The Recovery Theorem

• Published date: 01 April 2015
• DOI: http://doi.org/10.1111/jofi.12092
• Date: 01 April 2015
• Author: STEVE ROSS

THE JOURNAL OF FINANCE •VOL. LXX, NO. 2 •APRIL 2015

The Recovery Theorem

STEVE ROSS∗

ABSTRACT

We 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.

FINANCIAL MARKETS PRICE SECURITIES with payoffs extending out in time, and

the hope that they can be used to forecast the future has long fascinated both

scholars and practitioners. Nowhere is this more apparent than for the fixed

income markets, with an enormous literature devoted to examining the predic-

tive content of forward rates. However, with the exception of foreign exchange

and some futures markets, a similar line of research has not developed for

other markets. This absence is most notable for the equity markets.

While there exists a rich market in equity options and a well-developed theory

of how to use their prices to extract the martingale or risk-neutral probabilities

(see Cox and Ross (1976a,1976b)), there has been a theoretical hurdle to using

these probabilities to forecast the probability distribution of future returns,

that is, real or natural probabilities. Risk-neutral returns are natural returns

that have been “risk adjusted.” In the risk-neutral measure, the expected re-

turn on all assets is the risk-free rate because the return under the risk-neutral

measure is the return under the natural measure with the risk premium sub-

tracted out. The risk premium is a function of both risk and the market’s risk

aversion, and, thus, to use risk-neutral prices to estimate natural probabilities

we have to know the risk adjustment so we can add it back in. In models with

a representative agent this is equivalent to knowing both the agent’s risk aver-

sion and the agent’s subjective probability distribution, and neither is directly

∗Ross is with the Sloan School, MIT. I want to thank the participants in the UCLA Finance

workshop for their insightful comments as well as Richard Roll, Hanno Lustig, Rick Antle, Andrew

Jeffrey, Peter Carr, Kevin Atteson, Jessica Wachter,Ian Martin, Leonid Kogan, Torben Andersen,

John Cochrane, Dimitris Papanikolaou, William Mullins, Jon Ingersoll, Jerry Hausman, Andy

Lo, Steve Leroy, George Skiadopoulos, Xavier Gabaix, Patrick Dennis, Phil Dybvig, WillMullins,

Nicolas Caramp, Rodrigo Adao, Steve Heston, Patrick Dennis, the referee, Associate Editor, and

the Editor. All errors are my own. I also wish to thank the participants in the AQR Insight Award

and AQR for its support.

DOI: 10.1111/jofi.12092

615

616 The Journal of Finance R

observable. Instead, we infer them from fitting or “calibrating” market mod-

els. Unfortunately, efforts to empirically measure the aversion to risk have led

to more controversy than consensus. For example, measures of the coefficient

of aggregate risk aversion range from two or three to 500 depending on the

model and the macro data used. Additionally, financial data are less helpful

than we would like because we have a lengthy history in which U.S. stock

returns seemed to have consistently outperformed fixed income returns—the

equity premium puzzle (Mehra and Prescott (1985))—which has even given rise

to worrisome investment advice based on the view that stocks are uniformly

superior to bonds. These conundrums have led some to propose that finance

has its equivalent to the dark matter that cosmologists posit to explain their

models’ behavior for the universe when observables seem insufficient. The dark

matter of finance is the very low probability of a catastrophic event and the

impact that changes in that perceived probability can have on asset prices (see,

for example, Barro (2006) and Weitzmann (2007)). Apparently, however, such

events are not all that remote and “five sigma events” seem to occur with a

frequency that belies their supposed low probability.

When we extract the risk-neutral probabilities of such events from the prices

of options on the S&P 500, we find the risk-neutral probability of, for example,

a 25% drop in one month to be higher than the probability calculated from

historical stock returns. But since the risk-neutral probabilities are the nat-

ural probabilities adjusted for the risk premium, either the market forecasts

a higher probability of a stock decline than has occurred historically or the

market requires a very high risk premium to insure against a decline. Without

knowing which is the case, it is impossible to separate the two and infer the

market’s forecast of the event probability.

