Monte Carlo Methods applied to the Black-Scholes financial market model

Overview

This project implements a Monte Carlo simulation of the Black-Scholes financial model, using both the European and the Asian options. It contains an OpenCL C++ kernel, to be mapped to FPGA via SDAccel. It provides much better energy-per-operation than a GPU implementation, at a comparable performance level. Further details can be found in the following paper. If you find this work useful for your research, please consider citing:

@INPROCEEDINGS{7920245, 
  author={L. Ma and F. B. Muslim and L. Lavagno}, 
  booktitle={2016 European Modelling Symposium (EMS)}, 
  title={High Performance and Low Power Monte Carlo Methods to Option Pricing Models via High Level Design and Synthesis}, 
  year={2016},
  pages={157-162}, 
  doi={10.1109/EMS.2016.036}, 
  month={Nov},}

And

@ARTICLE{7859319, 
    author={F. B. Muslim and L. Ma and M. Roozmeh and L. Lavagno}, 
    journal={IEEE Access}, 
    title={Efficient FPGA Implementation of OpenCL High-Performance Computing Applications via High-Level Synthesis}, 
    year={2017}, 
    volume={5}, 
    pages={2747-2762}, 
    doi={10.1109/ACCESS.2017.2671881}, 
    }

Black-Scholes Model

The Black-Scholes model, which was first published by Fischer Black and Myron Scholes in 1973, is a famous and basic mathematical model describing the behaviour of investment instruments in financial markets. This model focuses on comparing the Return On Investment for one risky asset, whose price is subject to geometric Brownian motion and one riskless asset with a fixed interest rate.

The geometric Brownian behaviour of the price of the risky asset is described by this stochastic differential equation:

where S is the price of the risky asset (usually called stock price), r is the fixed interest rate of the riskless asset, is the volatility of the stock and is a Wiener process.

According to Ito's Lemma, the analytical solution of this stochastic differential equation is as follows:

where , (the standard normal distribution).

Call/Put Option

Entering more specifically into the financial sector operations, two styles of stock transaction options are considered in this project, namely the European vanilla option and Asian option (which is one of the exotic options). Call options and put options are defined reciprocally. Given the basic parameters for an option, namely expiration date and strike price, the call/put payoff price could be estimated as follows. For the European vanilla option, we have:

where S is the stock price at the expiration date (estimated by the model above) and K is the strike price. For the Asian option, we have:

where T is the time period (between now and the option expiration date) , S is the stock price at the expiration date, and K is the strike price.

The Monte Carlo Method

The Monte Carlo Method is one of the most widely used approaches to simulate stochastic processes, like a stock price modeled with Black-Scholes. This is especially true for exotic options, which are usually not solvable analytically. In this project, the Monte Carlo Method is used to estimate the payoff price of a given instrument using the Black Scholes model.

The given time period has to be partitioned into M steps according to the style of the option. M=1 for a European option, since the payoff price is independent of the price before the expiration date. At each time point of a Monte Carlo simulation of this kind, the stock price is determined by the stock price at the previous time point and by a normally distributed random number. The expectation of the payoff price can thus be estimated by N parallel independent simulations.

The convergence of the result produced by the Monte Carlo method is ensured in this case by running a very large number of simulation steps, namely , (which should be a very large number, e.g. ). Other convergence criteria (e.g. checking the difference between successive iterations) could be added.

Normally Distributed Random Number Generation

A key aspect of the quality of the results of the Monte Carlo method is the quality of the random numbers that it uses. The normally distributed random numbers that are used in this project are generated by the Mersenne-Twister algorithm followed by the Box-Muller transformation.

Mersenne-Twister

The Mersenne Twister is an algorithm to generate uniformly distributed pseudo random numbers. Its very long periodicity makes it a suitable algorithm for our application, since as discussed above the Monte Carlo method requires millions of random numbers.

