Monte Carlo European Option Pricing with RNG Interface for Intel® Xeon Phi™ Coprocessor

Download Available under the Intel Sample Source Code License Agreement license.

Background

Monte Carlo is a numerical method that uses statistical sampling techniques to approximate solutions to quantitative problems. The name comes from the famous casino in the principality of Monaco, where a roulette table provides uncertainty outcomes just like series of random numbers. The contemporary version of the Monte Carlo algorithm was first used by Stanislaw Ulam, while he was working on the Manhattan project in the mid-1940s. Nicholas Metropolis was first to make the connection between the casino and the algorithm and coined the term Monte Carlo to refer to any numerical simulation algorithm that involves a random number generator. John von Neumann was the first to implement Monte Carlo on the ENIAC Computer in the late 1940s. Since then, Monte Carlo has been widely used in engineering physics, molecular dynamics, and in calculating integrals with complicated boundary conditions.

In 1973, Fisher Black and Myron Scholes published their historical paper and introduced what later became known as the Black-Scholes Option Pricing model for financial derivatives. As the rest of the world was still trying to digest the Black-Scholes Model, an actuarial professor from the University of British Columbia, Phelim Boyle introduced the Monte Carlo method to Finance and successfully used it as an alternate way to get the same result as the Black-Scholes Model. In his article, he takes the example of a European call option and calculates its price using the Monte Carlo method.

In this paper, we use the same numerical problem as an example to highlight various techniques and practices to achieve high performance computing on Intel® Xeon® processors and Intel® Xeon Phi™ coprocessors.

Code Access

The Monte Carlo European Option with RNG interface is maintained by Shuo Li and is available under the BSD 3-Clause Licensing Agreement. The code supports the asynchronous offload of the Intel Xeon processor (referred to as “host” in this document) with the Intel Xeon Phi coprocessor (referred to as “coprocessor” in this document) in a single node environment.

To access the code and test workloads:

Go to source location to download the MonteCarloRNGsrc.tar file.

Build Directions

Here are the steps you need to follow in order to rebuild the program:

1. Install the Intel® Composer XE 2013 SP 2 on your system
2. Source the environment variable script file compilervars.csh under /pkg_bin
3. Untar the montecarlorng.tar file, type make to build the binary
4. Issue the make command, be unconditional, and be silent using the –Bs option

[prompt]$ make –Bs

Run Directions

Copy the following files to the Intel Xeon Phi coprocessor card.

[prompt]$ scp MonteCarloRNGSP.knc yourhost-mic0:
[prompt]$ scp MonteCarloRNGDP.knc yourhost-mic0:
[prompt]$ scp /opt/intel/composerxe/lib/mic/libiomp5.so yourhost-mic0:
[prompt]$ scp /opt/intel/composerxe/tbb/lib/mic/libtbbmalloc.so yourhost-mic0:
[prompt]$ scp /opt/intel/composerxe/tbb/lib/mic/libtbbmalloc.so.2 yourhost-mic0:
[prompt]$ scp /opt/intel/composerxe/mkl/lib/mic/libmkl_intel_lp64.so yourhost-mic0:
[prompt]$ scp /opt/intel/composerxe/mkl/lib/mic/libmkl_sequential.so yourhost-mic0:
[prompt]$ scp /opt/intel/composerxe/mkl/lib/mic/libmkl_core.so yourhost-mic0:

Turn on the turbo mode on your Intel Xeon Phi coprocessor card.

[prompt]$ sudo /opt/intel/mic/bin/micsmc --turbo enable
Information: mic0: Turbo Mode Enable succeeded.

Invoke the binary and set the environmental variable for the execution from the host.

[prompt]$ ssh yourhost-mic0 "export LD_LIBRARY_PATH=.;export OMP_NUM_THREADS=244;export KMP_AFFINITY='compact,granularity=fine';./MonteCarloRNGSP.knc"
Monte Carlo European Option Pricing Single Precision

Compiler Version  = 14
Release Update    = 2
Build Time        = Jun  2 2014 12:22:43
Path Length       = 262144
Number of Options = 999912
Block Size        = 16384
Worker Threads    = 244

Starting options pricing...
Parallel simulation completed in 21.439754 seconds.
Validating the result...
L1_Norm          = 4.812E-04
Average RESERVE  = 12.872
Max Error        = 8.035E-02
==========================================
Total Cycles = 28586338291
Cyc/opt      = 28588.854
Time Elapsed =   21.440
Options/sec  = 46638.222
==========================================
[prompt]$ ssh yourhost-mic0 "export LD_LIBRARY_PATH=.;export OMP_NUM_THREADS=244;export KMP_AFFINITY='compact,granularity=fine';./MonteCarloRNGDP.knc"
Monte Carlo European Option Pricing Double Precision

