Hardware Accelerators for Financial Mathematics - Methodology, Results and Benchmarking Christian de Schryver # , Henning Marxen ∗ , Daniel Schmidt # # Micrelectronic Systems Design Department, University of Kaiserslautern Erwin-Schroedinger-Strasse, 67663 Kaiserslautern, Germany schryver@eit.uni-kl.de schmidt@eit.uni-kl.de ∗ Stochastics and Financial Mathematics Department, University of Kaiserslautern Erwin-Schroedinger-Strasse, 67663 Kaiserslautern, Germany marxen@mathematik.uni-kl.de Abstract—Modern financial mathematics consume more and more computational power and energy. Finding efficient algo- rithms and implementations to accelerate calculations is therefore a very active area of research. We show why interdisciplinary cooperation such as (CM)2 are key in order to build optimal designs. For option pricing based on the state-of-the-art Heston model, no implementation on dedicated hardware is known, yet. We PC GPGPU FPGA are currently designing a highly parallel architecture for field programmable gate arrays based on the multi-level Monte Carlo Fig. 1. PC vs. GPGPU vs. FPGA method. It is optimized for high throughput and low energy consumption, compared to GPGPUs. In order to be able to Moving away from GPGPUs to dedicated hardware acceler- evaluate different algorithms and their implementations, we ators can help to drastically reduce the power consumption present a benchmark set for this application. We will show a very at the same or even higher throughput. For different applica- promising outlook on future work, including dedicated ASIPs, tion domains, some comparisons between CPU, GPGPU and fixed-point research and real-time applications. programmable hardware units (field programmable gate arrays, Index Terms—finance, benchmarking, hardware acceleration FPGAs) have already been shown in [13] and [5], highlighting the enormous potential of energy savings for FPGAs. Figure 1 I. I NTRODUCTION shows that standard software implementations require the Nowadays, financial markets are as vivid as never before. In least effort for implementation and can provide the highest modern electronic markets, stock prices may change several flexibility, while dedicated hardware solutions on FPGAs are times within a few milliseconds. Participating traders (that can hard to design and - once finished - not easy to be changed also be computers) have to evaluate the prices and react very again. From a different view, FPGAs can save up to about quickly in order to get the highest profit, which requires a lot 99% of energy compared to a software implementation on a of computational effort. standard PC and allow a much higher throughput. GPGPUs In general, running these computations on servers or clusters are located between standard PCs and FPGAs. Between each with standard CPUs is not feasible due to either long run times neighboring architectures, one can expect a difference of about or high energy consumption. Using general purpose graphics one order of magnitude on average for power consumption and processing units (GPGPUs) as accelerators helps to increase throughput [13]. Although most financial institutes are relying the speed, but still requires a lot of energy. Besides, at the on GPGPUs at the moment for the reason of standardized moment energy efficiency becomes more and more crucial for software development toolkits and their flexibility, FPGAs the reason of high energy costs and - even more critical - a are an interesting alternative because of their higher energy limited supply of energy that can be provided. For example, efficiency. in [16] it is stated that the City of London (with its new A big challenge is the complexity of many models used financial center Canary Wharf where a lot of leading institutes to estimate the future price behavior of financial products. In are located) does not provide additional energy until after many cases no mathematical closed-form solution exists so the Olympic winter games in 2012, that have higher priority. that approximation methods like Monte Carlo simulations or Financial institutes are currently outsourcing all computing the finite difference method must be employed. Though, it is systems not used for pricing computations (such as storage necessary to precisely specify a solution right at the beginning or backup) out of the critical area. of the design process. Re-designing a nearly finished hardware This leads to the dilemma of needing faster computations on implementation can require a very high amount of effort. The the one hand and limited energy resources on the other hand. Center of Mathematical and Computational Modeling (CM)2 of the University of Kaiserslautern is a perfect forum for an second stochastic differential equation for stochastic volatility interdisciplinary cooperation to tackle this issue. variations. This significantly increases the complexity of the For this project, we have developed a design methodology calculation and of the implementation thereof. Nevertheless, that helps to select a feasible parameter set for a hardware ac- the Heston model reflects the real behavior of current stock celerator in question that we present in Section III. In order to markets much better and is nowadays widely accepted in the make implementations transparently comparable, we propose financial mathematics community. But - to the best of our to use standardized benchmark sets - we elaborate on this in knowledge - no hardware accelerator for that