  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.