Compiler Version  = 14
Release Update    = 2
Build Time        = Jun  2 2014 12:22:44
Path Length       = 262144
Number of Options = 999912
Block Size        = 8192
Worker Threads    = 244

Starting options pricing...
Parallel simulation completed in 47.075885 seconds.
Validating the result...
L1_Norm          = 4.812E-04
Average RESERVE  = 12.920
Max Error        = 8.034E-02
==========================================
Total Cycles = 62767847297
Cyc/opt      = 62773.371
Time Elapsed =   47.076
Options/sec  = 21240.429
==========================================

The program priced about a million sets of option input data. If you divide 1 million among 244 threads, you will get 4098.36. Without losing generality, let’s simplify the number of options each thread runs to a round number of 4096, or 4k, and the total number of options the program will price is 999,912. For each option, the program first generates a random number sequence that is independently and identically distributed or i.i.d. for short. We cover details of generating random number sequences in a later section. We then use these random numbers as samples of stock movement with the European payoff formula and calculate the stock value and confidence interval in a formula shown in the next section.

The program was built on the host and executes on the Intel Xeon Phi coprocessor. For each option data set, it calculates the option values and the confidence intervals. Result validation is part of the benchmark. It measures the average error between the calculated result and the result from Black-Scholes-Merton ^[2] formula.

This benchmark runs on a single node of an Intel Xeon Phi coprocessor.  It can also be modified to run in a cluster environment. The program reports a latency measure in seconds spent in pricing activities and also a throughput measure using total number of options priced over the elapsed time, which was printed out as the last performance number in Options/sec.

Generating and Using Random Numbers in Monte Carlo Methods

Since Monte Carlo is a numerical method based on the simulation of random variables, the implementation of this algorithm starts with identifying a random number generator. VSL, the vector statistical library component of the Intel® Math Kernel Library (Intel® MKL), provides a variety of random number generators for different distributions. Our implementation uses the Mersenne Twister random number generator using the normal distribution. VSL is part of the Intel® C++ Composer XE 2013 that we use to build applications for Intel Xeon processor and Intel Xeon Phi coprocessors.

Using Random Number Generators

Inside VSL a random number sequence is identified as a stream. Each stream delivers random numbers in a given distribution in vector interface. To manage the complexity, VSL uses two implementation layers to support different RNGs and different distributions. At the lower level, all core random number generation routines are implemented to deliver random numbers in uniform distribution. At the higher level, transformation functions are applied to turn uniform distribution to the distribution the user desires.

To use VSL, follow this typical 5-step process:

1. Specify RNG streams
2. Initialize and create the random number streams
3. Request a vector of random numbers in a specific distribution
4. Consume the random number sequences in the simulation
5. Destroy the RNG streams

Here is how the process works for calculating our European Call options:

1. Specify a random number stream. In our benchmark we are going to use all 61 cores and create 4 threads per core. In total, we can have 244 threads. To allow each thread to price an option independently, we should give each thread an independent random number stream. We need to declare an RNG state descriptor for each thread. We can declare these data structures before we create any threads.
    

   #include <mkl_vsl.h>
       // Declare random number buffer and random number sequence descriptors
       float *samples[MAX_THREADS];
       VSLStreamStatePtr Streams[MAX_THREADS];

   VSLStreamStatePtr is a C/C++ opaque data structure and Streams is an array of still uninitialized opaque data.

2. Initialize and create the random stream and set up the stream with a basic random number generator and an integer seed.

   Once we have created worker threads, each thread will allocate its own buffer to receive the RNG sequences and initialize the stream descriptor so that it knows what basic random number generator to use and whether the threads need to work together to ensure mutual independency.

   samples[threadID] = (float *)scalable_aligned_malloc(RAND_BLOCK_LENGTH * sizeof(float), SIMDALIGN);
   vslNewStream(&(Streams[threadID]), VSL_BRNG_MT2203 + threadID, RANDSEED);

   Intel MKL provides the following routine to create and initialize the stream you declared:

   vslNewStream (VSLStreamStatePtr &Randomstream, int brng, int seed )

   Randomstream is a reference to the uninitialized random stream you just declared. It takes a reference because the routine passes back an initialized stream state descriptor. brng is an enumeration parameter specifying which basic random number to use. VSL_BRNG_MT2203 specifies a family of modified Mersenne Twister[^10] pseudo generators, each of which is i.d.d. In our problem, each thread uses its own random number generator identified by VSL_BRNG_MT2203 plus its thread ID. You can find more information on the basic random number generator here BRNG parameter definitions. seed is an integer for the random stream to ensure the reproducibility for debugging purposes.

