“‘So, a standard derivative in [classical] calculus is a limiting quotient of [differences]…So, the idea is: what if you [replace] the differences [with] ratios? At first, when you hear this you think

it’s nuts…but I’ve totally bought into this philosophy” ~ Financial engineer Peter Carr [1]

Bigeometric-calculus, developed by Grossman and Katz [2], is a non-Newtonian alternative to the “usual” calculus of Newton and Leibniz; It uses multiplication based differentiation and integration instead of addition. In classical calculus, differences measure changes and sums measure accumulations. In bigeometric calculus, ratios measure changes in arguments and values and products are used for accumulations.

Some authors call the bigeometric calculus the “Volterra calculus”. However, this is not correct: Volterra calculus is not non-Newtonian is quite different from bigeometric calculus [3].

## Why Use Bigeometric Calculus?

Bigeometric calculus can be used anywhere problems are exponential in nature. It is particularly suited to problems involving in growth related problems, numerical approximations problems and price elasticity; It has emerged as a useful tool for working with the Black-Scholes model in financial engineering. In some cases involving differential equations, bigeometric calculus has been found to be more accurate than classical calculus [4]. These derivatives also work in the world of fractals, where the ordinary derivative depending on fractal dimension doesn’t exist [5].

## Derivatives in Bigeometric Calculus

In Newtonian calculus, linear functions have a constant derivative; In bigeometric calculus, it is the power functions which have a constant derivative. The relationship between the classical and bigeometric derivative (G) is [4]:

The bigeometric derivative is formally defined as [6]:

Unlike the classical derivative, this derivative is scale invariant (or scale free). In other words, it is invariant for all changes of scale or unit in function arguments and values: if you took a function and doubled it, then compare derivatives for both the doubled and non-doubled functions, they would be the same [3].

## References

[1] Tudball, D. (2017). Peter Carr’s Hall of Mirrors. Wilmott. Volume 2017, Issue 89 p. 36-44.

[2] Grossman M., Katz R., (1972), Non-Newtonian Calculus, Lee Press, Piegon Cove, Massachusetts.

[3] Grossman, M. & Katz, R. (2021). Non-Newtonian Calculus. Retrieved May 2, 2021 from: https://sites.google.com/site/nonnewtoniancalculus/-multiplicative-calculus

[4] Boruah, K. & Hazarika, B. (2000). Bigeometric Calculus and its applications. Retrieved May 2, 2021 from: https://arxiv.org/pdf/1608.08088.pdf

[5] Aniszewska, D. & Rybaczuk, M. (2008). Lyapunov type stability and Lyapunov exponent for exemplary multiplicative dynamical systems, Nonlinear Dynamics, Volume 54, Issue 4, Springer, 2008

[6] Filip, & D. Piatecki, C. (2014). An overview on the non-Newtonian calculus and its potential applications to economics. ffhal-00945788

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