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

Tjeerd De Vries

TL;DR

The paper develops a projection-based method to recover risk-neutral moments from option prices, offering finite-sample optimality within the span of traded payoffs and direct replicability of observed prices. It contrasts this with the Carr–Madan approach, showing projection yields superior tail performance, avoids unstable numerical differentiation, and extends naturally to multivariate settings through ridge-function approximation and market-completeness theory. The authors derive convergence rates, distribution-estimation properties, and conditions for market completeness, and they demonstrate substantial finite-sample gains in simulations for SVIX/VIX and joint FX dependence. The empirical applications recover risk-neutral correlations and joint tail risk using FX and SPX option data, revealing time-varying dependence, crisis-period tail risk premia, and hedging implications for currency portfolios. Overall, projection provides a robust, model-free framework for estimating a wide range of risk-neutral quantities, including joint distributions, covariances, and tail events, with clear practical relevance for hedging and risk management.

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.

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

TL;DR

The paper develops a projection-based method to recover risk-neutral moments from option prices, offering finite-sample optimality within the span of traded payoffs and direct replicability of observed prices. It contrasts this with the Carr–Madan approach, showing projection yields superior tail performance, avoids unstable numerical differentiation, and extends naturally to multivariate settings through ridge-function approximation and market-completeness theory. The authors derive convergence rates, distribution-estimation properties, and conditions for market completeness, and they demonstrate substantial finite-sample gains in simulations for SVIX/VIX and joint FX dependence. The empirical applications recover risk-neutral correlations and joint tail risk using FX and SPX option data, revealing time-varying dependence, crisis-period tail risk premia, and hedging implications for currency portfolios. Overall, projection provides a robust, model-free framework for estimating a wide range of risk-neutral quantities, including joint distributions, covariances, and tail events, with clear practical relevance for hedging and risk management.

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.
Paper Structure (49 sections, 16 theorems, 167 equations, 8 figures, 6 tables)

This paper contains 49 sections, 16 theorems, 167 equations, 8 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Estimating nonlinear payoffs using projection
  3. Carr-Madan approach
  4. A simple illustration of the projection method
  5. General projection approach
  6. Continuous-state limit
  7. Completeness, convergence, and distribution estimation
  8. Market completeness
  9. Convergence rate
  10. Estimation of the risk-neutral CDF and PDF
  11. Completeness in multiple asset markets and joint dependence
  12. Identifying joint dependence from options on multiple portfolios
  13. Identification of risk-neutral covariances and correlations
  14. Projection and equicorrelation
  15. Completeness in FX markets
  16. ...and 34 more sections

Key Result

Proposition 1

Let Assumption asmp:compact hold and define an inner product on $L^2(A)$ by If $\max_i |s_{i+1}-s_i|\to 0$ as $n_s\to\infty$, then $\hat{\beta}_{n_s}\to\hat{\beta}$, where Moreover, $\hat{\beta}$ solves the minimization problem

Figures (8)

  • Figure 1: Replication of cubic payoff. The figure shows the function $g(R_{t \to T}) = (2/3)R_{t \to T}^3 - (37/40)R_{t \to T}^2 + (21/25)R_{t \to T}$ (black), together with the projection-based portfolio (blue) and CM portfolio (red). The approximations are based on 15 strike prices drawn from a uniform distribution. Dashed vertical lines indicate the minimum and maximum strike values used.
  • Figure 2: MSE of approximation. The figure shows the convergence rate as a function of the number of strikes (upper and middle panels) and as a function of the strike range (bottom panels). In the top panels, the strike grid is equally spaced, while in the middle panels the strikes are uniformly distributed.
  • Figure 3: Estimated correlation and joint tail risk in exchange-rate markets. Each point is one of 1,000 Monte Carlo simulations. Top: true correlation versus its projection-based estimate. Bottom: true joint left-tail probability $\mathbf{P}(S_{1,T} \le 0.95, S_{2,T} \le 0.95)$ versus its projection-based estimate.
  • Figure 4: VIX estimate. This figure shows the projection VIX estimate minus the VIX estimate obtained by CM. The solid orange line denotes the 60-day moving average of this difference. The blue dots indicate the 20 largest observed differences.
  • Figure 5: Daily risk-neutral correlation, volatility, and crash risk (30-day horizon). Panels (a)–(d) plot: (a) the 30-day risk-neutral correlation between EUR/USD and GBP/USD; (b) the corresponding annualized 30-day risk-neutral standard deviations for each exchange rate; (c) the 30-day joint crash probability under independence ($p_\text{EUR}\,p_\text{GBP}$) and under the option-implied dependence structure; (d) the option-implied (dependent) crash probability alongside the physical crash probability estimated from OLS. The estimates in this last panel are smoothed using a 30-day moving average.
  • ...and 3 more figures

Theorems & Definitions (41)

  • Example 1: Risk-neutral variance (SVIX)
  • Example 2: Risk-neutral entropy (VIX)
  • Example 3: Risk-neutral distribution
  • Example 4: Risk-neutral covariance and correlation
  • Example 5: Projection approach
  • Definition 1: Projection estimator
  • Remark 1: Constrained least squares
  • Remark 2: Weighted least squares
  • Remark 3: Redundancy of option-implied regressors
  • Proposition 1
  • ...and 31 more