (DP) Up to 25 GB/s memory B/W Up to 75 GB/s I/O B/W >300 GB/s EIB Top frequency >4GHz (observed in lab)

11 Cell/B.E. Features Heterogeneous multi-core system architecture Power Processor Element for control tasks Synergistic Processor Elements for data- intensive processing Synergistic Processor Element (SPE) consists of Synergistic Processor Unit (SPU) Synergistic Memory Flow Control (MFC) Data movement and synchronization Interface to highperformance Element Interconnect Bus SPE 16B/cycle SPU SXU LS MFC SPU SXU PPE LS MFC L2 SPU SXU LS MFC 16B/cycle PPU L1 SPU SXU LS MFC 32B/cycle 16B/cycle SPU SXU LS MFC EIB (up to 96B/cycle) PXU 64-bit Power Architecture with VMX SPU SXU LS MFC 16B/cycle MIC Dual XDR TM SPU SXU LS MFC SPU SXU BIC LS MFC FlexIO TM 16B/cycle (2x) 11 CDO Pricing on Cell/Lurng-Kuo Liu/Virat Agarwal

12 Profiling results of the CDO pricing algorithm Running time of various stages in CDO pricing Cholesky Decomposition Calculate Payments Sum Payments Statistics Generate Correlated Random numbers Generate Default Times Generate Normals Sorting Using 100 firms and 100,000 paths Computational Complexity of various stages: Generate Normals: O(Np) Cholesky Decomposition: O(N 3 ) Generate Correlated Random Numbers: O(N 2 p) Generate Default Times: O(Np) Sort: O(pN logn) Calculate Payments: O(Np) 12 CDO Pricing on Cell/Lurng-Kuo Liu/Virat Agarwal

13 Random Numbers: Mersenne Twister Astronomical period of , suitable for Monte Carlo Algorithm series of shift operations on x k+n generates the output random number 2 different parallelization strategies Optimize for a single SPE, use different (random) seeds. Fine-grain parallelism for generating a single stream. 13 CDO Pricing on Cell/Lurng-Kuo Liu/Virat Agarwal

14 Optimization for the SPE N = 624, M=397 Vector starting ti from location (i+1) or (i+m) may not be quadword aligned. Computation of latter part of array requires updated data from the first M entries Data dependence 14 CDO Pricing on Cell/Lurng-Kuo Liu/Virat Agarwal

15 Normalized Random Numbers: Polar Method 1. Generate Random Numbers a & b 2. V 1 2a-1 V 2 2b-1 3. R V V If R > 1, continue from STEP 1 - R 1 sqrt (-2 logr/r) - X V 1 R 1 - Y V 2 R 1 Optimization on SPE Use two random number vectors a & b Redo if condition fails for any pair of random numbers Overheard due to skipping of perfectly normal random numbers 15 CDO Pricing on Cell/Lurng-Kuo Liu/Virat Agarwal

16 Performance Comparison of RNG (MT) with other architectures Time (in seconds) to generate 100 million random numbers in sequential and block pattern on various architectures. * Source: 16 CDO Pricing on Cell/Lurng-Kuo Liu/Virat Agarwal

17 Performance Comparison of RNG (MT) with other architectures P erformance comparison of RNG (Mersene T w ister) on various architectures Block Sequential (seconds) Time Intel_1.4 Intel_3.0 AMD_2.4 PPC_1.33 Cell 17 CDO Pricing on Cell/Lurng-Kuo Liu/Virat Agarwal

18 Performance compared with other Cell/B.E. implementations Performance comparison of our optimized RNG (Mersene Twister) as compared with other Cell/B.E. implementations nning Time (seconds) Ru Another Cell RNG (MT) SDK RNG* Our RNG Time (in seconds) to generate 100 million normalized random numbers on a single SPE. * Vectorized ed Random Number generation e available with Cell SDK 2.1 Running Time (seco onds) Performance comparison of our optimized RNG (with Normalization) as compared with other Cell/B.E. implementatoins bit 64-bit Another Cell RNG w/n Our RNG w/n 18 CDO Pricing on Cell/Lurng-Kuo Liu/Virat Agarwal

19 Correlation Matrix : Cholesky Decomposition Cholesky decomposition on correlation matrix C -> LL T, where L is a NxN lower triangular matrix Modified d version of the Gauss Algorithm Initial optimized version for a single SPE Analyzing ways to further optimize and parallelize on the Cell. 19 CDO Pricing on Cell/Lurng-Kuo Liu/Virat Agarwal

20 Generating Correlated Random Numbers Compute N (number of firms) normalized random numbers. Vector V[0.. N-1]. Calculate V = LV, where L is a lower triangular matrix. Cell Optimization: Branch mispredicts compromise performance for small N. 2 load instructions (6 cycles) for each madd (6 cycles), inefficient use of the even pipeline. pp Initial performance results. 20 CDO Pricing on Cell/Lurng-Kuo Liu/Virat Agarwal

21 Generating Correlated Random Numbers Also working on utilizing the lower triangular property of the matrix L, to achieve better performance. 21 CDO Pricing on Cell/Lurng-Kuo Liu/Virat Agarwal

22 Conclusions CDO pricing is computationally intensive instead of communications intensive. We use Monte-Carlo simulation Highly scalable among various SPEs Initial Performance results Show substantial speedup for Mersenne Twister and Normalization as compared to other architectures Initial results for cholesky decomposition and generating correlated random numbers. Cell is a good fit for financial workloads. Double precision is essential for FSS workloads 22 CDO Pricing on Cell/Lurng-Kuo Liu/Virat Agarwal

23 Thank you Questions? 23 CDO Pricing on Cell/Lurng-Kuo Liu/Virat Agarwal