model has been Section IV. By applying this methodology and our benchmark, published up to now. For GPGPUs, the first implementations we have developed some reference implementation designs have been presented in the last year. Bernemann, Schreyer that we show in Section V, together with the status quo of our and Spanderen from the german bank WestLB [3] showed research and our contributions up to now. In Section VI we that they could achieve a speeup of 50 times over CPU by give an overview of open issues and what we plan to examine using GPGPUs for simulating the Heston model. Zhang and in the future. Section VII concludes the paper. Oosterlee published a technical report [19] in March 2010 where they even showed speedups of more than 100 times. II. S TATE - OF - THE -A RT AND R ELATED W ORK The presented speed-ups look very impressive. However, Mathematical finance basically has two different directions. unfortunately we were not able to fairly decide which solution One is concerned with the evaluation of optimal investment seems to be the most promising for further research and strategies under certain market conditions and the other di- refinements. We will go a bit more into the details of that rection is the pricing of derivatives. The basic idea of pricing problem in Section IV. options is to assume some sort of model for the underlying III. H OW T O C HOOSE T HE R IGHT D ESIGN price process and take the discounted expected value - under a certain measure - of an option as the option price. For many fields of applications, finding the most efficient A very common problem treated is the calculation of design under certain constraints is a difficult job. The main option prices based on the Black-Scholes model from 1973. reason for this is a large design space. The design space is This model relies on one stochastic differential equation and made up of all possible parameter choices for the design, that describes the price development of an option over the time, means all possible implementation instances. depending on market parameters such as riskless interest rate, Most parameters are not adjustable independently, since long term drift and a constant volatility. they are mutually linked. For example, fixing the target ar- Accelerator design for financial mathematics is a very active chitecture to FPGAs one the one hand has a large impact on research area, and several FPGA implementations have been the selection of suitable algorithms and number systems, and published in the past. At the FPL 2008 Woods and VanCourt on the other hand affects many performance metrics such as [17] presented a hardware accelerator for multiple, quasi- energy consumption, throughput and numerical precision. random, standard Brownian motions suitable for the accelera- Furthermore, the parameters within the design space are in tion of quasi-Monte Carlo simulation of financial derivatives. many cases not limited to a single domain of expertise, but For credit risk modelling, Thomas and Luk could gain a require interdisciplinary know-how and decisions. This makes speed-up of more than 90 times compared to a 2.4 GHz not only the choice of the right values a challenge, but also Pentium-4 Core2 [14]. An accelerator for Monte Carlo based the evaluation and comparison of different implementations. credit derivative pricing was developed by Kaganov, Chow Besides speedup, more characteristics such as energy effi- and Lakhany [10] in 2008 and showed to be 63 times faster ciency, convergence rate or numerical precision may be very than their software model. Wynnyk and Magdon-Ismail [18] important. This especially holds true for financial mathematics presented an FPGA accelerator for American option pricing accelerators. based on the Black-Scholes model in 2009 and could achieve During our research we have seen a lot of papers that a speedup of eleven up to 73 times compared to a software show elaborate implementations of a specific algorithm (see implementation running on a standard PC. Section II) that is not questioned in the papers anymore. However, nowadays the Black-Scholes model is no longer However, we claim that the algorithm itself is in fact not the up to date and does not provide an accurate reflection of most important selection. An accelerator should be designed to modern financial market behaviors, mostly because of the solve a specific problem - it does not matter which algorithm is volatility not being constant in reality. Furthermore, closed- used, as long as the result is calculated correctly. We therefore form solutions for the Black-Scholes model exist and it only propose to distinguish clearly between three terms: has demonstration purposes to apply stochastic solution meth- • the problem that is tackled (what to solve) ods such as Monte Carlo simulations or the finite difference • the employed model (how to solve) method [1]. Nevertheless, it is still very common to publish • the solution (how to build) accelerator implementations based on that model, at least in To clarify the situation, we use the problem “calculate the the electrical engineering community. price of an option with two barriers for a given duration” as In 1993, Steven L. Heston presented a more accurate model an example. European knock-out barrier options pay a certain [9] that extends the model from Black and Scholes by a amount of money at a fixed maturity time depending on the one of an electrical engineer who may not understand the