Quadratic Variation

Abstract

Time changes of Brownian motion impose restrictive jump structures in the motion of asset prices. Quadratic variations also depart from time changes. Joint Laplace Fourier transforms for quadratic variation and the stock are developed. They are used to study the multiple of the cap strike over the variance swap quote attaining a given percentage price reduction for the capped variance swap. Bootstrapped data and simulated paths spaces are used to study the multiple of the dynamic hedge return desired by a realized variance swap contract. The optimized hedge multiple in the bootstrapped data is consistently near unity. Simulated path spaces allow for departures matched by also similarly altering the position in the log contract. Value multiples of quadratic variation over variance swaps are consistently larger in simulations and bootstrapped data. The departures reflect negative dividend yields embedded in the skewness of risk neutral distributions.

Keywords: Bilateral Gamma, Sato Process, Volatility of Variance, Self Decomposable Laws, Sato Lévy Mixtures

JEL Classification: G10, G12, G13