An Introduction to Computational Finance

Abstract

Preface Mathematical finance theory has been established as one sector of finance theory. This theory is based on probability theory and many probabilists have contributed to this field. The Black-Scholes model is a typical model for the complete market. This model is an outstanding model convenient to analyze, but it is well-known that in the real world the completeness of the market is not usually satisfied. The distribution of the log return of an asset usually has a fat tail and asymmetry, and the market is usually incomplete. Therefore we need other models for the incomplete market. The geometric Lévy process (GLP) model is one of the most important models for the incomplete market. This model is able to possess a fat tail property, an asymmetric distribution and smile/smirk properties of implied volatility. An incomplete market has many martingale measures (or risk-neutral measures) by the second fundamental theorem of mathematical finance. So we have to select a suitable martingale measure among them in order to discuss option pricing based on arbitrage theory. Many kinds of martingale measures have been proposed for this. Among them the minimal entropy martingale measure (MEMM) is the most important candidate. Among the many GLP models, the importance of the geometric stable process (GSP) model is recognized by Fama [36] and Mandelbrot [77]. Since then many researchers have studied this model. The suitable martingale measure for the GSP model had not been evident for many years, but since the existence of the MEMM for GSP model was proved and an explicit form of it was obtained, the GSP model can be applied to a wide class of problems concerning option pricing.

An adaptive algorithm for high‐dimensional integrals on heterogeneous CPU‐GPU systems

“…However, we measure some significant differences in the performance gain due to the computational nature of the integrand functions. More precisely, the families F (2) and F (3) show a performance gain of about 5× because their analytic expressions are based only on floating point operations, without the use of trigonometric or exponential functions as for the families F (1) , F (4) , and F (5) . The worst case is represented by the family F (6) with a performance gain of about 1.8× because the thread divergence due to the presence of the selection structure in its analytic expression, that greatly limits the GPU performance when threads follow different paths in the control flow.…”

“…The computation of (1) is a key task in many scientific applications, ranging from computer graphics, 2 to the high energy physics, 3 to the computational finance, 4 so that several methods, algorithms and software are available to this problem. 5 In this paper, we pay special attention to the class of parallel adaptive algorithms because they are able to achieve high accuracy with an acceptable computational cost so that they are on the basis of several mathematical software libraries and routines.…”

Enhanced group analysis of a semi linear generalization of a general bond-pricing equation

“…In Financial Mathematics, the former describes the evolution of standard or "vanilla" products [11,7,12,13], while the latter an exotic type of products [14,15,16]. Moreover, a common assumption for the function H -termed also as the barrier function -is to have the exponential form…”

High-Order Compact Finite Difference Method for Black–Scholes PDE

“…For various purpose, there are many kinds of options, such as, vanilla options (European call or put option, American call or put option), Asian option, Bermudan option, exotic option, look-back option, barrier option, etc. [8,9]. Options that can be exercised only on the maturity date are called European option, while options that can be exercised at any time up to the maturity date are called American option.…”