mathematician’s concerns in detail, but is wondering what the best decisions with respect to hardware efficiency might be. IV. B ENCHMARKING - FAIRLY C OMPARING I MPLEMENTATIONS Comparing different implementations is a non-trivial task. Many attributes can be considered, including speed, accuracy and energy consumption. This becomes even more difficult when it is not clear which algorithm was used. Furthermore, in many cases it is not possible to distinct whether a presented algorithm or implementation has the displayed behavior only Fig. 2. Barrier testing for a Brownian motion with the employed example or in a more general setting. Nevertheless, it is important to be able to compare various value of the underlying asset. This amount is only paid when algorithms and different implementations, also over various the barrier is not crossed up to the maturity time. If one of the target architectures. barriers is hit, the option becomes worthless. Thus it needs to Therefore the need for a benchmark set arises. This set be checked whether the barrier was ever hit or not. Figure 2 should be independent of the algorithm and implementation illustrates the typical random behavior of different realizations used. For option pricing in financial mathematics, this need of an asset price over the time. has already been claimed by Morris and Aubury in 2007 [12]. It is obvious that the problem description itself does not We are not aware of any progress made since that paper was yet give any suggestions to the solution. Since the price of an published. We therefore have decided to develop a completely option is tightly coupled to the price of a certain stock at the new benchmark that will enable us to fairly compare different market, we need a model that provides the stock price behavior. algorithms, e.g. multi-level and single-level Monte Carlo, on For our chosen example, suitable models are for example different hardware. Thus we propose a benchmark based on the Black-Scholes model (outdated nowadays) or the Heston the problem/model combination. In our case it is the pricing model. The model in general gives a formal and abstract view of double barrier options in the Heston model. It is clear that of (a certain aspect of) the problem. independently of the used algorithm and implementation the The solution finally is a dedicated approach for solving a result must be the same. Therefore the final prices of the (modeled) problem. It is characterized by a specific algorithm different options in the benchmark set have to be provided. and its implementation. For evaluating the Heston model, With the benchmark set it is possible to use different for example, finite difference methods or stochastic Monte metrics, like speed (that is now the real time until the results Carlo simulations can be used. They may be implemented for are available), accuracy and power consumption, for the cal- example on standard PC clusters, GPGPUs or on FPGAs. culations leading to the right (or approximate) result without The parameters of the design space can basically be divided actually looking at the implementation and the algorithm itself. into two groups: the algorithmic parameters that are mostly se- This allows a fair and publicly traceable comparison of the lected by mathematicians, and the implementation parameters solution part of the problem/model/solution triplet. determined by the hardware designers. However, as mentioned The benchmark itself consists of different combinations before correlations exist between several parameters, so that of parameters for the Heston model and for barrier options, the selection should be optimally made by having a generative including the prices. The data for the Heston parameters is exchange between experts of both groups. taken from different recent publications ( [2], [11], [20]) and For the rest of the paper, we will use the problem-model- are enlarged by an extremer case. The benchmark parameters solution triple “calculate the price of an option with two span a wide range of possible combinations used in this field. barriers for a given duration with the Heston model by using For some options of the benchmark closed form solutions Monte Carlo methods” as a showcase. Even for that specific exist that allow to obtain the exact results. This is important selection, the design space is still very large. An extract of to verify that simulations converge to the correct values and the related design space is shown in Figure 31 . We cannot makes it easier to compare the results. For the other cases the explain every parameter here (for details see [11] [8]), but exact prices are not known and are therefore provided as close in a nutshell it is obvious that even for a very specific task approximations. a huge amount of possible accelerator implementations may For further publications we not only encourage the authors exist. In Figure 3 we see two different views of the same to use the presented benchmark but also give details of the tree. These trees are symbolic for the design space. The left algorithm and the implementation used. Thus it is possible one is the view of a mathematician that has mostly to do to see where an increase in performance comes from. This with algorithmic and numerical aspects. The right view is the is essential in order to evaluate the contribution of a certain 1 result and to find ways to improve it even further. To achieve http://www.sxc.hu/photo/285734 We thank sxc user