Quantum Physics and Options

Coming from the field of quantum many body physics I have always been interested in books and publications combining quantum physics with quantitative finance. Therefore I am going to write some blog posts on this topic. As a starting point I want to introduce the concept of the Hamiltonian  in the pricing of options. In quantum mechanics, the Hamiltonian is the operator corresponding to the total energy of the system. By analogy with classical mechanics, the Hamiltonian is commonly expressed as the sum of operators corresponding to the kinetic and potential energies of a system in the form

H=T+V

Can a Hamiltonian formulation provide new tools for obtaining solutions for op- tion pricing ? Two key concepts related to Hamiltonians are 

• Eigenfunctions
• Potentials

It turns out, that the knowledge of all the eigenfunctions of a Hamiltonian yields an exact solution for a large class of path-dependent and path-independent options. If we take, for an example, barrier options - they can modelled by placing constraints on the eigen- functions of the Hamiltonian. In our next blog post we will review those aspects of quantum mechanics that are relevant for the analysis of option pricing.