To forecast the market risk, assessing the stock price indices is the foundation. Multi-fractal has lots of advantage when explaining the volatility of the stock prices. The asset price returns is a multi-period (multi-fractal dimension) market depending on market scenarios which are the measure points. This paper considers the multi-fractal spectrum model (MSM) to measure the random character of asset price returns, aimed at deriving the MSM version of the random behaviour of equity returns of the existing ones in literature. We investigate the rate of returns prior to market signals corresponding to the value for packing dimension in fractal dispersion of Hausdorff measure. Furthermore, we give some conditions which determine the equilibrium price, the future market price and the optimal trading strategy.

This work deals with optimal trading using Black-Scholes equation with transaction costs. The partial differential equation for option pricing with transaction costs on the domain (P,T)∈(0,∞)×(0,T) with terminal condition C(P,T)=Max(P-E,0),P∈(0,∞) for European call options with strike price, E, and a suitable terminal condition for European puts was obtained and then solved to obtain the optimal value function.

The world of finance works better through logistics. We consider the logistics function in large market crashes corresponding to values of packing dimension of Rn  or  \(\alpha\) (max) by analyzing the fractal dispersion of Hausdorff prior to market signal with constraint of a zero heat capacity. We also advocate a stochastic Ito’s iterated procedure for locating optimal market crises signal relative to the Heat equation to give an early warning. The resultant differential equation is solve to obtain the recurrence formula.

This work investigates the rate of return from two portfolio management strategies. We first examine the return from total investment which includes both investment in the risky stock and investment in the risk-free asset. Secondly, we examine the return from investment in the risky stock only. We derive some optimality properties for the two portfolio management strategies. We show that the limiting behaviour of the rate of return on total investment is determined by the limiting behaviour of a related diffusion process.