ABSTRACT
Computational finance is a cross-disciplinary area that relies onmathematics, statistics, finance, and computational algorithms to make critical decisions. One of the core tasks in this area is to analyze and measure the risk component that a financial portfolio would create. Portfolio would generally comprise of stocks, bonds, and other instruments, such as derivatives. Derivatives are financial instruments that depend on some other assets, such as stocks. They are also referred commonly as options. An option is a contract in which the buyer (holder) has the right but no obligation to buy (call)
or sell (put) an underlying asset (for example, a stock) at a predetermined price (strike price) on or before a specified date (expiration date). The seller (also known as writer) has the obligation to honor the terms specified in the contract (option). The holder pays an option premium to the writer [44]. Option pricing is one of the fundamental problems in finance which has led to two Nobel prizes
to be awarded. In 1998, Myron S. Scholes and Robert C. Merton received the Nobel prize for the Black-Scholes-Merton (BSM) model [6]. Recently, Charles Engle received a Nobel prize in 2003 for his autoregressive conditional heteroskedasticity (ARCH) model [32]. There are a multitude of financial instruments traded on the markets. The class of instruments known as American options
are claims to payoffs at a time chosen by the holder of the security. These contracts give the holder the right to exercise the option prior to the expiration date in order to achieve maximum profit. Therefore, determining an optimum exercise policy is a key issue in pricing American securities. The solution for the optimal exercise policy must typically be performed numerically, and is usually a computationally intensive problem. There are many numerical techniques proposed in the literature for option pricing, such as
binomial lattice approach [68] for the asset price and then roll backward the tree checking at each node if early exercise is optimal. Binomial lattice approach is one of the traditionally used computing models [23] in the finance community due to its intuitiveness and the ease of implementation. Another traditional model is the Monte-Carlo (MC) [10] simulation of price movements. This is done by evaluating the option value from many simulated trajectories of the asset price and taking the average. The accuracy of the final price depends on the number of simulations performed. These trajectory calculations are independent of each other and hence lead to embarrassingly parallel situations (for example, see [66]). Some of the computational fluid dynamics techniques, such as finite-differencing [31,67] and finite-element methods [75], are emerging as important techniques for option pricing problem by solving the Black-Scholes and other models manifested as partial differential equations (PDE). Recently, fast Fourier transform (FFT) has been shown [16] to be suitable for option pricing problems. Also, it is shown [24] that FFT method can be used to study multifactor models for this problem and shown to be quite effective in terms of computational times when compared to other numerical techniques, such as MC and PDE approaches. Economic and financial models used for evaluation and forecasting purposes typically lead to
large dynamic, nonlinear problems that have to be solved for a certain time span. In particular, many of the problems in finance demand efficient algorithms and high-performance computing capabilities [2,74]. All the techniquesmentioned earlier are suitable for parallel computing.However, parallel computing is not yet widely popular in the field of finance, though there is recognition for its need. The prices and the price dynamics of goods and financial securities the investor purchases determine the investor’s overall budget. An investor has many assets to choose from over many periods and the market dynamics determine the price of each asset. Computational requirement under such conditions are large and when there are market frictions the computational requirements increase dramatically [41]. An important aspect of the portfolio optimization problem is the objective function that represents the investor’s preferences. The need of investors to manage the risks associated with their portfolios, vis-à-vis, their liabilities also prompted the development of mathematical models for portfolio management. These models trace their roots to Markowitz’s mean-variance optimization models [40]. Academics and practitioners are turning increasingly to mathematical models and computer simulations in order to understand the peculiarities of financial markets and to develop risk-management tools. One of the notable models is the highly celebrated Black-Scholes model for option pricing [6]. Immense computing power available now need to be harnessed in an effective manner to address the ever-increasing demand from the practitioners, for example, for solving the complex mathematical models for the risk management throughportfolio adjustments.However,more andmoremodels are developed these days for pricing the risks of equities, stocks, bonds, derivatives of these instruments without much attention to their solubility or tractability. The need to compute the solutions for such models dictates the design and development of efficient algorithms that not only solve the problem in hand but also try to utilize the supercomputing power of the present day. In other words, parallel algorithms are in great demand for financial problems, especially for long-dated options with many underlying assets. Many of the computational finance problems are shown to be well suited for solution on parallel architectures of many kinds [21,51]. However, the adoption of parallel computing to the modern day finance has been very modest [74]. Moreover, volatility of the market place demands the processing of information quickly and accurately in order to react in a very short time to avoid potential huge losses.
