2018/06/06 by Will Hicks, Hicks, Will
Economics, Econometrics and Finance · #81S25 #91B70 #Complex Systems and Time Series Analysis #Economic theories and models #FOS: Economics and business #Mathematical Finance (q-fin.MF) #Stochastic processes and financial applications #msc:81S25 #msc:91B70 #q-fin.MF
paper · pdf · doi:10.48550/arxiv.1806.07983
21 pages, 3 figures
openalex publication_date 2018/06/06 · arxiv created 2018/06/27 · arxiv updated 2018/06/28 · openalex created_date 2022/08/23 · openalex updated_date 2026/07/28
In this paper, we establish a link between quantum stochastic processes, and nonlocal diffusions. We demonstrate how the non-commutative Black-Scholes equation of Accardi & Boukas (Luigi Accardi, Andreas Boukas, 'The Quantum Black-Scholes Equation', Jun 2007, available at arXiv:0706.1300v1) can be written in integral form. This enables the application of the Monte-Carlo methods adapted to McKean stochastic differential equations (H. P. McKean, 'A class of Markov processes associated with nonlinear parabolic equations', Proc. Natl. Acad. Sci. U.S.A., 56(6):1907-1911, 1966) for the simulation of solutions. We show how unitary transformations can be applied to classical Black-Scholes systems to introduce novel quantum effects. These have a simple economic interpretation as a market `fear factor', whereby recent market turbulence causes an increase in volatility going forward, that is not linked to either the local volatility function or an additional stochastic variable. Lastly, we extend this system to 2 variables, and consider Quantum models for bid-offer spread dynamics.