3. Request a vector of random numbers of a specific distribution. Using the random stream descriptor, we can call one of the distribution generators to produce a sequence of random numbers with a certain probability distribution and a specific data type. The result will be placed in a user-provided buffer in the form of a C array.

   float *rand = samples[threadID];
   vsRngGaussian (VSL_RNG_METHOD_GAUSSIAN_ICDF, Streams[threadID], RAND_BLOCK_LENGTH, rand, MuByT, VBySqrtT);

   Intel MKL uses the following routine to generate normal distributed random numbers.
   vsRngGaussian(method, stream, n, r, a, sigma)where:
   method can be VSL_RNG_METHOD_GAUSSIAN_ICDF, and other values are listed here
   stream initialized random streams
   n the number of random numbers to be requested
   r address of the receiving buffer, usually a C array declared to hold the random number
   a first parameter to the distribution. For normal distribution, it’s the mean.
   sigma second parameter to the distribution. For normal distribution, it’s the std deviation.

4. Consume the sequence of random numbers.

   for(int i=0; i < RAND_BLOCK_LENGTH; i++)
       {
           float callValue  = Y * exp2f(rand[i]) - Z;
           callValue = (callValue > 0) ? callValue : 0;
           v0 += callValue;
           v1 += callValue * callValue;
       }

5. Delete the random stream.

   Use vslDeleteStream (VSLStreamStatePtr &stream) to delete the stream declared in step 1. Since we created the stream in the worker threads, it’s customary to destroy the stream in the worker thread.

Other Implementation Notes

In our implementation, worker threads are created using OpenMP* parallel directives. Each thread creates its own random number streams, generates the unique option input data, prices the option, and then validates the result. Each thread’s input data is generated by calling C runtime library rand_r in a unique sequence identified by its thread ID, which guarantees each thread will produce a unique and reproducible sequence.

The aligned memory allocation interface from Intel® Threading Building Blocks (Intel® TBB) are used to allocate the aligned memory blocks that are also cache-friendly to the worker threads. This means, these memory blocks have to be disposed of using the corresponding API. Intel TBB is part of our minimum build requirement.

OpenMP reduction operations are used to calculate the statistical properties of all the options. It’s also used to find the maximum error from the threads.

Source Code for MonteCarloRNG Core

The following is a core part of MonteCarloRNG using single precision data types. Double precision is almost identical.

// Declare random number buffer and random number sequence descriptors
float *samples[MAX_THREADS];
VSLStreamStatePtr Streams[MAX_THREADS];

// calculate the block number based on block size
const int nblocks = RAND_N/RAND_BLOCK_LENGTH;

#pragma omp parallel reduction(+ : sum_delta) reduction(+ : sum_ref) reduction(+ : sumReserve) reduction(max : max_delta)
{

#ifdef _OPENMP
    int threadID = omp_get_thread_num();
#else
    int threadID = 0;
#endif
    unsigned int randseed = RANDSEED + threadID;
    srand(randseed);
float *CallResultList     = (float *)scalable_aligned_malloc(mem_size, SIMDALIGN);
float *CallConfidenceList = (float *)scalable_aligned_malloc(mem_size, SIMDALIGN);
float *StockPriceList     = (float *)scalable_aligned_malloc(mem_size, SIMDALIGN);
float *OptionStrikeList   = (float *)scalable_aligned_malloc(mem_size, SIMDALIGN);
float *OptionYearsList    = (float *)scalable_aligned_malloc(mem_size, SIMDALIGN);
for(int i = 0; i < OPT_PER_THREAD; i++)
{
    CallResultList[i]     = 0.0f;
    CallConfidenceList[i] = 0.0f;
    StockPriceList[i]     = RandFloat_T(5.0f, 50.0f, &randseed);
    OptionStrikeList[i]   = RandFloat_T(10.0f, 25.0f, &randseed);
    OptionYearsList[i]    = RandFloat_T(1.0f, 5.0f, &randseed);
}

samples[threadID] = (float *)scalable_aligned_malloc(RAND_BLOCK_LENGTH * sizeof(float), SIMDALIGN);
vslNewStream(&(Streams[threadID]), VSL_BRNG_MT2203 + threadID, RANDSEED);

#pragma omp barrier
if (threadID == 0)
{
    printf("Starting options pricing...n");
    sTime = second();
    start_cyc = _rdtsc();
}

for(int opt = 0; opt < OPT_PER_THREAD; opt++)
{
    const float VBySqrtT = VLog2E * sqrtf(OptionYearsList[opt]);
    const float MuByT    = MuLog2E * OptionYearsList[opt];
    const float Y        = StockPriceList[opt];
    const float Z        = OptionStrikeList[opt];

    float v0 = 0.0f;
    float v1 = 0.0f;
    for(int block = 0; block < nblocks; ++block)
    {
        float *rand = samples[threadID];
        vsRngGaussian (VSL_RNG_METHOD_GAUSSIAN_ICDF, Streams[threadID], RAND_BLOCK_LENGTH, rand, MuByT, VBySqrtT);

