Working Paper Series

_______________________________________________________________________________________________________________________

National Centre of Competence in Research Financial Valuation and Risk Management

Working Paper No. 487

The Joint Behavior of Credit Spreads, Stock Options and Equity Returns when Investors Disagree

Andrea Buraschi Fabio Trojani

Andrea Vedolin

First version: March 2007 Current version: April 2008

This research has been carried out within the NCCR FINRISK project on

“New Methods in Theoretical and Empirical Asset Pricing”

___________________________________________________________________________________________________________

The Joint Behavior of Credit Spreads, Stock

Options and Equity Returns when Investors

Disagree

Andrea Buraschi, Fabio Trojani and Andrea Vedolin∗

Abstract

We study how the interaction of agents with different beliefs about a firm’s future cash flows determines the joint

behavior of credit spreads, option implied volatilities, and stock returns. Beliefs heterogeneity influences the pricing kernel

in a way that supports more realistic credit spreads and a co–movement with stock return volatility and option implied

volatility in line with the empirical evidence. These features also help to rationalize the negative (positive) delta of some

call (put) options. We test the model predictions using panel and Logit regressions and find strong empirical support for

our theory.

JEL Classification Codes: D80, G12, G13.

Keywords: Credit Risk, Credit Spreads, Heterogeneous Beliefs, Implied Volatility.

First Version: March 2007

This Version: April 2008

∗Andrea Buraschi is at Imperial College London, Tanaka Business School. Fabio Trojani and Andrea Vedolin are at the University

of St. Gallen. We would like to thank Turan Bali, Francis Breedon, Anna Cieslak, Martin Cremers, Alexander David, Joost Driessen,

Jerome Detemple, Lina El-Jahel, Axel Kind, Ilaria Piatti, Paolo Porchia, Suresh Sundaresan, Liuren Wu, and participants of the Arne

Ryde Workshop in Financial Economics at Lund University, the Annual Meeting of the Swiss Society of Economics and Statistics 2007,

SFI Conference on Portfolio Management and Derivatives, Lugano, 2007, the American Finance Association Meeting, New Orleans,

2008, the European Winter Summit in Hemsedal 2008, the Finance seminars of the University of Amsterdam, Baruch College, Bocconi

University, London School of Economics, the University of St. Gallen, and the University of Zurich for many helpful comments. Fabio

Trojani gratefully acknowledges the financial support of the Swiss National Science Foundation (NCCR FINRISK and grants 101312–

103781/1 and 100012–105745/1). Andrea Vedolin acknowledges the financial support of the Swiss National Science Foundation (grant

PBSG1–119230).

1

This paper extends the structural Merton’s (1974) model to an incomplete information economy, in

which investors have heterogeneous beliefs about a firm’s future cash flows. In equilibrium, changes in beliefs force

agents to change their investment in credit sensitive assets in order to equate their (individual specific) expected

marginal utilities. Therefore, belief heterogeneity is a risk factor that affects asset prices and credit spreads. The

interplay of disagreement and credit risk generates pricing restrictions that are very different from those of the

standard Merton (1974) model. In this paper, we study these restrictions, both theoretically and empirically.

To motivate the potential role of heterogeneous beliefs, it is convenient to illustrate a recent notorious event

that affected credit markets. The standard structural Merton model implies that credit spreads are monotonically

increasing in option-implied stock volatility and decreasing in the stock price. This feature suggests that one could

buy/sell credit default swaps (CDS) and use equities (or a derivative on the equity) to dynamically delta hedge

the position, a strategy that is being implemented by several hedge-funds. Unfortunately, this relationship is often

empirically violated. The most prominent example of this failure occurred in May 2005, when General Motors (GM)

and Ford got downgraded to junk status. Before May 2005, many hedge funds sold CDS on GM and hedged their

exposure by shorting the equity (or by creating a long volatility position in GM options). The rationale for this

strategy was that, consistent with the structural Merton model, wider credit spreads would be accompanied by a

drop in the share price (or an increase in the option implied volatility). After the downgrade of GM to junk status

by Standard & Poors, credit spreads on a 10 year CDS increased by almost 200 basis points in one month. The

share price, however, rose almost 25% to 32.75 USD, and the implied volatility of short-term at-the-money options

on GM increased by 50% to reach 62.73%. Many widely known hedge funds engaging in capital structure arbitrage

posted large losses and the state of the hedge fund industry obtained center stage in the financial press. Part of the

explanation for these features might include the role of margin calls in intrinsically illiquid markets. In this paper, we

focus on an explanation based on the role of belief heterogeneity in a frictionless market and argue that the classical

complete market argument implicitly neglects this key priced risk factor. For instance, during this major credit event

belief disagreement on GM’s future cash flows (see Section IV. for details) more than doubled from 0.21 to 0.49. If

disagreement is priced, Merton’s positive relationship between credit spreads and stock prices can break down. This

feature suggests a far less trivial link between credit and option markets, which might explain why early empirical

studies of the Merton model obtained mixed results. This paper analyzes this link in a model that yields realistic

patterns for the joint behavior of credit spreads, option-implied and realized volatilities, and stock returns.

