# Performance of tail hedged portfolio with third moment variation swap

## TL;DR

This paper investigates the effectiveness of third moment variation swaps in hedging skew and tail risks of financial assets, analyzing the distribution of hedged portfolios under complex stochastic models.

## Contribution

It introduces a PDE-based approach to analyze the distribution of portfolios hedged with third moment variation swaps under stochastic volatility models.

## Key findings

- Hedged portfolios exhibit more Gaussian-like, thin-tailed distributions.
- The PDE approach effectively models the distribution of the hedged portfolio.
- Numerical analysis demonstrates the potential of third moment swaps in risk management.

## Abstract

The third moment variation of a financial asset return process is defined by the quadratic covariation between the return and square return processes. The skew and fat tail risk of an underlying asset can be hedged using a third moment variation swap under which a predetermined fixed leg and the floating leg of the realized third moment variation are exchanged. The probability density function of the hedged portfolio with the third moment variation swap was examined using a partial differential equation approach. An alternating direction implicit method was used for numerical analysis of the partial differential equation. Under the stochastic volatility and jump diffusion stochastic volatility models, the distributions of the hedged portfolio return are symmetric and have more Gaussian-like thin-tails.