A Nonlocal Approach to The Quantum Kolmogorov Backward Equation and Links to Noncommutative Geometry

The Accardi-Boukas quantum Black-Scholes equation can be used as an alternative to the classical approach to finance, and has been found to have a number of useful benefits. The quantum Kolmogorov backward equations, and associated quantum Fokker-Planck equations, that arise from this general framework, are derived using the Hudson-Parthasarathy quantum stochastic calculus. In this paper we show how these equations can be derived using a nonlocal approach to quantum mechanics. We show how nonlocal diffusions, and quantum stochastic processes can be linked, and discuss how moment matching can be used for deriving solutions.Comment: 19 page

Path integral approach to Asian options in the Black-Scholes model

We derive a closed-form solution for the price of an average price as well as an average strike geometric Asian option, by making use of the path integral formulation. Our results are compared to a numerical Monte Carlo simulation. We also develop a pricing formula for an Asian option with a barrier on a control process, combining the method of images with a partitioning of the set of paths according to the average along the path. This formula is exact when the correlation is zero, and is approximate when the correlation increases.Comment: 13 pages, 3 figures, updated version has added references to path integral literatur

Levy processes - from probability theory to finance and quantum groups

Stochastic processes are families of random variables; Lévy processes are families indexed by the positive reals which are independent with stationary increments and are stochastically continuous. The author reviews the basic properties of Lévy processes and considers some of their applications