We study a simple extension of the structural–form Merton (1974) model, in which a firm has a capital structure

consisting of equity and corporate debt. We depart from the standard model in a single dimension, by assuming

two groups of investors, with identical preferences and endowments, who form heterogeneous expectations about the

growth rate of a firm’s cash flows. This simple generalization implies a pricing kernel that is directly affected by the

degree of difference in beliefs. This feature has several important asset pricing implications. First, an increase in

the difference in beliefs induces wider credit spreads. The main economic intuition for this effect is that when belief

disagreement increases it is optimal for the more pessimistic agent to reduce the overall exposure to credit risk in her

portfolio, by demanding default protection. This protection is supplied by the optimist at a cost which reduces, in

1

equilibrium, the market value of the firm. At the same time, it is possible to show that belief disagreement increases

the volatility of the firm value. Together, these effects decrease corporate bond prices and raise credit spreads because

defaultable bonds are proportional to a short put option on firm value. Since disagreement increases the market price

of the default event, we find that at the calibrated parameters more realistic credit spreads arise without implying

an unrealistic physical probability of default. Second, the model suggests that the higher volatility of the firm value

implies a larger volatility of stock returns and a higher implied volatility of individual stock options. This finding is

interesting since it offers a simple structural explanation for the positive co–movement of option-implied volatilities

and credit spreads, without introducing other exogenous assumptions such as jump risk.1 Third, belief disagreement

also impacts the shape of the option–implied volatility smile. The model generates, depending on the level of firm

leverage, both a negative and a positive slope for the option-implied volatility smile. This is a noteworthy result

because the evidence of an inverted volatility skew for some individual stock options is often considered an empirical

puzzle in option pricing literature.2 In our model, this feature may be caused by the endogenous co-dependence

of stock returns and stock volatility (i.e. skewness) and can be either positive or negative: For standard levels

of leverage, skewness is negative because disagreement reduces the stock price and increases, at the same time,

stock volatility. However, for some other leverage levels the model generates a positive risk-neutral skewness, which

implies the inverted option-implied volatility skew. Fourth, as disagreement increases, the price of the corporate

bond decreases, but the price of equity can increase if the firm is moderately levered. It follows that in our model

higher credit spreads do not have to be associated with a lesser distance to default or a high default probability, a

feature which plays a key role in the potential explanation of the credit risk puzzle.3

To empirically study the joint link between belief heterogeneity and the price of contingent claims issued by the

firm, we build a firm specific index of belief disagreement based on earning forecasts from the Institutional Brokers

Estimate System (I/B/E/S). We also use a panel of monthly data on US corporate bond prices, individual stock

prices, and option implied volatility smiles of 337 firms, for the period 1996 – 2005. To the best of our knowledge,

we are the first to match these four databases and to explore the joint behavior of corporate credit spreads, stock

returns, option implied volatility smiles, and investors’ divergence of opinions. The main contributions of our analysis

are the following.

First, when we confront the model implications with the data, we find that differences in beliefs can support

more realistic levels of credit spreads. Moreover, when we regress credit spreads on the difference in beliefs index,

the estimated coefficients are positive as well as statistically and economically significant. This is, in a simple

regression of credit spreads on our disagreement proxy, a one standard deviation change in disagreement implies a

change of approximately 12 basis points in senior credit spreads, which is about one third of the sample standard

deviation of senior credit spreads in our data set. Disagreement also consistently improves the explanatory power of

1See, among others, Cremers, Driessen, Maenhout, and Weinbaum (2006), Zhang, Zhou, and Zhu (2006), and Carr and Wu (2006),for credit risk models based on different assumptions on the underlying jump process.

2See, e.g., Bakshi, Kapadia, and Madan (2003).3Structural credit risk models, such as the Merton (1974) and Black and Cox (1976) models, are known to predict unrealistically low

credit spreads relative to measures of distance to default or the physical default probability, especially for firms of high credit quality;see, e.g., Eom, Helwege, and Huang (2004), Huang and Huang (2004). See David (2007) for a discussion on the issue and a proposal foran alternative model that relaxes this link.

2

credit spreads regressions that include additional explanatory risk factors suggested in the literature. In a regression

including option–related variables the R2 is about 10% higher when disagreement is accounted for. Similarly, in

a regression with credit ratings the R2 is about 16% higher when we include a disagreement proxy. These results

are robust even after controlling for several other explanatory variables, such as proxies for liquidity risk (Longstaff,

2004), Fama-French factors (Schaefer and Strebulaev, 2005) and macro– or business–cycle variables (Collin-Dufresne,

Goldstein, and Martin, 2001). At the same time, the robustness of the coefficients of other explanatory variables in

our regressions is rather weak, after we control for differences in beliefs. These results extend the findings of the most

recent empirical literature on the credit spreads puzzle. Longstaff and Schwartz (1995), Leland and Toft (1996),

and Collin-Dufresne and Goldstein (2001), among many others, extend standard structural models by introducing

additional exogenous risk factors that should help explain credit spreads. However, even these models find it difficult

to explain several features of the data and Collin-Dufresne, Goldstein, and Martin (2001) argue in favor of a missing

risk factor driving credit spreads.4 Our findings suggest that the missing factor might be related to belief disagreement

among investors.

Second, our model offers a simple structural explanation for the positive empirical link between the volatility of

stock returns, the implied volatility of individual stock options, and corporate credit spreads. Campbell and Taksler

(2003) show that firm-level equity volatility explains as much of the variation in corporate credit spreads as credit

ratings do. Cremers, Driessen, and Maenhout (2007) use the at-the-money implied volatility of individual stock

options as a proxy for volatility risk, the implied volatility of out-of-the-money puts as a proxy for jump risk, and

link them to the dynamics of credit spreads.5 In our structural economy, credit spreads and stock (option-implied)

volatilities are endogenously driven by belief disagreement. Consistent with the model, in our panel regressions we

find that option-related variables lose their significance for credit spreads, after controlling for belief disagreement.