Box-Muller transform

The Box Muller transformation transforms a pair of independent, uniformly distributed random numbers in the interval (0,1) into a pair of independent normally distributed random numbers, which are required for simulating the Black Scholes model. Given two independent ,

then , also independent.

Getting Started

The repository contains two directories, "blackEuro" and "blackAsian", implementing the European option and the Asian option respectively. They are written using C++, rather than OpenCL, because there is no workgroup-level memory that is worth sharing. All Monte Carlo simulations are independent. In this implementation, since the complexity of the random number generation process is simpler than the complexity of the Monte Carlo simulation step, 1 random number generator feeds a group of NUM_SIMS simulations, by utilizing BRAMs storing the intermediate results. The iterations over NUM_RNGS are fully unrolled (as if they were executed by an independent work item in a work group) and the iterations over NUM_SIMS and NUM_SIMGROUPS are pipelined. These two parameters should be chosen to fully utilize the resources on an FPGA. In order to ensure that enough simulations of a given stock are performed, NUM_SIMGROUPS can then be tuned.

The total number of simulations is and each simulation group assigned to a given RNG runs NUM_SIMS simulations. The best value for NUM_SIMS can be chosen by maximizing the use of BRAM blocks. By default it is 512 (which fills one BRAM block with 512 float values).

File Tree

blackScholes
│   README.md
│
└── common
│   │   defTypes.h
│   │   RNG.h
│   │   RNG.cpp
│   │   stockData.h
│   │   stockData.cpp
│   │   blackScholes.h
│   │   main.cpp
│   └─  ML_cl.h
│
└── blackEuro
│   │   solution.tcl
│   │   blackEuro.cpp
│   └─  blackScholes.cpp
│
└── blackAsian
    │   solution.tcl
    │   blackAsian.cpp
    └─  blackScholes.cpp

File/Dir name	Information
blackEuro.cpp	Top function of the European option kernel
blackAsian.cpp	Top function of the Asian option kernel
solution.tcl	Script to run sdaccel
blackScholes.h	It declares the blackScholes object instantiated in the top functions (same methods for European and Asian option).
blackScholes.cpp	It defines the blackScholes object instantiated in the top functions. Note that the definitions of the object methods are different between the European And Asian options.
stockData.cpp	Basic stock datasets. It defines an object instantiated in the top functions
RNG.cpp	Random Number Generator class. It defines an object instantiated in the blackSholes objects.
main.cpp	Host code calling the kernels, Input parameters for the kernels can be changed from the comman line.
ML_cl.h	CL/cl.hpp for OpenCL 1.2.6

Note that in the repository we had to include the OpenCL header for version 1.2.6, instead of the version 1.1 installed by sdaccel, because the latter causes compile-time errors. SDAccel and Vivado HLS work perfectly well with this header.

Parameters

The values of the parameters for a given stock and option are listed in the namespace Params in "main.cpp". The values of these parameters, except for the name of the kernel, have a default value and can be changed via the corresponding command-line option at runtime. The kernel name must be "blackAsian" for the Asian option and blackEuro for the European one. The call price and the put price are used for functional verification only (they are the expected output values for a given set of input values)

For example, the RTL emulation can be executed as follows to use the Asian option with the default parameter values:

run_emulation -flow hardware -args "-n blackAsian"

This TCL command in solution.tcl, on the other hand, uses a different set of values for the input parameters, and specifies the expected call and put output values:

run_emulation -flow hardware -args "-n blackAsian -s 100 -k 105 -r 0.1 -v 0.15 -t 10 -c 24.95 -p 0.283"

Argument	Meaning and default value
-t	time period (1.0)
-r	interest rate of riskless asset (0.05)
-v	volatility of the risky asset (0.2)
-s	initial price of the stock (100)
-k	strike price for the option (110)
-n	the kernel name, to be passed to the OpenCL runtime (no default; must be blackAsian or blackEuro)
-a	the binary name, to be passed to the OpenCL runtime (blackScholes.xclbin)
-c	expected call price (for verification)
-p	expected put price (for verification)

The number of simulations N, and the number of time partitions M, as well as all the other parameters related to the simulation, are listed in "blackScholes.cpp". They are compile-time constants in order to generate an optimized implementation. However, NUM_SIMGROUPS could also be set via a run-time command line option, like those above.

