Monte Carlo financial models on Amazon F1 instances

Introduction

This repository includes F1-optimized implementations of four Monte Carlo financial models, namely:

• the European and the Asian options of the the Black-Scholes model,
• the European and the European barrier options of the Heston model.

Theory

Black-Scholes Model

The Black-Scholes model, which was first published by Fischer Black and Myron Scholes in 1973, is a well known 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).

Entering more specifically into the financial sector operations, two styles of stock transaction options are considered in this implementation of the Black-Scholes model, 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.

Heston Model

The Heston model, which was first published by Steven Heston in 1993, is a more sophisticated 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 and volatility are subject to geometric Brownian motion and one riskless asset with a fixed interest rate.

Two styles of stock transaction options are considered in this implementation of the Heston model, namely the European vanilla option and European barrier 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 can be estimated as discussed in this article.

Usage

Black-Scholes models

Examples of usage for the European and Asian options:

> blackeuro 
> blackasian 

The outputs of both commands are the expected call and put prices. The -b <binary_file_name> option can be used to specify a binary file name different from the default <kernel_name>.hw.xilinx_xil-accel-rd-ku115_4ddr-xpr.awsxclbin

The model parameters are specified in a file (in protobuf form) called blackEuro.parameters and blackAsian.parameters respectively. The meaning of the parameters is as follows.

Parameter	Meaning
time	time period
rate	interest rate of riskless asset
volatility	volatility of the risky asset
initprice	initial price of the stock
strikeprice	strike price for the option

Heston models

Examples of usage for European and European with barrier options:

> hestoneuro 
> hestoneurobarrier

The outputs of both commands are the expected call and put prices. The -b <binary_file_name> option can be used to specify a binary file name different from the default <kernel_name>.hw.xilinx_xil-accel-rd-ku115_4ddr-xpr.awsxclbin

The model parameters are specified in a file (in protobuf form) called hestonEuro.parameters and hestonEuroBarrier.parameters respectively. The meaning of the parameters is as follows (see also this article for more details).

Parameter	Meaning
time	time period
rate	interest rate of riskless asset
volatility	volatility of the risky asset
initprice	initial price of the stock
strikeprice	strike price for the option
theta	long run average price volatility
kappa	rate at which the volatility reverts to theta
xi	volatility of the volatility
rho	covariance
upb	upper bound on price
lowb	lower bound on price

Performance on Amazon F1 FPGA

Target frequency is 250MHz. Target device is 'xcvu9p-flgb2104-2-i'

Model	Option	N. random number generators	N. simulations	N. simulation groups	N. steps	Time C5 CPU [s]	Time F1 CPU [s]	Time F1 FPGA [s]	LUT	LUTMem	REG	BRAM	DSP
Black-Scholes	European option	64	512	1024	1	125	114	0.23	31%	2%	15%	26%	43%
Black-Scholes	Asian option	64	512	65536	256	376	497	0.83	31%	2%	16%	26%	43%
Heston	European option	32	512	512	256	226	330	1.52	18%	2%	9%	11%	26%
Heston	European barrier option	32	512	512	256	32	40	0.75	18%	2%	9%	11%	26%

The results on the CPUs use a single thread. For n threads with independent resources, the speedup would be exactly n, since Monte Carlo simulations are completely independent.

Further information and recompilation

Further informations about the optimizations used in this implementation can be found in the paper High Performance and Low Power Monte Carlo Methods to Option Pricing Models via High Level Design and Synthesis. 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},}

In all cases, the enclosed Makefile can be used to compile the models. For example:

cd blackScholes_model/europeanOption
source <path to SDSoc v2017.1>/.settings64-SDx.sh
export SDACCEL_DIR=<path to aws-fpga>/SDAccel
export COMMON_REPO=$SDACCEL_DIR/examples/xilinx/
export PLATFORM=xilinx_aws-vu9p-f1_4ddr-xpr-2pr_4_0
export AWS_PLATFORM=$SDACCEL_DIR/aws_platform/xilinx_aws-vu9p-f1_4ddr-xpr-2pr_4_0/xilinx_aws-vu9p-f1_4ddr-xpr-2pr_4_0.xpfm
make TARGETS=hw DEVICES=$AWS_PLATFORM all

compiles the code and generates the F1-targeted bitstream for the European option of the Black-Scholes model. Environment variable COMMON_REPO must point to the examples/xilinx sub-directory of the folder where the AWS development kit github has been checked out.

For purely SW execution, a target pure_c has been added to the Makefiles. It compiles the main and the kernel using purely C++ and runs it on the local CPU.

Note that, for the sake of efficient implementation on the FPGA, the simulation parameters which directly affect the amount of parallelism in the implementation are set as compile-time constants in blackScholes.cpp, hestonEuro.h and hestonEuroBarrier.h respectively. They are listed below and can be changed by recompiling the kernels and re-generating the AFI.

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

See also these repositories for more information about the compilation and optimization of the kernels:

• the European and the Asian options of the the Black-Scholes model,
• the European and the European barrier options the Heston model.