This result is robust to the inclusion of other control variables.

Third, we provide a simple economic explanation for why the slope of individual stock-option smiles sometimes

reverses sign. Bakshi, Kapadia, and Madan (2003) document in detail these differences and find on average a far

less negatively skewed risk-neutral distribution of individual OEX stock options, relative to index options. More

surprisingly, the slopes of individual smiles are not always negative; 20% of the thirty largest stocks in the S&P100

have a positively sloped smile in approximately 30% (40%) of the cases for short term options (long term options).

They emphasize that this feature is difficult to explain with a traditional structural option pricing model and conclude

saying: “What causes the slope of individual smiles to reverse sign? The differential pricing in the cross-section of

strikes, and in the cross-section of stocks is puzzling”;6 In our data there is a significant fraction of observations

4Eom, Helwege, and Huang (2004), for example, find that the Merton (1974) model predicts spreads that are on average too low,but the Longstaff and Schwartz (1995) and the Collin-Dufresne and Goldstein (2001) models predict credit spreads that are on averagetoo high. Ericsson, Jacobs, and Oviedo (2005) document that a significant fraction of swap spread variation cannot be explained bystructural models.

5Additional studies emphasizing the link between credit spreads and single-stock options implied volatilities and skewness are Hull,Nelken, and White (2004), Carr and Wu (2006), and Tauchen and Zhou (2006). Hull, Nelken, and White (2004) link stock optionsand credit default swaps in the standard Merton (1974) model and propose a new estimation technique. Carr and Wu (2006) allow forstochastic default in a joint framework including stock options and credit risk. Tauchen and Zhou (2006) extract jump risk premia fromrealized jumps.

6Related results are discussed in Toft and Prucyk (1997) who produce empirical evidence supporting a positive relationship betweenthe implied volatility skew and leverage.

3

for which the smile of individual stock options is positive. This can be explained by the endogenous co-movement

between stock price and volatility (skewness) in our model, which has a sign that depends on the firm’s leverage:

For some leverage levels belief disagreement generates a positive risk-neutral skewness that reverses the slope of

the option–implied volatility smile in our model. Our panel regressions confirm that disagreement has significant

explanatory power for explaining both the positive and the negative slope of the smile of individual stock options.

Fourth, we find that disagreement can explain the no-arbitrage violations implied by single-factor models for

credit spreads and single-stock options. Single-factor models imply a negative relationship between credit spreads

and stock prices, and a positive (negative) link between the prices of a call (put) option and the price of the underlying

stock. These relations are often violated in our data. One example discussed above is the downgrade of General

Motors in May 2005. Bakshi, Cao, and Chen (2000) and Perignon (2006) find that prices of index option calls (puts)

do not always move in the same (opposite) direction as their underlying: The delta of a call (put) option in the

data is sometimes negative (positive). In our model, these monotonic relations between credit spreads, option prices,

and stock prices can be violated due to the time-varying endogenous skewness. In our sample, the percentage of

these no–arbitrage violations for individual call (put) stock options ranges between 3 and 10 percent (between 2 and

10 percent), depending on the option’s maturity and moneyness. For credit spreads, the percentage of arbitrage

violations varies between 14% and 19%, depending on leverage. Moreover, we observe a very clear pattern in the

structure of these violations, both in credit and option markets, as a function of leverage. Consistent with the data,

our model yields credit market violations that are more likely for low-leverage firms, and option market violations

that are more likely for out-of-the-money calls (puts) of firms with high (low) leverage. Moreover, we find that belief

disagreement has a statistically significant impact on the likelihood of these violations when they are regressed on

our disagreement proxy in a Logit panel regression.

Finally, our model implies a simple structural explanation also for the relationship between stock returns and

belief disagreement. Recent empirical asset pricing literature has deliberated over the sign of the relation between

stock returns and the divergence of opinions. Diether, Malloy, and Scherbina (2002) find a negative relation between

dispersion in analysts’ earnings forecasts and stock returns. They interpret this finding as evidence in support of

Miller’s (1977) conjecture that binding short-selling constraints and disagreement generate stock prices reflecting the

view of the more optimistic investor. Using a different data set and empirical methodology, Anderson, Ghysels, and

Juergens (2005) get the opposite result. Our model offers a structural explanation for these mixed results, which

is based on the interaction between leverage and belief disagreement, and does not require further assumptions in

terms of market frictions: For normal levels of leverage, disagreement increases expected stock returns, but for low

leverage regions it can decrease them. The economic intuition is related to the different sensitivities of the value of

the default option which is embedded in stock returns. For low leverage companies, this option is far out-of-the-

money and its value is more sensitive to changes in skewness. For high leverage companies, this option is closer to be

in-the-money and the opposite holds. Consistent with this explanation, in the empirical analysis we get a significant

positive (negative) relation between disagreement and stock returns for low (high) leverage firms. The statistical and

economic significance of these findings is robust with respect to different control variables.

4

Our model borrows from the theoretical literature that studies the equilibrium impact of heterogeneity in beliefs.

