Diffusion Processes on Solvable Groups of Upper Triangular 3x3 matrices. Applications in Asian and Basket Options

One of the questions in algebraic groups is about the asymptotic behavior of the probability of return of a random walk, which closely related on the growth rate of a group. Upper-triangular matrices form a group. Solvable groups have an exponential growth rate and it was shown that the asymptotic behavior of the probability of return on these groups has a fractional-exponential decal. The results in the paper by Molchanov and others, are different from the previous finding. They showed that in the case of solvable groups of upper-triangular 2x2 matrices the return probability of the Brownian motions has a polynomial decay. In this dissertation, we extended this research to the case of solvable groups of upper-triangular 3x3 matrices. The elements in the 3x3 matrices that define a Brownian motion on these groups contain integrals of geometric Brownian motions. These integrals have an important role in Asian and Asian-Basket options. We proved some properties of these integrals and showed that certain cases of geometric Asian-basket call options with two assets have a higher risk that the same type of put options. Which implies that some trading strategies might benefit from a reevaluation using a new risk assessment of geometric Asian-Basket.