Determining the market’s forecast for returns is important for other reasons

as well. The natural expected return of a strategy depends on the risk premium

for that strategy, and, thus, it has long been argued that any tests of efficient

market hypotheses are simultaneously tests of both a particular asset pricing

model and the efficient market hypothesis (Fama (1970)). However, if we knew

the kernel, we could estimate the variation in the risk premium (see Ross

(2005)), and a bound on the variability of the kernel would limit how predictable

a model for returns could be and still not violate efficient markets. In other

words, it would provide a model-free test of the efficient market hypothesis.

A related issue is the inability to find the current market forecast of the

expected return on equities. Unable to obtain this directly from prices as we do

with forward rates,1we are left to using historical returns and opinion polls of

economists and investors, asking them to reveal their estimated risk premiums.

It certainly does not seem that we can derive the risk premium directly from op-

tion prices because by pricing one asset (the derivative) in terms of another (the

underlying), the elusive risk premium does not appear in the resulting formula.

But all is not quite so hopeless. While quite different, the results in this paper

are in the spirit of Dybvig and Rogers (1997), who showed that if stock returns

follow a recombining tree (or diffusion), then we can reconstruct the agent’s

1Although these too require a risk adjustment.

The Recovery Theorem 617

utility function from an agent’s observed portfolio choice along a single path.

Borrowing their nomenclature, we call these results recovery theorems as

well. Section Ipresents the basic analytic framework tying the state price

density to the kernel and the natural density. Section II derives the Recovery

Theorem, which allows us to estimate the natural probability of asset returns

and the market’s risk aversion—the kernel—from the state price transition

process alone. To do so, two important nonparametric assumptions are intro-

duced in this section. Section III derives the Multinomial Recovery Theorem,

which offers an alternative route for recovering the natural distribution for

binomial and multinomial processes. Section IV examines the application of

these results to some examples and highlights important limitations of the

approach. Section Vestimates the state price densities at different horizons

from S&P 500 option prices on a randomly chosen recent date (April 27, 2011),

estimates the state price transition matrix, and applies the Recovery Theorem

to derive the kernel and the natural probability distribution. We compare the

model’s estimate of the natural probability to the histogram of historical stock

returns. In particular, we shed some light on the dark matter of finance by

highlighting the difference between the odds of a catastrophe as derived from

observed state prices and the odds obtained from historical data. The analysis

of Section Vis meant to be illustrative and is far from the much needed

empirical analysis, but it provides the first use of the Recovery Theorem to

estimate the natural density of stock returns. Section VI outlines a model-free

test of efficient market hypotheses. Section VII concludes the paper, and points

to future research directions.

I. The Basic Framework

Consider a discrete-time world with asset payoffs g(θ)at time T, contingent

on the realization of a state of nature, θࢠ. From the Fundamental Theorem of

Asset Pricing (see Dybvig and Ross (1987,2003)), no arbitrage (NA) implies the

existence of positive state space prices, that is, Arrow-Debreu (Arrow (1952),

Debreu (1952)) contingent claims prices, p(θ)(or in general spaces, a price

distribution function, P(θ)), paying $1 in state θand nothing in any other states.

If the market is complete, then these state prices are unique. The current value,

pg,ofanassetpayingg(θ)in one period is given by

pg=g(θ)dP(θ).(1)

Since the sum of the contingent claims prices is the current value of a dollar

for sure in the future, letting r(θ0)denote the riskless rate as a function of the

current state, θ0, we can rewrite this in the familiar form

pg=g(θ)dP(θ)=(∫dP (θ)) g(θ)dP (θ)

∫dP (θ)

≡e−r(θ0)Tg(θ)dπ∗(θ)≡e−r(θ0)TE∗[g(θ)]=E[g(θ)φ(θ)],(2)