# Tail Option Pricing Under Power Laws

## TL;DR

This paper introduces a novel method for extending tail option prices based on Pareto laws, enabling analysis of the volatility surface and testing tail overpricing without requiring finite variance assumptions.

## Contribution

It develops a new approach to extend tail option prices using Pareto tail assumptions, providing a tool to scrutinize volatility surfaces and challenge existing models.

## Key findings

- Method effectively extends tail option prices using Pareto laws.
- Allows testing of tail overpricing in volatility surfaces.
- Does not require finite variance assumptions.

## Abstract

We build a methodology that takes a given option price in the tails with strike $K$ and extends (for calls, all strikes > $K$, for puts all strikes $< K$) assuming the continuation falls into what we define as "Karamata Constant" over which the strong Pareto law holds. The heuristic produces relative prices for options, with for sole parameter the tail index $\alpha$, under some mild arbitrage constraints. Usual restrictions such as finiteness of variance are not required. The methodology allows us to scrutinize the volatility surface and test various theories of relative tail option overpricing (usually built on thin tailed models and minor modifications/fudging of the Black-Scholes formula).