vxdigital for sharing this image of the oak tree and allowing the use of it in this paper. He holds all copyrights to this image. a higher transparency we will publish the code we used to Path-Serial Quasi Path-Parallel MC Simulation Heston Model Calculation (a) Algorithmic parameters (b) Implementation parameters Fig. 3. Two views of the same solution tree analyze the algorithms and the one we implemented on the more than one underlying asset. Nevertheless, for our project FPGA. we will stick to one asset. The benchmark was directly used when comparing different A. A New Random Number Generator for Non-Uniform Dis- Monte Carlo algorithms with the metric of computational tributions complexity. In our special problem/model combination there are a lot more adjustments to the algorithms than seen in Inherently, Monte Carlo simulations always consume a huge figure 3(a). Many of them can be combined, what leads to amount of random numbers. To obtain the maximum hardware a huge design space that now can be handled by applying efficiency for our implementation, we have developed a new the benchmark set, so that we haven been able to choose a random number generator for non-uniform distributions tai- specific algorithm to be efficiently implemented on dedicated lored to our application. hardware. The multi-level Monte Carlo method provides a bet- For our option pricing accelerators, we need two inde- ter asymptotic convergence behavior, using our benchmark we pendent, normally distributed random numbers for each time checked whether this method is beneficial for our application. step of a single simulated stock price path. In general, non- We will show our first results in the next section. uniformly distributed random numbers are generated in two steps: First, a uniform random number generator creates uni- V. S TATUS Q UO AND F IRST R ESULTS formly distributed values, and in a second step this number is transformed into the desired target distribution. We have started this cooperation within (CM)2 about one For the uniform random number generation, a lot of research year ago now. Both participating chairs have experience of has already been made leading to efficient and well-proven more than ten years in their respective field of research, so implementations, such as the Mersenne Twister MT19937 that that we can profit from a lot of knowledge in the areas of we use. The three main approaches for obtaining non-uniform efficient hardware design respectively stochastics and financial distributions are transformation, rejection, and inversion meth- mathematics. ods [15]. After evaluating the state-of-the-art, we decided to focus on For FPGAs, inversion methods are the usual way to go. accelerators for option pricing based on the Heston model - it They combine many desireable properties: by applying the seems to be a very promising topic since no implementations respective inverse cumulative distribution function (ICDF), (either on GPGPUs or FPGAs) have been available one year they transform every input sample x ∈ (0, 1) from a uniform ago. In contrast to that, the Heston model is already widely distribution to one output sample y = icdf (x) of the desired spread within the financial community. From former research output distribution by using piecewise polynomial approxima- done in the group of Prof. Korn, multi-level Monte Carlo tion of the ICDF. The works of Woods and VanCourt [17] and methods [7] seemed to provide a better convergence behavior Cheung et al. [4] show FPGA implementations of the inversion than standard single-level Monte Carlo or finite difference method. methods. Monte Carlo methods also have the advantage of However, both implementations use fixed-point number being very flexible. A barrier that is only relevant on a certain representations at the input. This leads to a loss of precision time interval to evaluate an option price for example can be in the tail regions where the probability of a value lying there easily implemented. Furthermore, multi-dimensional problems is very low. But these extreme events can have a large impact, can also be solved. This is needed in the case that an option has for example for options with barriers it is crucial to know if a barrier was hit or not, since it completely changes the refund values for a higher step width, this means for a lower Monte conditions. We have therefore developed a new implementa- Carlo simulation level. In the configuration memory, all model tion based on floating-point representation that provides the parameters are stored. same precision over the whole ICDF implementation at much Due to the inherent parallelism of Monte Carlo simulations, lower hardware costs. This work has been presented at the it is not only feasible but self-evident to instantiate as many 2010 International Conference on ReConFigurable Computing of these circuits as possible on an FPGA in order to increase and FPGAs (ReConFig) in December in Cancún, Mexico [6]. the simulation throughput. Our random number converter unit requires only about half The accelerator is implemented on a Xilinx ML-605 evalula- of the area compared to other state-of-the-art implementations tion board equipped with a Xilinx Virtex-6 FPGA. The board by even higher numerical precision. is connected via Gigabit ethernet to a host PC running the To validate our work, it was crucial to develop a new testing user interface and the program that calculates and sets the methodology, since