Article by: Anatoliy Swishchuk
Published by: Department of Mathematics & Statistics, York University
“A new probabilistic approach is proposed to study variance and volatility swaps for financial markets with underlying asset and variance that follow the Heston (1993) model. We also study covariance and correlation swaps for the financial markets. As an application, we provide a numerical example using S&P60 Canada Index to price swap on the volatility.”
Full article (PDF): Link
Published by: FINCAD
Date: 2008
“The classic derivatives that allow investors to take a view on volatility are straddles or strangles. A long position in a straddle, for example, will generate a profit if the underlying asset price moves up or down, or if the implied volatility rises. However, these options are also sensitive to the underlying asset price, as the delta of a straddle or a strangle is zero only when the option is at-the-money.
Unlike these options, variance and volatility swaps provide pure exposure to volatility. A volatility swap is essentially a forward contract on future realized price volatility. At expiry the holder of a long position in a volatility swap receives (or owes if negative) the difference between the realized volatility and the initially chosen volatility strike, multiplied by a notional principal amount. A variance swap is analogously a forward contract on future realized price variance, which is the square of future realized volatility.
In both cases, at inception of the swap the strike is chosen such that the fair value of the swap is zero. This strike is then referred to as fair volatility and fair variance, respectively.”
Full article: Link
Article by: Peter Carr, Roger Lee
Published by: Risk Magazine
Date: 26 Oct 2007
“In this paper we develop strategies for pricing and hedging options on realized variance and volatility. Our strategies have the following features.
• Readily available inputs: We can use vanilla options as pricing benchmarks and as hedging instruments. If variance or volatility swaps are available, then we use them as well. We do not need other inputs (such as parameters of the instantaneous volatility dynamics).
• Comprehensive and readily computable outputs: We derive explicit and readily computable formulas for prices and hedge ratios for variance and volatility options, applicable at all times in the term of the option (not just inception).
• Accuracy and robustness: We test our pricing and hedging strategies under skew-generating volatility dynamics. Our discrete hedging simulations at a one-year horizon show mean absolute hedging errors under 10%, and in some cases under 5%.
• Easy modification to price and hedge options on implied volatility (VIX).
Specifically, we price and hedge realized variance and volatility options using variance and volatility swaps. When necessary, we in turn synthesize volatility swaps from vanilla options by the Carr-Lee [4] methodology; and variance swaps from vanilla options by the standard log-contract methodology.”
Full article (PDF): Link
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