Recovery Theorem Research Articles

Uniqueness Theorems for Generalized Haar Systems

A uniqueness theorem and a recovery theorem for the coefficients of series in generalized Haar systems are proved under the assumption that the series converge in measure and satisfy a certain necessary condition on the distribution function of the majorant of partial sums.

State Price Density Estimation with an Application to the Recovery Theorem

I introduce a model to estimate the risk-neutral density. Current estimation techniques use a single mathematical model to interpolate option prices on two option dimensions: strike price and time-to maturity (TTM). I propose to use B-splines with at-the-money knots for the strike price interpolation and a function that depends on the option expiration horizon for the TTM interpolation. The results of this “hybrid” methodology are significantly better than other risk-neutral density extrapolation methods applied to the Recovery Theorem.

Recovering Recovery: On the Relationship between Gauge Symmetry and Trautman Recovery

This article (i) uncovers a foundational relationship between the ‘gauge symmetry’ of a Newton-Cartan theory and the celebrated Trautman Recovery Theorem and (ii) explores its implications for recent philosophical work on Newton-Cartan gravitation.

Estimating option-implied distributions in illiquid markets and implementing the Ross recovery theorem

In this research we describe how forward-looking information on the statistical properties of an asset can be extracted directly from options market data and demonstrate how this can be practically applied to portfolio management. Although the extraction of a forward-looking risk-neutral distribution is well-established in the literature, the issue of estimating distributions in an illiquid market is not. We use the deterministic SVI volatility model to estimate weekly risk-neutral distribution surfaces. The issue of calibration with sparse and noisy data is considered at length and a simple but robust fitting algorithm is proposed. We further attempt to extract real-world implied information by implementing the recovery theorem introduced by Ross (2015). Recovery is an ill-posed problem that requires careful consideration. We describe a regularisation methodology for extracting real-world implied distributions and implement this method on a history of SVI volatility surfaces. We analyse the first four moments from the implied risk-neutral and real-world implied distributions and use them as signals within a simple tactical asset allocation framework, finding promising results.Keywords: Option-implied distributions; SVI volatility model; Ross recovery theorem; Tikhonov regularisation; illiquid derivative markets

The Ross Recovery Theorem with a Regularised Multivariate Markov Chain

Recently, Ross (2015) derived a theorem, namely the Recovery Theorem, that allows for the recovery of the pricing kernel and real-world asset distribution, under particular assumptions, from a forward-looking risk neutral distribution. However, recovering the real-world distribution involves solving two ill-posed problems. In this paper, we introduce and test the accuracy of a regularised multivariate mixture distribution to recover the real-world distribution. In addition, we show that this method improves the estimation accuracy of the real-world distribution. Furthermore, we carry out an empirical study, using weekly South African Top40 option trade data, to show that the recovered distribution is in line with economic theory.

Calibration of the Ross Recovery Theorem to Real-world Data, and Tests of its Practical Value

Calibration of the Ross Recovery Theorem to Real-world Data, and Tests of its Practical Value

Exact Simultaneous Recovery of Locations and Structure from Known Orientations and Corrupted Point Correspondences

Let \(t_1,\ldots ,t_{n_l} \in {\mathbb {R}}^d\) and \(p_1,\ldots ,p_{n_s} \in {\mathbb {R}}^d\) and consider the bipartite location recovery problem: given a subset of pairwise direction observations \(\{(t_i - p_j) / \Vert t_i - p_j\Vert _2\}_{i,j \in [n_\ell ] \times [n_\text {s}]}\), where a constant fraction of these observations are arbitrarily corrupted, find \(\{t_i\}_{i \in [n_\ell ]}\) and \(\{p_j\}_{j \in [n_\text {s}]}\) up to a global translation and scale. This task arises in the Structure from Motion problem from computer vision, which consists of recovering the three-dimensional structure of a scene from photographs at unknown vantage points. We study the recently introduced ShapeFit algorithm as a method for solving this bipartite location recovery problem. In this case, ShapeFit consists of a simple convex program over \(d(n_l + n_s)\) real variables. We prove that this program recovers a set of \(n_l+n_s\) i.i.d. Gaussian locations exactly and with high probability if the observations are given by a bipartite Erdős–Rényi graph, d is large enough, and provided that at most a constant fraction of observations involving any particular location are adversarially corrupted. This recovery theorem is based on a set of deterministic conditions that we prove are sufficient for exact recovery. Finally, we propose a modified pipeline for the Structure for Motion problem, based on this bipartite location recovery problem.