Detemple and Murthy (1994, 1997) and Zapatero (1998) analyze the implications of divergence in opinions on financial

innovation and interest rate volatility. Kurz (1994) emphasizes that belief disagreement can arise from the difficulty to

distinguish one model from another using existing data. Basak (2000) shows that in an economy with heterogeneous

beliefs the risk of an extraneous process uncorrelated with fundamentals can be priced in equilibrium. Scheinkman

and Xiong (2003) study a two-period partial equilibrium economy, in which disagreement is caused by some agents

that are more confident than others about the accuracy of a signal. This feature generates different agents’ reactions

to the same signal in the economy. Dumas, Kurshev, and Uppal (2007) study the relation between disagreement in

beliefs, driven by overconfidence of some agents in the economy, and the excess volatility of stock returns. Xiong and

Yan (2006) model belief heterogeneity by assuming that the parameters of the learning models of different agents

are different. Buraschi and Jiltsov (2006) assume that disagreement arises due to different initial priors about the

parameters of a fundamental process and a signal. Berrada (2006) proposes a number of specifications of risk aversion

and beliefs that provide potential answers to the equity premium and risk-free rate puzzle. David (2007) develops

a setting that generates counter-cyclical consumption volatility, earnings forecast, and cross-sectional consumption

dispersion. At the calibrated parameters and for a risk aversion coefficient less than one, his model produces a more

realistic equity premium and interest rate. The common feature of all these models is that agents with different beliefs

react differently to the same fundamental shock or the same piece of information. Therefore, belief disagreement

influences individual consumption plans, optimal portfolios, the equilibrium pricing kernel, and asset prices. We

contribute to this literature by highlighting the equilibrium impact of belief disagreement in a structural economy

with credit risk.

The paper is organized as follows. Section I. introduces first our structural equilibrium model of credit risk with

belief disagreement. It then presents and discusses the equilibrium solutions for asset prices. Section II. investigates

in detail the empirical predictions of our model. Section III. describes the panel data set and Section IV. presents

the results of the empirical study. Section VI. concludes. All proofs can be found in the Appendix.

I. The Economy with Heterogenous Beliefs

A. The Model

We start from the simple structural model of Merton (1974) and take the dynamics of the asset cash-flows of the firm

as a primitive.7 The main point of departure from the Merton (1974) model is the introduction of heterogeneous

investors who disagree about the future cash-flows. Investors are identical in all other aspects, such as preferences,

endowments, and risk aversion. As in Merton’s (1974) model, the firm has a very simple capital structure, consisting

of debt and equity, where debt is the sum of two defaultable zero bonds with different seniority but identical maturity.

Since the cash-flows are stochastic, the firm may default on its obligations, so that the equilibrium value of the debt

7Merton’s (1974) model of credit risk assumes an exogenous firm value process with constant volatility. Even if we treat the firm valueas endogenous, the predictions of our model for the case, in which there is no disagreement are identical to those of Merton (1974).

5

depends on the probability that the firm will be able to generate enough cash-flows to cover its liabilities. Let the

assets’ cash-flows follow the process:

d log A(t) = µA(t)dt + σAdWA(t),

dµA(t) = (a0A + a1AµA(t))dt + σµAdWµA

(t),

where µA(t) is the cash flow’s expected growth rate, σA > 0 its volatility, a0A ∈ R the growth rate of expected cash

flow growth, a1A < 0 its mean-reversion parameter and σµA> 0 the volatility. The vector composed of WA(t) and

WµA(t) is assumed to be a standard two–dimensional Brownian motion.

The process A(t) is observable by all investors in the economy, but in contrast to Merton (1974), we assume that

µA(t) is unknown and it must be estimated. Let us specify with mA(t) the estimated value of the true growth rate

µA(t) of firm cash-flows. In practice, this is the goal of financial analysts and the data presents great dispersions

in beliefs about firms’ future earnings. This is not surprising since, as analysts’ reports show, expected future cash-

flows depend on analysts’ expectations about future sales, costs, the regulatory environment, and general business

conditions. Thus, agents may eventually disagree and have different values for mA(t). We consider an economy, in

which analysts have different miA(t), for i = 1, 2.

This apparently innocuous departure from the basic Merton’s model has some important implications. First,

while in Merton’s model the value of assets can be taken as exogenous, when agents disagree on future cash-flows

the asset value is endogenous, and depends on agents’ relative demands, which are a function of their subjective

beliefs. This simple feature makes markets incomplete and the equilibrium value of the firm becomes a function of

the degree of belief disagreement. Second, due to market incompleteness, contingent claims on firm value cannot

be priced by standard replication arguments: A dynamic portfolio investing in firm assets and the risk-free bond

does not replicate the payoffs of equity and corporate bonds. Additional financial assets are needed to complete the

market and their prices depend on equilibrium portfolio demands. It follows that belief disagreement is a priced risk

factor with real effects on the price of all contingent claims.

Because future cash-flows depend on a variety of factors that are related to the business cycle and the competitive

landscape, analysts use information that goes beyond simple evidence on historical cash-flows. Moreover, firm-specific

accounting information is typically observed infrequently and with noise because of possible biases due to earning

management, accruals and carry forwards, and accounting frauds.8 To reproduce this aspect, we consider a variable

8Malmendier and Tate (2005) find that especially during the new economy boom in 2000 the accounting values of some companieswere not very reliable. An imprecisely observed firm value is treated in Duffie and Lando (2001), who show that the quality of thefirm’s information disclosure can affect the term structure of corporate bond yields. Yu (2005) finds empirically that firms with higherdisclosure rankings tend to have lower credit spreads. An imperfectly observed firm value is also modeled by Cetin, Jarrow, Protter, andYildirim (2004), who assume that investors can access only a coarsened subset of the manager’s information set. Giesecke (2004) developsa model with an imperfectly observed default boundary. Collin-Dufresne, Goldstein, and Helwege (2003) assume that firm values areobserved with one time-lag. An industry implementation of these academic concepts is presented in CreditGradesTM, as described in theRiskMetrics (2002) technical document, which models the unobservable distance to default by a latent process explicitly.

