Derivation of Worth Growth Rate of an Investor’s Portfolio Under Multi-fractal Analysis

The fractal dimension is the basic notion for describing structures that have a scaling symmetry. In finance, multi-fractality is one of the well known facts which characterized non-trivial properties of financial time series. The stock price (or index) fluctuations can be described in terms of long-range temporal correlations by a spectrum of the Holder exponents and a set of fractal dimensions. To forecast the market risk, assessing the stock price indices is the foundation. Multi-fractal has lots of advantages when explaining the volatility of the stock prices. The asset price returns are multi-period market depending on market scenarios which are the measure points. In this work, we use some tools of multi-fractal analysis to derive the worth growth rate of an investor’s portfolio for particular and general cases. For the particular case, we considered the situation when the mean interest rate of some stocks does not depend on other stocks in the market. That is, an investor has invested his money in a stock with a linear mean return. Under the general case, we considered a market comprising some units of assets in long position and a unit of the option in short position. Using Ito’s formula on the present value of the market, we derived the growth rate of investor’s portfolio. Our model equations, which are based on multiplicative processes, capture all the features of the returns. They are tested using data from Zenith Bank of Nigeria stock prices. From our graphs, the worth of investment grows as stock price increases and also decreases with stock price.