This chapter illustrates the valuation of volatility futures and options according to Grünbichler and Longstaff (1996), abbreviated in the following by GL96. They derive a semi-analytical (“closed”) pricing formula for European volatility call options which is as easy to use as the famous Black-Scholes-Merton formula for equity options pricing. They model volatility directly and make the assumption that volatility follows a square-root diffusion process. The model is quite simple and parsimonious, such that it lends itself pretty well to serve as a starting point.
This chapter introduces the financial model, the futures and option pricing formulas of Grünbichler and Longstaff as well as a discretization of the model for Monte Carlo simulation purposes. For both the formulas and the Monte Carlo simulation approach, Python implementations are presented. Finally, the chapter shows in detail how to calibrate the GL96 model to market quotes for European call options on the VSTOXX volatility index.
6.2 The Valuation Framework
Grünbichler and Longstaff (1996) model the volatility process (e.g. the process of a volatility index) in direct fashion by a square-root diffusion or CIR process – named after the authors John C. Cox, Jonathan E. Ingersoll and Stephen A. Ross, who first introduced this type of stochastic process to finance; see Cox et al. (1985). The stochastic differential equation describing the evolution of the volatility ...