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Volatility risk is the risk of a change of price of a portfolio as a result of changes in the volatility of a risk factor. It usually applies to portfolios of derivatives instruments, where the volatility of its underlyings is a major influencer of prices.
Sensitivity to Volatility
A measure for the sensitivity of a price of a portfolio (or asset) to changes in volatility is vega, the rate of change of the value of the portfolio with respect to the volatility of the underlying asset.
This kind of risk can be managed using appropriate financial instruments whose price depends on the volatility of a given financial asset (a stock, a commodity, an interest rate, etc.). Examples are Futures contracts such as VIX for equities, or caps, floors and swaptions for interest rates.