Pricing Bounds for VIX Derivatives via Least Squares Monte Carlo

TL;DR

This paper introduces model-independent upper and lower bounds for pricing VIX derivatives using a stochastic duality approach and least squares Monte Carlo, providing reliable bounds without direct variance calculation.

Contribution

It develops new model-independent bounds for VIX derivatives leveraging stochastic duality and least squares Monte Carlo, addressing challenges with the square root of expected variance.

Findings

Bounds are tight for VIX futures, calls, and puts.

Method performs well across various parameters.

Single regression estimates both bounds efficiently.

Abstract

Derivatives on the Chicago Board Options Exchange volatility index (VIX) have gained significant popularity over the last decade. The pricing of VIX derivatives involves evaluating the square root of the expected realised variance which cannot be computed by direct Monte Carlo methods. Least squares Monte Carlo methods can be used but the sign of the error is difficult to determine. In this paper, we propose new model independent upper and lower pricing bounds for VIX derivatives. In particular, we first present a general stochastic duality result on payoffs involving concave functions. This is then applied to VIX derivatives along with minor adjustments to handle issues caused by the square root function. The upper bound involves the evaluation of a variance swap, while the lower bound involves estimating a martingale increment corresponding to its hedging portfolio. Both can be achieved simultaneously using a single linear least square regression. Numerical results show that the method works very well for VIX futures, calls and puts under a wide range of parameter choices.