Volatility Library

Article by: Kresimir Demeterfi, Emanuel Derman, Michael Kamal, Joseph Zou
Published by: Goldman, Sachs & Co.
Date: 1999

“Trading in derivatives has caused investors, and especially market makers, to be concerned with the volatility of asset returns along with their direction. Uncertain and time-varying volatility imparts risk to an otherwise hedged position, and volatility risk is not easy to manage with ordinary instruments. Volatility swaps are a new class of derivative, for which an asset’s volatility itself is the underlying. This article describes how volatility swaps work, and derives pricing and hedging equations for them. Interestingly, the natural derivative instrument in this family would be based on variance, rather than volatility, since a variance swap can be replicated (pretty well) by a static portfolio of ordinary European calls and puts on the price of the underlying asset. The authors also show how to set up a volatility hedge when the available traded options exhibit a smile or skew pattern.”

Full article (PDF, subscription required): Link

Article by: Kresimir Demeterfi, Emanuel Derman, Michael Kamal, Joseph Zou
Published by: Goldman, Sachs & Co.
Date: 1999

“Volatility swaps are forward contracts on future realized stock volatility. Variance swaps are similar contracts on variance, the square of future volatility. Both of these instruments provide an easy way for investors to gain exposure to the future level of volatility.

“Unlike a stock option, whose volatility exposure is contaminated by its stock-price dependence, these swaps provide pure exposure to volatility alone. You can use these instruments to speculate on future volatility levels, to trade the spread between realized and implied volatility, or to hedge the volatility exposure of other positions or businesses.

“In this report we explain the properties and the theory of both variance and volatility swaps, first from an intuitive point of view and then more rigorously. The theory of variance swaps is more straightforward. We show how a variance swap can be theoretically replicated by a hedged portfolio of standard options with suitably chosen strikes, as long as stock prices evolve without jumps. The fair value of the variance swap is the cost of the replicating portfolio. We derive analytic formulas for theoretical fair value in the presence of realistic volatility skews. These formulas can be used to estimate swap values quickly as the skew changes.

“We then examine the modifications to these theoretical results when reality intrudes, for example when some necessary strikes are unavailable, or when stock prices undergo jumps. Finally, we briefly return to volatility swaps, and show that they can be replicated by dynamically trading the more straightforward variance swap. As a result, the value of the volatility swap depends on the volatility of volatility itself.”

Full article (PDF): Link