A Recovery that We Can Trust? Deducing and Testing the Restrictions of the Recovery Theorem

How reliable is the recovery theorem of Ross (2015)? We explore this question in the context of options on the 30-year Treasury bond futures, allowing us to deduce restrictions that link the physical and risk-neutral return distributions. Our empirical results undermine the implications of the recovery theorem. First, we reject an implicit assumption of the recovery theorem that the martingale component of the stochastic discount factor is identical to unity. Second, we consider the restrictions between the physical and risk-neutral return moments when the recovery theorem holds, and reject them in both forecasting regressions and generalized method of moments estimations. Received November 7, 2016; editorial decision July 24, 2017 by Editor Stijn Van Nieuwerburgh. Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web site next to the link to the final published paper online.

Functional Ross Recovery: An Operator Approach

Recently, Ross (2015) showed that the real-world probability distribution of a discrete Markovian state variable can be recovered from observed option prices. The so-called recovery theorem follows from Perron-Frobenius matrix theory when the pricing kernel is transition independent. In this paper, we generalize the recovery theorem to unbounded continuous state spaces using Perron-Frobenius operator theory. Building on our theoretical results, we devise a nonparametric estimation approach to empirically recover the pricing kernel and real-world probability density in closed form. Using S&P 500 index options, we analyze recovered pricing kernels empirically and find evidence that Ross recovery is misspecified.

Does the Ross Recovery Theorem Work Empirically?

Starting with the fundamental relationship that state prices are the product of physical probabilities and the stochastic discount factor, Ross (2015) shows that, given strong assumptions, knowing state prices suffices for backing out physical probabilities and the stochastic discount factor at the same time. We find that such recovered physical distributions based on the S&P 500 index are incompatible with future realized returns and fail to predict realized returns and variances. These negative results remain even when we add economically reasonable constraints. Simple benchmark methods based on a power utility agent or on the historical return distribution cannot be rejected.

Recursive Utiity and Thompson Aggregators

We reconsider the theory of Thompson aggregators proposed by Marinacci and Montrucchio. First, we prove a variant of their Recovery Theorem estabilishing the existence of extremal solutions to the Koopmans equation. Our approach applies the constructive Tarski-Kantorovich Fixed Point Theorem rather than the nonconstructive Tarski Theorem employed in their paper. We verify the Koopmans operator has the order continuity property that underlies invoking Tarski-Kantorovich. Then, under more restrictive conditions, we demonstrate there is a unique solution to the Koopmans equation. Our proof is based on υ0- concave operator techniques as first developed by Kransosels'kii. This differs from Marinacci and Montrucchio's proof as well as proofs given by Martins-da-Rocha and Vailakis.

A Recovery that We Can Trust? Deducing and Testing the Restrictions of the Recovery Theorem

How reliable is the recovery theorem of Ross (2015)? We explore this question in the context of options on the 30-year Treasury bond futures, allowing us to deduce restrictions that link the risk-neutral and physical distributions. The backbone of these restrictions is that the martingale component of the stochastic discount factor is unity. Our approach and empirical results provide an agnostic view of the claims of the recovery theorem in the long-term bond futures market. Moreover, the martingale component (extracted from convex optimization problems) exhibits substantial dispersion and is positively correlated with the transitory component.

A categorical equivalence between generalized holonomy maps on a connected manifold and principal connections on bundles over that manifold

A classic result in the foundations of Yang-Mills theory, due to Barrett [Int. J. Theor. Phys. 30, 1171–1215 (1991)], establishes that given a “generalized” holonomy map from the space of piece-wise smooth, closed curves based at some point of a manifold to a Lie group, there exists a principal bundle with that group as structure group and a principal connection on that bundle such that the holonomy map corresponds to the holonomies of that connection. Barrett also provided one sense in which this “recovery theorem” yields a unique bundle, up to isomorphism. Here we show that something stronger is true: with an appropriate definition of isomorphism between generalized holonomy maps, there is an equivalence of categories between the category whose objects are generalized holonomy maps on a smooth, connected manifold and whose arrows are holonomy isomorphisms, and the category whose objects are principal connections on principal bundles over a smooth, connected manifold. This result clarifies, and somewhat improves upon, the sense of “unique recovery” in Barrett’s theorems; it also makes precise a sense in which there is no loss of structure involved in moving from a principal bundle formulation of Yang-Mills theory to a holonomy, or “loop,” formulation.

Estimating Option-Implied Distributions in Illiquid Markets and Implementing the Ross Recovery Theorem

We describe how forward-looking information on the statistical properties of an asset can be extracted directly from options market data and how this can be used practically in portfolio management. Although the extraction of a forward-looking risk-neutral distribution is well-established in the literature, the issue of estimation in an illiquid market is not. We use the deterministic SVI volatility model to estimate weekly risk-neutral distribution surfaces. The issue of calibration with sparse and noisy data is considered at length and a simple but robust fitting algorithm is proposed. Furthermore, we attempt to extract real-world implied information by implementing the recovery theorem introduced by Ross (2015). Recovery is an ill-posed problem that requires careful consideration. We describe a regularization methodology for extracting real-world implied distributions and implement this method on a history of SVI volatility surfaces. We analyse the first four moments from the implied risk-neutral and real-world implied distributions and use them as signals within a simple tactical asset allocation framework, finding promising results.

