Valuing futures and options on volatility

The Valuation of Volatility Options

“…Then V;t 2 E = V 0 ; t 2 E for all V 0 V . 15 Proof of Lemma 8: Property i follows from the fact that the holder of a long maturity option can implement a n y exercise policy chosen by the holder of a shorter maturity option. For property ii note that C V 0 ; t C V;t + sup 2S t;T E t e ,r V 0 ,V where S t; T denotes the set of stopping times of the ltration with values in t; T , which follows from the inequality a + b + a + + b + .…”

Pricing variance swaps under stochastic volatility and stochastic interest rate

“…Results showed that prices of variance swaps implied by the regime switching Heston stochastic volatility model were significantly higher than those without switching regimes. Grunbichler and Longstaff [54] also developed pricing model for options on variance based on the Heston stochastic volatility model. One important finding was the contrast characteristics between volatility derivatives and options on traded assets.…”

A jump diffusion model for VIX volatility options and futures

“…Volatility derivatives are considered by some to "have the potential to be one of the most important new financial innovations" (Grünbichler and Longstaff, 1996). Traditionally, derivatives have allowed investors and firms to hedge against factors such as market volatility, interest rate volatility and foreign exchange volatility.…”

“…In response to the developments in the industry and academia,, Grünbichler and Longstaff (1996) developed the first models for the valuation of futures and European-style options written on instantaneous volatility. The authors assumed that the underlying volatility followed a mean reverting square root process, similar to that used earlier by Heston (1993).…”

“…This analysis is particularly useful in understanding the dynamics of the VIX and building an appropriate model for pricing options and futures. More specifically, we estimate the square root mean reverting process proposed by Grünbichler and Longstaff (1996) along with various jump diffusion variations. Overall, in line with previous research, we find that the addition of jumps improves fitting.…”