Abstract
This study deals with the arbitrage problem on the financial market when the underlying asset follows a mixed fractional Brownian motion. We prove the existence and uniqueness theorem for the mixed geometric fractional Brownian motion equation. The semi-martingale approximation approach to mixed fractional Brownian motion is used to eliminate the arbitrage opportunities.
Highlights
The classic Black-Scholes-Merton model was introduced in 1973
It is well known that the basis of Option pricing problems is how to describe the behavior of the underlying asset’s price
Since the fractional Brownian motion has two important properties, it has the ability to capture the behavior of underlying asset price
Summary
The classic Black-Scholes-Merton model was introduced in 1973. Option pricing problems have been one of the hotest issues for researchers and practitioners from the academia and industry. Since the fractional Brownian motion has two important properties (self-similarity and long-range dependence), it has the ability to capture the behavior of underlying asset price. The fractional Brownian motion is neither a Markov process nor a semi-martingale as well as it cannot use the usual stochastic calculus to analyze it. To eliminate the arbitrage opportunities and to reflect the long memory of the financial time series, many scholars have proposed the use of mixed fractional Brownian motion. The arbitrage problem still exists for H [ The issue with this is that the extensively used Itô calculus, developed from semi-martingales to solve stochastic integral, does not apply here. The non semi-martingale property of mixed fractional Brownian motion indicates that arbitrage opportunities are possible.
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