Arbitraj bakış açısından finansta kesirli brown hareketi

Fractional Brownian motion is a centered Gaussian process with stationary increments that is stochastically self-similar. It is suggested as a model in various disciplines, one of which is finance. Arbitrage is a trading strategy where positive earning is guaranteed with no risk. It is not expected in fair markets. Despite the fact that fractional Brownian motion allows for arbitrage, it has found a place in finance by capturing the long-range dependence observed in stock prices. We review the results recently obtained for arbitrage strategies when the stock price process is based on fractional Brownian motion. These are fractional Bachelier and fractional Black-Scholes models for the stock price or its logarithm. The suggested modifications in the model or in the trading to avoid arbitrage opportunities are analyzed. Existing stock price models which approximate a fractional Brownian motion in the limit are also studied. We construct two agent based stock price models as integrals with respect to a Poisson random measure. These processes are analyzed as the trading occurs more frequently and in smaller quantities. Fractional Brownian motion is obtained in the limit in the sense of finite dimensional distributions. We show that our simplified scaling is equivalent to time scaling used frequently for such limits.