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.

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},
    }

Or

@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}, 
    }

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

Build the project

In any directory such as

> cd FinancialModels_AmazonF1/blackScholes_model/europeanOption/
> make TARGETS=hw PLATFORM=$PLATFORM all 

Upload to AWS

See repo https://github.com/aws/aws-fpga/tree/master/SDAccel

Run the application

Examples of usage for the European and Asian options by BlackSchole model:

> blackeuro -b <binary_file_name> [-n number_of_repeat] [-s number_of_simulation] [-k number_of_time_partition] [-c expect_call_price] [-p expect_put_price] 
> blackasian -b <binary_file_name> [-n number_of_repeat] [-s number_of_simulation] [-k number_of_time_partition] [-c expect_call_price] [-p expect_put_price] 

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 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

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. threads	N. simulations	N. simulation groups	N. steps	Time C5 96-core CPU [ms]	Time F1 FPGA [ms]	LUT	LUTMem	REG	BRAM	DSP
Black-Scholes	European option	64	65536	32	1024	23.75	2.84	66%	7%	38%	23%	71%
Black-Scholes	Asian option	64	65536	32	1024	25	2.81	70%	8%	42%	31%	80%
Heston	European option	52	65536	64	1024	56.67	7.2	62%	8%	37%	20%	77%
Heston	European barrier option	52	65536	64	1024	36.25	6.4	63%	9%	39%	20%	77%