Managing Volatility Risk: An Application of Karhunen-Lo\`eve Decomposition and Filtered Historical Simulation

TL;DR

This paper applies Karhunen-Loève decomposition and Filtered Historical Simulation to model and predict volatility risk in interest rate options, specifically swaptions, by analyzing the dynamics of implied volatility surfaces over time.

Contribution

It introduces a novel combination of Karhunen-Loève decomposition with Filtered Historical Simulation for volatility risk management in interest rate derivatives.

Findings

Effective decomposition of volatility surface dynamics.

Accurate VaR predictions validated by standard tests.

Ensures non-arbitrage conditions in risk estimates.

Abstract

Implied volatilities form a well-known structure of smile or surface which accommodates the Bachelier model and observed market prices of interest rate options. For the swaptions that we study, three parameters are taken into account for indexing the implied volatilities and form a "volatility cube": strike (or moneyness), time to maturity of the option contract, duration of the underlying swap contract. It should be noted that the implied volatility structure changes across time, which makes it important to study its dynamics in order to well manage the volatility risk. As volatilities are correlated across the cube, it is preferable to decompose the dynamics on orthogonal principal components, which is the idea of Karhunen-Lo\`eve decomposition that we have adopted in the article. The projections on principal components are investigated by Filtered Historical Simulation in order to predict the Value at Risk (VaR), which is then examined by standard tests and non-arbitrage condition to ensure its appropriateness.