Beyond Carr Madan: A Projection Approach to Risk-Neutral Moment Estimation

TL;DR

This paper introduces a projection-based method for estimating risk-neutral moments from option prices, offering finite-sample optimality and improved accuracy over traditional approaches, with applications to FX options and joint tail risk analysis.

Contribution

It develops a novel projection approach for risk-neutral moment estimation that outperforms existing methods like Carr--Madan and extends to multivariate settings.

Findings

Projection estimator achieves minimal pricing error within traded payoffs.

Simulation results show significant accuracy improvements for VIX and SVIX.

Empirical analysis recovers risk-neutral correlations and predicts joint currency crashes.

Abstract

We propose a projection method to estimate risk-neutral moments from option prices. We derive a finite-sample bound implying that the projection estimator attains (up to a constant) the smallest pricing error within the span of traded option payoffs. This finite-sample optimality is not available for the widely used Carr--Madan approximation. Simulations show sizable accuracy gains for key quantities such as VIX and SVIX. We then extend the framework to multiple underlyings, deriving necessary and sufficient conditions under which simple options complete the market in higher dimensions, and providing estimators for joint moments. In our empirical application, we recover risk-neutral correlations and joint tail risk from FX options alone, addressing a longstanding measurement problem raised by Ross (1976). Our joint tail-risk measure predicts future joint currency crashes and identifies periods in which currency portfolios are particularly useful for hedging.