# A Black--Scholes inequality: applications and generalisation

## TL;DR

This paper explores a noncommutative semigroup structure on call price functions, deriving inequalities for implied volatility, and establishing a connection between semigroups, log-concave densities, and arbitrage-free market models.

## Contribution

It introduces a novel algebraic framework for call prices, linking semigroup theory with market models and providing new parametrizations of call surfaces.

## Key findings

- Derived a Black--Scholes implied volatility inequality.
- Established a correspondence between semigroups and log-concave densities.
- Provided an explicit example illustrating the theoretical framework.

## Abstract

The space of call price functions has a natural noncommutative semigroup structure with an involution. A basic example is the Black--Scholes call price surface, from which an interesting inequality for Black--Scholes implied volatility is derived. The binary operation is compatible with the convex order, and therefore a one-parameter sub-semigroup gives rise to an arbitrage-free market model. It is shown that each such one-parameter semigroup corresponds to a unique log-concave probability density, providing a family of tractable call price surface parametrisations in the spirit of the Gatheral--Jacquier SVI surface. An explicit example is given to illustrate the idea. The key observation is an isomorphism linking an initial call price curve to the lift zonoid of the terminal price of the underlying asset.