Spectral methods for volatility derivatives

TL;DR

This paper introduces a spectral method-based framework for pricing various volatility derivatives, including options on realized variance and VIX, by extending the generator of a regime switching model with jumps and local volatility.

Contribution

It proposes a novel spectral approach with a semi-analytic block-diagonalization algorithm to efficiently evaluate joint distributions for complex volatility derivatives.

Findings

Successfully calibrates the model to S&P 500 options data.

Enables consistent pricing of European, forward-starting, and VIX options.

Provides a computationally feasible method for complex derivatives pricing.

Abstract

In the first quarter of 2006 Chicago Board Options Exchange (CBOE) introduced, as one of the listed products, options on its implied volatility index (VIX). This created the challenge of developing a pricing framework that can simultaneously handle European options, forward-starts, options on the realized variance and options on the VIX. In this paper we propose a new approach to this problem using spectral methods. We use a regime switching model with jumps and local volatility defined in \cite{FXrev} and calibrate it to the European options on the S&P 500 for a broad range of strikes and maturities. The main idea of this paper is to "lift" (i.e. extend) the generator of the underlying process to keep track of the relevant path information, namely the realized variance. The lifted generator is too large a matrix to be diagonalized numerically. We overcome this difficulty by applying a new semi-analytic algorithm for block-diagonalization. This method enables us to evaluate numerically the joint distribution between the underlying stock price and the realized variance, which in turn gives us a way of pricing consistently European options, general accrued variance payoffs and forward-starting and VIX options.