Parameter	information
NUM_STEPS	number of time steps (M)
NUM_RNGS	number of RNGs running in parallel, proportional to the area cost
NUM_SIMGROUPS	number of simulation groups (each with simulations) running in pipeline, proportional to the execution time
NUM_SIMS	number of simulations running in parallel for a given RNG (512 optimizes BRAM usage)

The area cost is proposrtional to NUM_RNG.

How to run an example

In each sub-directory, there is a script file called "solution.tcl". It can be used as follows:

sdaccel solution.tcl

The result of the call/put payoff price estimation will be printed on standard IO.

Note that RTL simulation can take a very long time for the Asian option. In order to obtain (imprecise) results quickly, the computation cost C can be reduced. For instance, NUM_SIMGROUPS has been set to 2 for the Asian option.

Sample Output

For the European option:

Option parameter	value
T	1
S0	100
K	110
rate	5%
volatility	20%

Simulation parameter	value
NUM_RNGS	8
NUM_SIMS	512
NUM_SIMGROUPS	4
NUM_STEPS	1

Area utilization	value
LUT	28%
FF	14%
BRAM	3%
DSP	13%

Output	value
call price	6.048
put price	10.65

For the Asian option,

Option parameter	value
T	10
S0	100
K	105
rate	1%
volatility	15%

Simulation parameter	value
NUM_RNGS	2
NUM_SIMS	64 (to keep RTL simulation under control; it should ideally be 512)
NUM_SIMGROUPS	2
NUM_STEPS	128

Area utilization	value
LUT	7%
FF	4%
BRAM	1%
DSP	3%

Output	value
call price	23.67
put price	0.22

Performance Metrics

As discussed above, the computational cost is a key factor that affects both the performance of the simulation and the quality of the result. The time complexity of the algorithm is O(C), so that we analyze the performance of our implementation as the total simulation time (number of clock cycles times clock period) per step:

The time taken by the algorithm is so for each step,

Basic Simulation procedure:

• Outer loop (N iterations in total)

• Inner loop (M iterations)

• Generate random numbers (unrolled loop)
• Estimate stock price at each time partition point

• Calculate the payoff price

• Count the sum of payoff prices

• Estimate the average

We can see that is related to the latency of the inner loop. Since each iteration of the inner loop requires random numbers, one of factors that limit the latency is the latency of generating a random number. The other factor are the mathematical operations of one Black Sholes step.

At frequencies below 100MHz on modern FPGAs, two random numbers are produced every two clock cycles (pipeline with Initiation Interval 2). By considering also the unrolling factor NUM_RNGS, the time for each step on the FPGA . For instance, at the frequency of 100MHz with NUM_RNGS=8,

Performance Comparison

• Intel HD Graphics 4400 laptop GPU, with 80 cores, 1100MHz
• GeForce GTX 960 with 1024 cores, 1178MHz, default power 120W, idle power 8W
• Quadro K4200 with 1344 cores, 784MHz, default power 108W, idle power 13W
• GeForce GTX Titan Z with 5760 cores, 876MHz(extrapolated)
• Virtex 7 xc7vx690tffg1157-2, using the sin/cos functions

platform	t(ns)	power(W)	energy/step(nJ)	notes
HD 4400	3.13	15	46.9
GTX 960	0.163	98-8	14.67
Quadro K4200	0.204	105-13	20.81
GTX Titan	0.0389	375	14.61	extrapolated
Virtex 7 sin/cos	0.315	24.4	7.69
Virtex 7 sinf/cosf	0.0958	21.2	1.94