# 1. INTRODUCTION

The price of many financial derivative securities can be expressed as intractable integrals of very high dimensionality [1]. For example, it is not unusual today for the risk-neutral price of an exotic derivative security to be a function of 20 or more underlying assets valued at 100 or more points in time, yielding an integral over 2,000+ dimensions.

Generally for integration problems of such high dimensionality, Monte Carlo (MC) and the related Quasi-Monte Carlo (QMC) methods are the only practical solution. Typically, millions of simulations are required to achieve an estimate of the price to the desired accuracy. Notoriously computationally intensive, repetitive, and embarrassingly parallel, such simulations are an interesting candidate for hardware acceleration.

A large body of work has been compiled over the years demonstrating that field-programmable gate arrays (FPGAs) are fast and efficient devices for generating high-quality pseudo-random numbers (see for example [2]-[4]). Some authors have also investigated the hardware acceleration of MC simulation [5]. However, we know of no work that investigates the efficacy of FPGAs for accelerating an entire MC options simulation, nor have we seen any direct comparisons of the performance of CPUs, FPGAs, and GPUs for this task. In this paper, we present an FPGA accelerator suitable for the pricing of a class of financial instruments by MC simulation. All of the components of the MC simulation are accelerated, including asset pathway generation, asset pathway valuation, and expectation. We use IEEE 754 double-precision floating-point arithmetic exclusively.
This paper is organized as follows. In section 2, we briefly review the MC method. In section 3, we present a mathematical description of European options pricing via Monte Carlo. This is followed in section 4 by a description of the hardware accelerator. We present hardware resource utilization and performance results in section 5. In section 6, we conclude with a discussion of potential improvements in the design and future work.

# 2. MONTE CARLO SIMULATION

We begin with a short review of MC methods. The following summary closely follows the introduction in [6]. Consider the MC estimation of an integral over the s-dimensional hypercube (with no loss of generality) of the form

where u = [u0, u1, … us-1]T is a vector of uniform random numbers and f represents the transformation from that vector to the simulation output f(u), assumed an unbiased estimator of μ. The form of the estimator of μ considered here is given by