21
Long Memory Processes and Benoit Mandelbrot
Do not go where the path may lead, go instead where there is no path and leave a trail.
Ralph Waldo Emerson

21.1 INTRODUCTION

The assumption of a Gaussian process is a cornerstone of modern mathematical finance and is present in almost every book from theory of investment to the theory of option pricing. It is common mathematical behavior of almost every professional quantitative analyst working in financial institutions to assume that underlying any observed financial instrument there is a martingale and, therefore, an equivalent martingale measure.
Hence, before looking at data or analyzing graphs to gain insight about the underlying process, the community of quantitative analysts working in banks begin any model by postulating the absence of arbitrage and axiomatically assuming market completeness. Additionally, for convenience purpose - that is, for ease of use or mathematical tractability - the chosen martingale is almost always the Gaussian process.
In this chapter we discuss some of the problems and highlight the inappropriatedness of the current Gaussian framework largely used by academic and financial market practitioners alike. This chapter is based on the seminal and original work of Benoit Mandelbrot (private communication, 2009). In Section 21.2 we describe some properties of the price processes of financial instruments and we present evidence against the Brownian motion assumption. In Section 21.3 we describe the ...

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