Empirical Ross recovery without discretization

• Published date: 01 May 2022
• Author: Joseph Golec,Yuewu Xu,Xiangkun Yao
• Date: 01 May 2022
• DOI: http://doi.org/10.1111/fire.12295

DOI: 10.1111/fire.12295

ORIGINAL ARTICLE

Empirical Ross recovery without discretization

Joseph Golec1Yue wu X u2Xiangkun Yao3

1School of Business, University of

Connecticut, Storrs, Connecticut, USA

2Gabelli School of Business, Fordham

University, New York,New York, USA

3School of Finance, Nankai University, Tianjin,

China

Correspondence

XiangkunYao, School of Finance, Nankai Uni-

versity,Tianjin 300350, China.

Email:xiangkun.yao@nankai.edu.cn

Abstract

We propose a new continuous-state method for empirically

recoveringthe key objects in the Ross recovery theory which

avoids discretization. The new method is based on a key

bivariate orthogonal Hermite representation of the state

price transition kernel, which leads to an elegant correspon-

dence of the eigenvalue–eigenfunction system of the tran-

sition kernel and the eigenvalue–eigenvector system of the

expansion coefficient matrix. Using S&P 500 index option

prices, we demonstrate how our method can generate well-

behaved stateprice transition kernels and physical densities.

Our method can also be used to compute the key objects in

the Hansen and Scheinkman’s factorization theory.

KEYWORDS

asset pricing, continuous states, recovery theory

JEL CLASSIFICATION

G12, G13

1INTRODUCTION

The Ross (2015) recovery theory establishes asurprising asset pricing result under mild conditions: one only needs an

asset’s risk-neutral density to determine its physical density and pricing kernel. The result is important because if it

holds empirically,then one can forecast the physical density in a fully forward-looking fashion using only information

contained in option prices! The validity of Ross recovery also has important implications for other important topics in

finance, such as the study of preferences (Dybvig & Rogers, 1997), etc.

Carr and Yu (2012) extend Ross recoveryto the continuous time setup, and show that it continues to hold if the

range of the diffusion process is bounded. Dubynskiy and Goldstein (2012) study the effect of boundary conditions

on recovery.Walden (2017) extends the theory to unbounded continuous state spaces for asset prices following dif-

fusions. In a subsequent work, Sydow and Walden(2017) examine the Ross recovery problem in the diffusion process

Financial Review. 2022;57:345–367. wileyonlinelibrary.com/journal/fire ©2022 The Eastern Finance Association 345

346 GOLEC ET AL.

setupas well as its approximate version using PDE techniques. Generally speaking, these studies show that Ross recov-

ery can be extended to situations where the assumptions are rathermild.1

Empirical studies test the theory using option prices in various markets (e.g., stocks, bonds, bond futures, etc.) or

apply it in various ways. Ross (2015) himself recovers the physicaldistribution of the S&P 500 Index and its moments

at quarterly horizon using option prices. Jackwerth and Menner (2020) test the recovery theory using extensive S&P

500 options data, and forecast the physical density,but reject the theory. Audrino et al. (2021) study the effectiveness

of market timing strategiesbased on recovered physical moments of the S&P 500, and find the strategies outperform

those using only the risk-neutral distribution. This suggests that recovered moments add useful information overand

above the risk-neutraldistribution.

The relatively successful theoretical extensions of recoverytheory, but mixed empirical results, suggest that addi-

tionalattention to empirical methods could be worthwhile. Therefore, we examine the common empirical method used

in most empirical analyses: discretization. The method typically follows four steps: (1) start with a continuous risk-

neutral density,(2) discretize that density, (3) apply the theory to recover the associated discretized physical density,

and (4) translate that back into a continuous physicaldensity.

These back-and-forth translations between continuous and discrete densities can have serious drawbacks: Ross

(2015) notes that a coarse grid for the state space generates a poorly-behaved transition pricing matrix. Subsequent

researchers use relatively large discretized grids (Jackwerth & Menner (2020) use about 120 states, and Audrino et al.

(2021) use 149 states) to minimize discretization error,but they still find irregular (narrow spikes) and unstable pricing

kernels or physical probability densities. In fact, regardless of the discretization method, it is difficult to avoid spikes

as the risk-neutral density is obtained using numerical derivatives over the discretization grid. Our paper proposes a

recovery method that avoidsdiscretization, and hence, its spurious effects. This is a key contribution of our paper.

Hansen and Scheinkman (2009) propose a general factorization framework that nests Ross (2015) recovery as a

special case. The close relationship between Ross recovery and the Hansen–Scheinkman framework is illuminated in

the extensive study of Borovicka et al. (2016), where they show that recovery holds if the martingale component in

the factorization theory is degenerate (i.e., unity). In a recent extensive study,Dillschneider and Maurer (2019)also

extend the Ross recovery to the continuous state setup and establish some insightful results clarifying the relation-

ship between the Ross recovery and the Hansen–Scheinkman framework.Qin and Linetsky (2017) further extend the

Hansen–Scheinkman framework to the semi-martingale setting, in which empirical application would require com-

puting the eigenvalues and eigenfunctions of the associated operators.Our empirical method can also be used to com-

pute the largest eigenvalue–eigenfunction pair in the factorization theory,and hence answers in part the call for new

research in Hansen and Scheinkman (2009).

Our new empirical method for Ross recoverycompletely avoids the discretization process. It is a continuous formu-

lation for continuous stock prices. Its advantage is that the recovered densities are automatically smooth, with many

fewer parameters to estimate. Under the more general framework of Hansen and Scheinkman (2009), our method

estimates the key objects in their factorization theory such as the eigenvalue–eigenfunction pairs of the conditional

expectations’ operator.

The key to our new method is a bivariate orthogonal expansionof the full transition kernel (under the risk-neutral

measure).2This expansion reduces the determination of the transition kernelto the determination of the associated

(infinite) coefficient matrix. This matrix is then estimated using option prices with various strikes and maturities.

We then show that the physical density is essentially the reciprocal of the eigenfunction corresponding to the

largest eigenvalue of the full transition kernel, under our continuous formulation of the theory. To facilitate the

computation of these eigenvalue–eigenfunction pairs, we establish a one to one correspondence between the

1Forexample, Sydow and Walden (2020) point out that Ross recovery works in a complete marketsetup, which includes many work-horse models in finance,

asdiscussed in Walden (2017).

2Hermite expansion is used by Aït-Sahalia (2002) in his seminal study of transition density for stochastic differential equations. It has also been used by

Xiu (2014) to obtain expansionformulas for European option prices. The difference between his and our use of the Hermite expansion is ours is a bivariate

orthogonalexpansion. This bivariate orthogonal expansion is critical for establishing the key eigenvalue–eigenfunction correspondence.