6

z(t), which we call a signal or a leading indicator, that analysts can use to improve the estimation of the cash-flow

growth rate. The dynamics of z(t) follows a Gaussian process whose drift is related to the cash-flow drift:

dz(t) = (αµA(t) + βµz(t))dt + σzdWz(t),

dµz(t) = (a0z + a1zµz(t))dt + σµzdWµz

(t),

where σz > 0 is the volatility of the signal, a0z ∈ R the long-term growth rate of expected signal growth, a1z < 0

the mean-reversion parameter and σµz> 0 the volatility. (Wz(t), Wµz

(t))′ is a standard two-dimensional Brownian

motion independent of (WA(t), WµA(t))′.

Investors use observations of both A(t) and z(t) to make inferences about µA(t). The parameters α and β measure

the strength of the relation between µA(t) and the expected change in z(t). When β = 0, the expected change in

z(t) is perfectly correlated with µA(t), and z(t) contains information exclusively about the expected growth rate of

firm’s cash flows. When β 6= 0, the expected change in z(t) is a mixture of µA(t) and the growth rate µz(t) of an

additional risk factor, which could include, among other things market–wide information about the aggregate state

of the economy.

B. Disagreement

We consider a simple specification for the uncertainty and disagreement in the economy. We assume that agents

update their beliefs using all available information according to Bayes’ rule. The difference in the posterior can be

generated either by a difference in the distribution of agents’ priors or in the subjective value of some parameter in

the dynamics for cash flows and signals. In the latter case, a parsimonious model can be based on the assumption

that the volatility parameter σµAin the dynamics of cash flows’ expected growth rate is agent dependent. In both

cases, given that the stochastic process of the drift in these dynamics is conditionally Gaussian, the heterogeneity

can be summarized by the difference in their means mi(0) and covariance matrices γi(0). This assumption allows

us to characterize the dynamics of the individual posterior distribution and to obtain predictions about optimal

portfolio choices of different agents and their effect on equilibrium asset prices. Let mi(t) := (miA(t), mi

z(t))′ :=

Ei((µA(t), µz(t))

′|FYt

), where FY

t := FA,zt is the information generated by A(t) and z(t) up to time t, and Ei(·)

denotes expectation relative to the subjective probability of investor i = 1, 2. To specify the disagreement process in

our model, let Y (t) = (log A(t), z(t)), b1 = diag(σ1µA

, σ1µz

), a0 = (a0A, a0z)′, a1 = diag(a1A, a1z), B = diag(σA, σZ)

and A =

(

1 0

α β

)

.

The (posterior) belief dynamics of agent one can be obtained as an application of the Kalman-Bucy filter and are

given by:

dm1(t) = (a0 + a1m1(t))dt + γ1(t)A′B−1dW 1

Y (t), (1)

dγ1(t)/dt = a1γ1(t) + γ1(t)a′

1 + b1b1′ − γ1(t)A′(BB′)−1Aγ1(t), (2)

7

with initial conditions m1(0) = m10 and γ1(0) = γ1

0 , where dW 1Y (t) := B−1

(dY (t) −

(Am1(t)

)dt)

is the innovation

process induced by the first investor’s belief and filtration.9 To completely characterize the dynamic disagreement

structure in our economy, we need to specify the learning dynamics of agent two. It is convenient to do this

indirectly, by specifying the disagreement process implied by this learning dynamics. This process is the key state

variable driving all equilibrium quantities in our economy. In our model, the disagreement process is defined by the

two dimensional process:

Ψ(t) :=

ΨA(t)

Ψz(t)

=

(m1

A(t) − m2A(t))/σA

(m1z(t) − m2

z(t))/σz

.

The first component of Ψ(t) measures disagreement about the expected growth rate of the future firm cash flows.

The second component measures the disagreement about the additional signal z(t). Both components are normalized

by their local standard deviation. The dynamics for Ψ(t) are given by:

dΨ(t) = B−1(a1B + γ2(t)A′B−1

)Ψ(t)dt + B−1(γ1(t) − γ2(t))A′B−1dW 1

Y (t), (3)

dγ2(t)/dt = a1γ2(t) + γ2(t)a′

1 + b2b2′ − γ2(t)A′(BB′)−1Aγ2(t), (4)

with initial conditions Ψ(0) = (m1A(0)−m2

A(0))/σA, (m1z(0)−m2

z(0))/σz and γ2(0) = γ20 , where b2 := diag(σ2

µA, σ2

µz)

emphasizes the potential dependence of this parameter on the subjective belief of agent two.

The dynamics of m1(t) and Ψ(t) completely characterize the beliefs induced by priors and filtrations of investors.

In this model, heterogeneity in the prior belief Ei((µA(0), µz(0))′|FY0 ) is sufficient to let investors disagree about

µA(t) at all times, as in Buraschi and Jiltsov (2006), even if they agree on the dynamics of cash flows and signals

(i.e. σ1µA

= σ2µA

). Beliefs heterogeneity can arise naturally also under other realistic assumptions. For instance,

disagreement can arise under identical initial priors if investors have different subjective perceptions about the

precision of the signal. Xiong and Yan (2006) assume that the subjective dynamics of A(t) and z(t) are different

across investors. Dumas, Kurshev, and Uppal (2007) follow Scheinkman and Xiong (2003) and model a general

equilibrium economy with overconfident investors that perceive the Brownian motions WA(t) and Wz(t) as correlated.

