Tolerance to arbitrage, inclusion of fractional Brownian motion to model stock price fluctuations
Abstract (Summary)We propose a continuous-time model of stock price fluctuations. This model includes fractional Browrllan motion (fBm) as a possible representation of stock price changes. We provide mathematical definitions of trading strategies, self-financing strategies, and attainability of contingent daims in this setting. Under certain conditions, we construct arbitrage opportunities using properties of continuous h c - tions with bounded pvariation, with 1 < p < 2. It is shown that almost all sample paths of fBm with 112 < H < 1 have bounded pvariation with 1/H < p < 2. Therefore, in a hancial market where stock price changes behave like geometric fBm with 112 < H < 1, there are arbitrage opportunities. In addition, rve show that functions of bounded pmiation with p 2 1 may have the property of idbite crossing of zero. This is used to show that stoploss start-gain strategy may not be self-financing for stock price processes having bounded pvariation with 1 < p < 2. Thus, by definition, the stop-loss start-gain strategy may not parantee arbitrage under the assumption of geometric fBm. This complements the result of Carr and Jarrow (1990) concerning standard geometric Brownian motion. Simulations and statistical analyses are also included. In particular, we adapt certain modifications of quadratic variation to test the index H under the assurnption that stock price process having a distribution of geometric mm.
Source Type:Master's Thesis
Date of Publication:01/01/1997