#pragma vector aligned
#pragma simd reduction(+:v0) reduction(+:v1)
#pragma unroll(4)
        for(int i=0; i < RAND_BLOCK_LENGTH; i++)
        {
            float callValue  = Y * exp2f(rand[i]) - Z;
            callValue = (callValue > 0) ? callValue : 0;
            v0 += callValue;
            v1 += callValue * callValue;
        }
    }
    const float  exprt      = exp2f(RLog2E*OptionYearsList[opt]);
    CallResultList[opt]     = exprt * v0 * INV_RAND_N;
    const float  stdDev     = sqrtf((F_RAND_N * v1 - v0 * v0) * STDDEV_DENOM);
    CallConfidenceList[opt] = (float)(exprt * stdDev * CONFIDENCE_DENOM);
} //end of opt

#pragma omp barrier
if (threadID == 0) {
    end_cyc = _rdtsc();
    eTime = second();
    printf("Parallel simulation completed in %f seconds.n", eTime-sTime);
    printf("Validating the result...n");
}

double delta = 0.0, ref = 0.0, L1norm = 0.0;
int max_index = 0;
double max_local  = 0.0;
for(int i = 0; i < OPT_PER_THREAD; i++)
{
    double callReference, putReference;
    BlackScholesBodyCPU(
        callReference,
        putReference,
        StockPriceList[i],
        OptionStrikeList[i], OptionYearsList[i],  RISKFREE, VOLATILITY );
        ref   = callReference;
        delta = fabs(callReference - CallResultList[i]);
        sum_delta += delta;
        sum_ref   += fabs(ref);
        if(delta > 1e-6)
             sumReserve += CallConfidenceList[i] / delta;
        max_local = delta>max_local? delta: max_local;
}
max_delta = max_local>max_delta? max_local: max_delta;
vslDeleteStream(&(Streams[threadID]));
scalable_aligned_free(CallResultList);
scalable_aligned_free(CallConfidenceList);
scalable_aligned_free(StockPriceList);
scalable_aligned_free(OptionStrikeList);
scalable_aligned_free(OptionYearsList);

}//end of parallel block

Appendix

About the Author

Shuo Li works for the Intel Software and Service Group. His main interests are parallel programming and application software performance. In his recent role as a staff software performance engineer covering the financial service industry, Shuo works closely with software developers and modelers and helps them achieve high performance with their software solutions.

Shuo holds a Master's degree in Computer Science from the University of Oregon and an MBA degree from Duke University.

References and Resources

^[1]Option Pricing: A Simplified Approach (1979) by John C. Cox, Stephen A. Ross, and Mark Rubinstein:

^[2]Theorie de la Speculation, Annales Scientifiques de l´ Ecole Normale Sup´erieure, 21–86. Bachelier, L. (1900). reprinted 1995 Editions Jacques Gabay

^[3]Hull, John C, Options, Futures, and other Derivatives, 7th Edition Prentice-Hull, 2009

^[4]Wilmott, P., Derivatives: The Theory and Practice of Financial Engineering. Chichester: Wiley, 1998

^[5]Cox, J. C. Ross, S. A. and Rubinstein, M. Option Pricing: A simplified Approach Journal of Financial Economics 7 (October 1979): 229-64

^[6]Black, F., and M. Scholes, The Pricing of Options and Corporate Liabilities Journal of Political Economy, 81(May/June 1973): 637-59

^[7]Merton, R. C. Theory of Rational Option Pricing, Bell Journal of Economics and Management Science, 4(Spring 1973): 141-83

^[8]Boyle, P. P., Options: A Monte Carlo Approach Journal of Financial Economics, 4 (1977) 323-38

^[9]Black, Fischer and Scholes, Myron The Pricing of Options and Corporate Liabilities (May-Jun 1973)

^[10]Matsumoto, M., and Nishumira T. Mersenne Twister: A 623-Dimensionally Equidistributed Uniform Pseudo-Random Number Generator, ACM Transactions on Modeling and Computer Simulation, Vol. 8, No. 1, Pages 3-30, January 1998

^[11]Intel Xeon processor: http://www.intel.com/content/www/us/en/processors/xeon/xeon-processor-e7-family.html

^[12]Intel Xeon Phi coprocessor:  https://software.intel.com/en-us/articles/quick-start-guide-for-the-intel-xeon-phi-coprocessor-developer

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