These models imply a different conditional variance of the disagreement process, but have similar implications

on the direction of the impact of disagreement on credit spreads, option prices, and stock returns. However, in

contrast to a setting with overconfidence, the fully rational economy implies a belief disagreement process that

converges asymptotically to zero, as subjective posterior variances converge to their steady state level. In our

setting, overconfidence arises for σ1µA

6= σ2µA

. In this case, the disagreement process has a non–trivial steady state

distribution. Moreover, note that since µA(t) is unobservable, the “true” parameter σµAis unknown to all investors

in the economy.

9A formal proof of this result can be found in Liptser and Shiryaev (2000); see also the technical Appendix.

8

C. Investors’ Preferences, Financial Markets and Equilibrium

The economy is populated with two investors with different subjective beliefs, but who are identical in all other

aspects. They maximize their life-time expected power utility subject to the relevant budget constraint:

V i = supci

Ei

(∫∞

0

e−ρt ci(t)1−γ

1 − γdt∣∣∣FY

0

)

, (5)

where ci(t) is the consumption of agent i = 1, 2 at time t, γ > 0 is the relative risk aversion coefficient, and ρ ≥ 0

is the time preference rate. Investors finance their consumption plans by trading in financial assets. In contrast to

Merton’s (1974) structural model, trading in the risk-less bond and the assets of the firm is not sufficient to complete

the market because the disagreement between investors implies in equilibrium a non zero market price of risk for both

A(t) and z(t). Disagreement influences the prices of all financial assets in the economy, which – from the perspective

of agent one – can be written as functions of the two-dimensional filtered Brownian motion W 1Y (t) in equation (3).

It follows that at least one additional financial asset is needed to complete the market and to determine a unique

stochastic discount factor. We focus on an economy which includes corporate bonds, equity, and European options

on equity. We denote with r(t) the interest rate of the risk-less bond in the model, assumed to be in zero net supply,

with S(t) the price of the stock of the firm, in positive net supply, with O(t) the price of a European option on

the stock, assumed in zero net supply, and with Bs(t) (Bj(t)) the price of the senior (junior) bond, also in positive

supply. V (t) denotes the value of the single firm in our economy.

Definition 1 (Equilibrium). An equilibrium consists of a unique stochastic discount factor such that: (I) given

equilibrium prices, all agents in the economy solve the optimization problem (5), subject to their budget constraint.

(II) Good and financial markets clear.

To solve for the equilibrium, we can apply the martingale approach, originally developed by Cox and Huang

(1986), in its extension to the case of heterogeneous beliefs; see, among others, Cuoco and He (1994), Karatzas

and Shreve (1998), and Basak and Cuoco (1998). Our equilibrium can be conveniently attained by constructing

a representative investor with a stochastic weighting process that captures the impact of belief disagreement. The

utility function of the representative investor is defined by:

U (c(t), λ(t)) = supc(t)=c1(t)+c2(t)

{c1(t)

1−γ

1 − γ+ λ(t)

c2(t)1−γ

1 − γ

}

, (6)

where λ(t) > 0 is the stochastic weight that captures the impact of belief heterogeneity. The optimal consumption

policy of agent i is ci(t) = (yiξi(t))−1/γ , where yi is the Lagrange multiplier associated with the static budget

constraint of agent i and ξi is the state-price density of this agent. Under our assumptions, it is possible to obtain

closed-form expressions for ci(t) and ξi(t), in terms of the exogenous state-variables.

Proposition 1. In equilibrium, the individual state price densities of agent one and two are:

ξ1(t) =e−ρt

y1A(t)−γ

(

1 + λ(t)1/γ)γ

, ξ2(t) =e−ρt

y2A(t)−γ

(

1 + λ(t)1/γ)γ

λ(t)−1,

9

where the weighting process λ(t) = y1ξ1(t)/(y2ξ

2(t)) follows the dynamics:

dλ(t)

λ(t)= −ΨA(t)dW 1

A(t) −

(

αΨA(t)σA

σz+ βΨz(t)

)

dW 1z (t) . (7)

The individual optimal consumption policies are:

c1(t) = A(t)(

1 + λ(t)1/γ)−1

, c2(t) = A(t)λ(t)1/γ(

1 + λ(t)1/γ)−1

.

The individual state price densities in the economy are functions of the exogenous cash flow process A(t) and the

weighting process λ(t). The dynamics of the weighting process is completely determined by the disagreement process

Ψ(t), which determines the volatility of λ(t). It follows that, in contrast to the classical Merton (1974) model, in

our economy the stochastic discount factor is a function of both A(t) and Ψ(t). This property has important asset

pricing implications.

When agents agree, i.e. Ψ(t) = 0 and σ1µA

= σ2µA

, we obtain as a special case Merton’s economy with identical

agents. In this case, the relative weight λ(t) is constant and equal to one, so that the optimal consumption plans are

identical across investors.10 When agents have different subjective beliefs about the probabilities of future states,

i.e. Ψ(t) 6= 0, agents implement different optimal consumption plans. Optimistic investors tend to consume more in

states associated with high aggregate cash flows, at a lower marginal utility of consumption because they perceive

these states as more likely. Similarly, pessimistic investors consume more and have a lower marginal utility in

states associated with low aggregate cash flows. It follows that the relative consumption share varies over time in a

stochastic way. The consumption share of the optimist is higher (lower) in states of high (low) aggregate cash flows.

