Option Pricing Under Multifractional Process and Long-Range Dependence

Abstract

We introduced a new method to compute the European Call (and Put) Option price under the assumption of multifractional Brownian motion (mBm). The reason why we need a procedure for estimating the Option price is due to the absence of a closed formula for this process. To compute the Option price, we first simulated the logarithmic price under mBm and, by using a discount factor, we computed the option’s pay-off. Then, we fitted the best probability distribution associated to the discounted pay-off, computing the European Call Option price as its average.

Anomalous diffusion, aging, and nonergodicity of scaled Brownian motion with fractional Gaussian noise: overview of related experimental observations and models

“…122 Multifractional FBM-based processes with a Hurst exponent varying in time 49,123,124 were also considered recently. 125–127 Such processes were used, e.g. , in mathematical-finance models with (multi-)fractional stochastic volatility.…”

A Composite Index for Measuring Stock Market Inefficiency

“…Another important issue lies on the fact that assuming a constant value for the Hurst exponent h is unrealistic (e.g., see Bianchi [32], Bianchi et al [31], and Mattera and Sciorio [42]) and explicitly contradicts the adaptive market hypothesis (AMH). To overcome this limitation, we consider an additional efficiency measure based on a time-varying Hurst exponent.…”

A Nonparametric Approach for Testing Long Memory in Stock Returns’ Higher Moments

“…The presence of long memory in stock returns has important implications in finance. For instance, portfolio decisions may become extremely sensitive to the investment horizons [5], the pricing of options under standard assumption is not anymore reliable [6] and the same applies to traditional tests of market efficiency that are no longer valid for time series affected by self-similarity.…”