Extra-Dimensional Approach to Option Pricing and Stochastic Volatility

TL;DR

This paper introduces a novel extra-dimensional framework for option pricing, deriving a generalized 5D Black-Scholes equation with stochastic volatility, and interprets the Merton-Garman equation as a first excited state within this higher-dimensional model.

Contribution

It presents a new higher-dimensional approach to derive and interpret the stochastic volatility models in option pricing, linking extra dimensions to financial dynamics.

Findings

Derived the 5D Black-Scholes differential equation with stochastic volatility.

Interpreted the Merton-Garman equation as the first excited state in an infinite family.

Showed projections from higher-dimensional space can reproduce known stochastic models.

Abstract

The generalized 5D Black-Scholes differential equation with stochastic volatility is derived. The projections of the stochastic evolutions associated with the random variables from an enlarged space or superspace onto an ordinary space can be achieved via higher-dimensional operators. The stochastic nature of the securities and volatility associated with the 3D Merton-Garman equation can then be interpreted as the effects of the extra dimensions. We showed that the Merton-Garman equation is the first excited state, i.e. n=m=1, within a family which contain an infinite numbers of Merton-Garman-like equations.