The cyclical behavior of the consumption share is reflected in the dynamics of the stochastic weight λ(t): If agent

one (agent two) is optimistic about future cash flows, the stochastic weight is counter-cyclical (pro-cyclical) because

the individual marginal utilities of consumption are proportional to the price of the underlying state, with direct

implications on the equilibrium stochastic discount factor.

Another important difference with respect to the classical Merton (1974) model, is the shape of the risk premia.

From the state price density expressions in Proposition 1, the equilibrium market prices of risk can be derived using

Ito’s Lemma.

Proposition 2. Let θiA and θi

z be the subjective market prices of risk for cash flow and signal shocks of agent i. It

then follows:

θ1A(t) = γσA + ΨA(t)

(1 + λ(t)1/γ

)−1λ(t)1/γ , θ1

z(t) =(1 + λ(t)1/γ

)−1λ(t)1/γ

(

αΨA(t)σA

σz+ βΨz(t)

)

,

θ2A(t) = γσA − ΨA(t)

(1 + λ(t)1/γ

)−1, θ2

z(t) = −(1 + λ(t)1/γ

)−1(

αΨA(t)σA

σz+ βΨz(t)

)

.

10To focus on the impact of disagreement, we always assume identical initial endowments of the total cash flow stream A(t) acrossinvestors.

10

The market prices of cash flow risk θiA(t) is the sum of two terms. The first term is the standard price of risk that

is also found in the homogeneous Merton (1974) economy and it is given by the product of relative risk aversion and

cash flow volatility. The second term is due to the heterogeneity in beliefs. Since the stochastic discount factors are

functions of the weighting process λ(t), the market prices of risk contain a term related to the volatility of this process,

which is proportional to the disagreement index Ψ(t). This is due to the fact that when beliefs are heterogeneous,

agents face relative consumption share risk. This additional source of risk makes the stochastic discount factor

dependent on the stochastic weight λ(t), whose dynamics depends on the difference in beliefs. Belief-dependent state

prices are necessary in order to support belief-dependent consumption shares.

The second aspect that makes the asset pricing implications different from Merton’s (1974) is that when the

difference in beliefs is not zero, the signal is priced (θiz(t) 6= 0, i = 1, 2). When the signal is related to aggregate

cash flow growth (α, β > 0), agents use the signal to improve their beliefs on future cash flows. This implies that

unexpected innovations in the signal affect trading between agents and subsequently market prices. Thus, higher

signal volatility generates higher asset price volatility. Since the signal is not an explicit argument of consumer

preferences, the economy generates dynamics that could be interpreted or labeled as having “excess volatility”.

The market prices of risk perceived by the optimist are greater than those perceived by the pessimist. Initially,

this might seem surprising. The economic intuition, however, is simple: In equilibrium, the optimist experiences a

lower marginal utility of consumption in states associated with large cash flows, which she perceives as more likely.

When investors disagree, trading occurs between investors, who have to finance their different optimal consumption

plans. The optimist has to finance a higher level of optimal consumption in better future states. The pessimist has to

finance a higher level of consumption in bad future states. Therefore, the pessimist buys financial protection against

low cash flow states from the optimist. This excess demand implies a lower price for securities with positive exposure

to cash flow shocks and the risk implied by bad cash flow states is transferred from the pessimist to the optimist. It

follows that if a negative state occurs, the more optimistic agent is hit twice: First, because the aggregate endowment

is lower; second, since her relative consumption share is lower due to the protection agreement. The size of this risk

transfer is proportional to the size of the disagreement among agents Ψ(t). Therefore, beliefs disagreement has real

effects on the market price of the default event in our economy.

D. Pricing of Financial Assets

Given the expressions for the individual state price densities ξ1(t) and ξ2(t) in Proposition 1, we can easily price

any contingent claim in the economy by computing expectations of its contingent payoffs weighted by state price

densities. For simplicity, we give the relevant pricing expressions from the perspective of agent one.

The equilibrium firm value and the price of a default-free zero coupon with maturity T are given by:

V (t) = A(t)E1t

(∫∞

t

e−ρ(u−t) ξ1(u)

ξ1(t)

A(u)

A(t)du

)

, B(t, T ) = E1t

(ξ1(T )

ξ1(t)

)

.

11

We assume that the firm repayment structure satisfies a strict priority rule, in which payments to the junior bond

holders are made only if the contractual payments to the senior bond holders have been made. As in the Merton

(1974) model, default occurs only at maturity of the corporate bonds, if the value of the assets of the firm is less

than the face value of these bonds. We assume zero-bankruptcy costs. Therefore, in the event of default equity

holders are left with a zero price of equity and the corporate bond holders share the residual firm value according

to the pre-specified seniority rules. To focus on the implications of disagreement in credit risk, we keep the default

structure as simple as possible and do not explore more general default rules or more flexible default and liquidation

procedures.11 In this setting, it follows that the price of the senior bond is the sum of the prices of the zero-coupon

bond and the price of a short put option on the firm value:

Bs(t, T ) = K1B(t, T ) − E1t

(ξ1(T )

ξ1(t)(K1 − V (T ))

+

)

, (8)

where K1 is the face value of the senior bond. The price of the junior bond is the price of a call spread on the firm

value:

Bj(t, T ) = E1t

(ξ1(T )

ξ1(t)(V (T ) − K1)

+

)

− E1t

(ξ1(T )

ξ1(t)(V (T ) − (K1 + K2))

+

)

,

where K2 is the face value of the junior bond. The price of equity is the firm value residual in excess of the price of

the total corporate debt,

S(t) = V (t) − Bs(t, T ) − Bj(t, T ) . (9)

