Discretely sampled variance and volatility swaps versus their continuous approximations

TL;DR

This paper investigates the validity of using continuous approximations for discretely sampled variance and volatility swaps, providing conditions for finiteness and convergence, with applications to the 3/2 stochastic volatility model.

Contribution

It offers new theorems characterizing when discretely sampled swap values are finite and when they converge to continuous approximations, highlighting limitations of the approximation method.

Findings

Discretely sampled swap prices may not exist for some models.

Finiteness of swap values depends on specific process conditions.

Convergence to continuous values is only guaranteed under certain parameters.

Abstract

Discretely sampled variance and volatility swaps trade actively in OTC markets. To price these swaps, the continuously sampled approximation is often used to simplify the computations. The purpose of this paper is to study the conditions under which this approximation is valid. Our first set of theorems characterize the conditions under which the discretely sampled swap values are finite, given the values of the continuous approximations exist. Surprisingly, for some otherwise reasonable price processes, the discretely sampled swap prices do not exist, thereby invalidating the approximation. Examples are provided. Assuming further that both swap values exist, we study sufficient conditions under which the discretely sampled values converge to their continuous counterparts. Because of its popularity in the literature, we apply our theorems to the 3/2 stochastic volatility model. Although we can show finiteness of all swap values, we can prove convergence of the approximation only for some parameter values.