Fractional Brownian Motions in Financial Models, Simulation and Pricing

Fractional Brownian motion (fBM) was first introduced within a Hilbert space framework by Kolmogorov [Kol40], while studying spiral curves in Hilbert space; the process was further studied and was coined the name ’fractional Brownian motion’ in the 1968 paper by Mandelbrot and Van Ness [MVN68]. It has been widely used in various scientific fields, most notability in hydrology as first suggested in [Man65]. It also plays an important role in communication technology by enriching the queuing theory in terms of simulating real network traffic. In recent years, it has been steadily gaining traction in finance. This is due to the fact that, traditional stochastic volatility model driven by ordinary Brownian motion implies exponential decay of the implied volatility smile, where empirical studies shows the decay is of power order. This can be explained by the long-memoryness found in autocovariance of the time-series, and this long-memory feature in instantaneous volatility process is called the volatility persistence. Even though this phenomenon is commonly observed on market implied volatility surface, it has been largely ignored because of the difficulty to capture it within the ordinary stochastic volatility framework. Within the framework of time-series analysis, the volatility persistence is concluded as the long-range dependence displaying in the instantaneous volatility time-series, or depicted by prominent components at low frequency under spectral density of the autocovariance function. This phenomenon is so commonly found, that Granger [Gra66] considered such phenomenon as the ”typical spectral shape of an economic variable”. The presence of the long-memory phenomenon has important implication in financial economics and financial engineering, especially in the area of the portfolio optimization: the choice of optimal consumption/saving because now the optimal decision might become very sensitive to the investment horizon instead of an asymptotically time-homogeneous problem if the asset return now is long-range dependent, also in the area of derivative pricing. The introduction of long-memory process is inconsistent with the pre-existing continuous-time process framework commonly utilized in derivative pricing (see, [Mah90b], [Mah90a]), and in [Mer87], [LeR89], Merton and Leroy respectively discussed the relationship of the efficient market hypothesis argument and the presence of the long-memory phenomenon. For detail discussion about the statistical test for the existence of long-range memory, see [Lo91] (the author has also proposed a robust extension of the R/S statistics for the purpose of robust statistical inference). Abudant empirical studies have been done with real data, most exmplemary the study done by Greene and Fielitz [GF77]. The studies was done on securities listed on the New York Stock Exchange, and many were found displaying long-range dependence in their daily returns. Several modeling approaches have been suggested capturing this persistence in conditional variance either via a unit-root or long memory process. In order to keep the pricing-framework largely intact, it is more interesting to study the long memory process, and fBM offers a simplistic extension particular for this purpose, owning to its similarity to the ordinary Brownian motion and its Gaussian properties, on the familiar process. In this paper, several approaches to simulate fractional Brownian motion with H > 1/2 are outlined, including the exact methods and approximate methods, where the Hurst Index H is a parameter used in literature to generalize Brownian motion into fractional Brownian motion, first made popular by Benoit Mandelbrot. Brief introduction of the truncated fractional Brownian motion (long-memory model in continuous time) is also included, as proposed in [CR96], [CR98], which is shown to be inadequate to replicate the fractional Brownian motion. Through the full Monte-Carlo simulation scheme, the implied volatility surface is constructed. One of the main result in the research is that, imposing correlation between the fractional Brownian motion driven stochastic volatility and the ordinary Brownian motion driven asset process does not translates into skewness of the implied volatility surface, this is further