standardized test suites do not exist for configuration values for the accelerator based on the retrieved non-uniform distributions. This work has been carried out in simulation results. cooperation with Elke Korn who has a lot of knowledge in the Synthesis and benchmarking results will be available soon. field of random numbers. The methodology and the validation We are currently supporting single- and double-precision results for our implementation are also presented in the paper. floating-point computations and are working on a fixed-point Our random numbers did not show any noticeable problems implementation as well. in the stochastic tests and also perfectly passed two different application simulations. VI. O UTLOOK AND F UTURE W ORK Also for the intended future work a close cooperation B. Fully Parallel Hardware Accelerator between financial mathematics and electrical engineering will To the best of our knowlege, no hardware implementations be mandatory, since we are planning to research aspects out of option price accelerators based on the Heston model exist of both fields. at the moment. We have therefore started with the first imple- One characteristic of the Monte Carlo method is the inherent mentation, that is nearly finished now. The hardware is fully- capability to parallelize the calculation. It therefore makes no parallel, fully-pipelined and designed for high throughput. difference whether several calculations are done in parallel but slowly or only one calculation is done at high speed, as long Uniform Random Uniform Random as the number of calculations remains equal altogether. At the Number Generator Number Generator moment, our parallel implementation allows simulating one User Interface time step on many assets simultaneously. The whole procedure Floating-Point Floating-Point of calculating one time step is fixed on the FPGA and limited Gaussian Converter Gaussian Converter to a single algorithm that is set at design time. price variations volatility One possible way to improve this is to sequentially com- Heston Step Generator Unit Multi-Level pute the basic calculations needed for one time step on a Monte Carlo Application-Specific Instruction-Set Processor (ASIP) within Controller Configuration the FPGA, i.e. it is runtime-programmable. This procedure Temporary Coarse Memory can reduce the required area and allows to calculate various Values Step Memory Unit algorithms since the functionality is defined in a corresponding program. It is therefore sufficient to load a different program Ethernet Ethernet Interface Stack without changing the hardware. The ASIP will occupy much less area than the parallel implementation presented in Sec- FPGA Board Host PC tion V-B, therefore many ASIPs can be instantiated in parallel. We are currently investigating the necessary instruction set. Fig. 4. Fully Parallel Accelerator Structure To increase the speed of the implementation even more, the floating-point computation can be replaced by fixed point Figure 4 shows the structure of one pipelined accelerator computations. In order to do so, errors resulting from the use circuit. In each clock cycle, our unit consumes two normally of fixed-point calculations have to be approximated. This task distributed random numbers, one for the stock price variation will also require both theoretical and practical expertise. and one for the volatility variation. The Heston step generator Besides working on the implementation, the benchmark unit calculates the price and volatility values for the next is very important to evaluate the designs. It will also allow time step based on a multi-level Monte Carlo algorithm. The to research fixed-point solutions with respect to necessary pipeline depth is about 60 stages. In order to maximally precision. Further steps will be to publish the analysis of the utilize the pipelined hardware, it computates one time step algorithms with the benchmark in a journal. To increase the for 60 assets in parallel, before moving to the next time transparency even more, we are currently setting up a web step. The values for the respective time steps are stored in site offering all the program code used to create the analysis. a memory temporarily. The coarse step memory holds interim It will contain an implementation of the multi-level and the crude Monte Carlo algorithm with a focus on the calculation ACKNOWLEDGEMENTS complexity rather than the implementation. We gratefully acknowledge the partial financial support To provide more benchmarking results, also for different from Center of Mathematical and Computational Modeling architectures, we are working on a GPGPU implementation. (CM)2 of the University of Kaiserslautern. It provides more flexibility and less implementation work than hardware designs do, but also requires higher energy consump- R EFERENCES tion. In order to verify this assumption the implementation is [1] R. L. Akers, I. Bica, E. Kant, C. Randall, and R. L. Young. SciFinance: A required. The Monte Carlo simulation algorithm for the Heston Program Synthesis Tool for Financial Modeling. AI Magazine, 22(2):27, 2001. model is currently carried out as a student work. [2] L. Andersen. Efficient simulation of the heston stochastic volatility Moreover, we are currently researching on real-time accel- model. 2006. eration of financial calculations. This means that hardware or [3] A. Bernemann, R. Schreyer, and K. Spanderen. 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