Empirical Recovery: Hansen-Scheinkman Factorization and Ross Recovery from High Frequency Option Prices

Recent research has shown that the Perron-Frobenius eigenfunction of a Markov risk neutral state price transition matrix has an interesting economic interpretation. Yet, the application to actual market prices presents significant challenge. For instance, even at the intraday frequency market data, has lots of gaps and can contain unpredictable levels of noise. As a consequence, the identification of the risk neutral state transition matrix often results in a matrix that violates the basic properties of the Markov chain presumed to be driving the evolution of asset prices. We provide a fast non-linear programming approach to the Recovery Theorem such that the attained minimum formally satisfies the desired mathematical and economical constraints (e.g. the de facto discount factor being smaller than unity and unimodality of the transition matrix). We demonstrate the empirical effectiveness of the methodology on S&P 500 index options and appeal to recent theoretical results to extend this approach to individual stocks.

Positive Eigenfunctions of Markovian Pricing Operators: Hansen-Scheinkman Factorization, Ross Recovery, and Long-Term Pricing

This paper develops a spectral theory of Markovian asset pricing models where the underlying economic uncertainty follows a continuous-time Markov process X with a general state space (Borel right process, or BRP) and the stochastic discount factor (SDF) is a positive semimartingale multiplicative functional of X. A key result is the uniqueness theorem for a positive eigenfunction of the pricing operator such that X is recurrent under a new probability measure associated with this eigenfunction (recurrent eigenfunction). As economic applications, we prove uniqueness of the Hansen and Scheinkman factorization of the Markovian SDF corresponding to the recurrent eigenfunction; extend the Recovery Theorem from discrete time, finite state irreducible Markov chains to recurrent BRPs; and obtain the long-maturity asymptotics of the pricing operator. When an asset pricing model is specified by given risk-neutral probabilities together with a short rate function of the Markovian state, we give sufficient conditions for the existence of a recurrent eigenfunction and provide explicit examples in a number of important financial models, including affine and quadratic diffusion models and an affine model with jumps. These examples show that the recurrence assumption, in addition to fixing uniqueness, rules out unstable economic dynamics, such as the short rate asymptotically going to infinity or to a zero lower bound trap without possibility of escaping.

Recovering Conditional Return Distributions by Regression: Estimation and Applications

I propose a regression approach to recovering the return distribution of an individual asset conditional on the return of an aggregate index based on their marginal distributions. This approach relies on the identifying assumption that the conditional return distribution of the asset given the index return does not vary over time. I show how to empirically implement this approach using option price data. I then apply this approach to examine the cross-sectional equity risk premium associated with systematic disaster risk, to estimate the exposure of banks to systemic shocks, and to extend the Ross (Journal of Finance, 2014) recovery theorem to individual assets.

The generation of band information law and the dynamic hiding of information law

Function packet sets (P-sets) is a novel model with law and dynamic characteristics, which is composed of function internal P-set $$S^{{\bar{F}}}$$ and function outer P-set $$S^{F}$$ . Using the structure, law and dynamic characteristic of function P-sets, we present the concepts of P-information law and band information law and their generation. Subsequently, the attribute theorem of P-information, the hiding and hiding theorem of P-information in band information law and the recovery theorem of information law are studied. Based on the above theory results, an application example of information law hiding is given and validated.

The Recovery Theorem

ABSTRACTWe can only estimate the distribution of stock returns, but from option prices we observe the distribution of state prices. State prices are the product of risk aversion—the pricing kernel—and the natural probability distribution. The Recovery Theorem enables us to separate these to determine the market's forecast of returns and risk aversion from state prices alone. Among other things, this allows us to recover the pricing kernel, market risk premium, and probability of a catastrophe and to construct model‐free tests of the efficient market hypothesis.

State Prices and Implementation of the Recovery Theorem

It is generally held that derivative prices do not contain useful predictive information, that is, information relating to the distribution of future financial variables under the real-world measure. This is because the market’s implicit forecast of the future becomes entangled with market risk preferences during derivative price formation. A result derived by Ross [1], however, recovers the real-world distribution of an equity index, requiring only current prices and mild restrictions on risk preferences. In addition to being of great interest to the theorist, the potential practical value of the result is considerable. This paper addresses implementation of the Ross Recovery Theorem. The theorem is formalised, extended, proved and discussed. Obstacles to application are identified and a workable implementation methodology is developed.