Option Pricing for Processes Driven by Mixed Fractional Brownian Motion With Superimposed Jumps

Abstract

We propose a geometric mixed fractional Brownian motion model for the stock price process with possible jumps superimposed by an independent Poisson process. Option price of the European call option is computed for such a model. Some special cases are studied in detail. Mathematics Subject Classification: 60G22, 91B28

Pricing Geometric Asian Options under Mixed Fractional Brownian Motion Environment with Superimposed Jumps

“…We obtained the European call option price for such models in Prakasa Rao's work. [5] We now introduce the Poisson process as a model for the jump times in the stock price process. Let N(0) = 0 and let N(t) denote the number of jumps in the process that occur by time t for t > 0.…”

“…[4] Computation of the European call option price for price processes driven by mfBm with superimposed jumps is given in Prakasa Rao's work. [5] We now propose a geometric mfBm model for the stock price process with possible jumps superimposed by an independent Poisson process.…”

A New Stopping Problem and the Critical Exercise Price for American Fractional Lookback Option in a Special Mixed Jump-Diffusion Model

“…However, continuous assumptions on the dynamics of assets ignore sudden shocks to asset returns due to the arrival of important information, since the financial crisis and significant business always result in sudden changes in firm values, which cannot be captured by continuous sample paths. To get around this problem and to take into account the long memory property, it is reasonable to use the mfBm with jumps model to capture fluctuations of the financial asset (see [27,30,31,33,36]).…”

Large Deviation Probabilities for Maximum Likelihood Estimator and Bayes Estimator of a Parameter for Mixed Fractional Ornstein-Uhlenbeck Type Process

“…Pricing geometric Asian options under mixed fractional Brownian motion was studied in Prakasa Rao (2015b). Option procing for processes driven by a mixed fractional Brownian motion with superimposed jumps was discussed in Prakasa Rao (2015a). Rudomino-Dusyatska (2003) and more recently Prakasa Rao (2009Rao ( , 2017 investigated problems of statistical inference for processes modeled via stochastic differential equations driven by a mixed fractional Brownian motion among others.…”