An analytical perturbative solution to the Merton–Garman model using symmetries

Abstract

In this paper, we introduce an analytical perturbative solution to the Merton–Garman model. It is obtained by doing perturbation theory around the exact analytical solution of a model which possesses a two‐dimensional Galilean symmetry. We compare our perturbative solution of the Merton–Garman model to Monte Carlo simulations and find that our solutions perform surprisingly well for a wide range of parameters. We also show how to use symmetries to build option pricing models. Our results demonstrate that the concept of symmetry is important in mathematical finance.

Spontaneous symmetry breaking in quantum finance

“…It can be easily proved that if we replace the option C(x, t) in eq. (5) by the security S(x, t) = e x , then we obtain the vacuum condition (4). Note however that although the operator defined aŝ pC(x, t) = ∂C(x, t)/∂x is a conserved quantity (symmetry of the Hamiltonian), the same operator is not a generator for the symmetry of the ground (martingale) state sincep…”

“…The new terms together must evidently obey the Martingale condition. Note that previously some authors considered some symmetry breaking mechanism in the MG case [4], however, the kind of symmetries as well as the methods developed were different to the case under study in the present paper. Finally, in a financial setting, the methods analyzed here can be used for improving some predictions in the market.…”

“…The spontaneous symmetry breaking phenomena is able to explain a significant number of observations, including superconductivity, ferromagnetism and the electroweak theory [1][2][3]. Although some symmetry breaking approaches were analyzed in [4] by using Financial equations, the generic phenomena of spontaneous symmetry breaking has not yet been explored in the scenario of Quantum finance. What we know about Quantum finance is that some financial equations such as the Black-Scholes (BS) [5] and the Merton-Garman (MG) [6,7], can be expressed in a Hamiltonian (eigenvalue) form after doing some transformation of variables [8][9][10][11].…”

Revenue Management in Airlines and External Factors Affecting Decisions: The Harmonic Oscillator Model

“…Collective decisions can be interpreted as spontaneous symmetry breaking of the system if we consider the price and the demand as Quantum Fields. Similar situations emerge in finance and other research areas where collective decisions emerge naturally [11,12]. Here we will analyze the standard Hamiltonian of the form…”