Finally, the price of an European call option on the stock is:

O(t, T ) = E1t

(ξ1(T )

ξ1(t)(S(T ) − Ke)

+

)

, (10)

where Ke is the option’s strike price.12

To compute the expectations in pricing expressions (8)–(10), we need the joint density of A(t), λ(t), and the

contingent claim payoff, since the stochastic discount factor, ξ1(t), is a function of both A(t) and λ(t). Unfortunately,

the joint distribution of A(t) and λ(t) is typically unavailable in closed-form. However, we can calculate their joint

Laplace transform in closed-form. This Laplace transform can be used, in a second step, to price more efficiently all

securities in our economy by Fourier Transform methods. In this way, we can avoid a pricing approach that relies

exclusively on Monte Carlo simulation methods, which would be highly computationally intensive. For convenience,

we compute the joint Laplace transform of A(t) and λ(t) in the case σ1µA

6= σ2µA

, which admits a steady state

distribution for the relevant state variables in our model.

11See, e.g., Black and Cox (1976) for a model with premature default. Anderson and Sundaresan (1996) and Mella-Barral and Perraudin(1997) model strategic debt service. Broadie, Chernov, and Sundaresan (2007) propose a setting that incorporates Chapter 7 and Chapter11 issues.

12Put option prices follow by put-call parity.

12

Proposition 3. The joint Laplace transform of A(t) and λ(t) under the belief of agent one is given by:

E1t

((A(T )

A(t)

)ǫ (λ(T )

λ(t)

)χ)

= Fm1

A(m1

A, t, T ; ǫ)FΨA,Ψz(ΨA, Ψz, t, T ; ǫ, χ) , (11)

where

Fm1

A(m1

A, t, T ; ǫ) = exp

(ǫ

a1A

(

−a0Aτ +

(a0A

a1A+ m1

A

)

(ea1Aτ − 1)

)

+1

2ǫ(ǫ − 1)σ2

Aτ

+ǫ2

4a21A

(γ2

A

σA

)2(3 − 4ea1Aτ + e2a1Aτ + 2a1Aτ

)

+ǫ2γ2

A

a1A

(

−τ +1

a1A(ea1Aτ − 1)

))

, (12)

with τ = T − t and

FΨA,Ψz(ΨA, Ψz, t, T ; ǫ, χ) = eA0(τ)+B1(τ)ΨA+B2(τ)Ψz+C1(τ)Ψ2

A+C2(τ)Ψ2

z+D0(τ)ΨAΨz , (13)

for functions A0, B1, B2, C1, C2 and D0 detailed in the proof in the Appendix.

The Laplace transform in Proposition 3 is a function of m1A(t), ΨA(t), and Ψz(t). The dependence on m1

A(t)

is exponentially affine. The dependence on ΨA(t) and Ψz(t) is exponentially quadratic. By computing the closed–

form characteristic function of A(t) and λ(t) we can now price the contingent claims in the economy by Fourier

inversion methods. The spirit of this approach is similar to the one used to price derivatives in stochastic volatility

models, such as Heston (1993), Duffie, Pan, and Singleton (2000), and Carr, Geman, Madan, and Yor (2001), or

in interest-rate models, such as Chacko and Das (2002). Dumas, Kurshev, and Uppal (2007) initially applied this

method to compute the prices of financial assets and the trading portfolios of a general equilibrium economy with

excess volatility generated by the presence of overconfident investors.

The pricing expressions implied by the Fourier Transform approach for all contingent claims in our economy are

summarized in the next Proposition. The results are used to study the dependence of credit spreads, option-implied

volatilities, and stock returns on differences in beliefs.

Proposition 4. Let

G(t, T, x;ΨA, Ψz) ≡∫ ∞

0

(1 + λ(T )1/γ

1 + λ(t)1/γ

)γ[

1

2π

∫ +∞

−∞

(λ(T )

λ(t)

)−iχ

FΨA,Ψz(ΨA, Ψz, t, T ; x, iχ)dχ

]

dλ(T )

λ(T ).

1. The equilibrium firm value is:

V (t) := V (A, m1A, ΨA, Ψz) = A(t)

∫ ∞

t

e−ρ(u−t)Fm1

A

(m1A, t, u; 1 − γ)G(t, u, 1 − γ; , ΨA, Ψz)du.

13

2. The equilibrium price of the corporate zero-coupon bond is:

B(t, T ) := B(t, T ;m1A, ΨA, Ψz) = e−ρ(T−t)Fm1

A

(m1A, t, T ;−γ)G(t, T,−γ; ΨA, Ψz).

3. The equilibrium price of the senior defaultable bond is:

Bs(t, T ) := Bs(t, T ; A, m1A, ΨA, Ψz) = B(t, T ) − E1

t

(

e−ρ(T−t)

(A(t)

A(T )

1 + λ(T )1/γ

1 + λ(t)1/γ

)γ

(K1 − V (T ))+)

.

4. The equilibrium price of the junior defaultable bond is:

Bj(t, T ) := Bj(t, T ; A,m1A, ΨA, Ψz),

= E1t

(

e−ρ(T−t)

(A(t)

A(T )

1 + λ(T )1/γ

1 + λ(t)1/γ

)γ((V (T ) − K1)

+ − (V (T ) − (K1 + K2))+)

)

.

5. The equilibrium price of equity is:

S(t) := S(t, T ; A,m1A, ΨA, Ψz) = V (t) − Bs(t, T ) − Bj(t, T ).

6. The equilibrium price of a European call option on equity is:

O(t, T ) := O(t, T ;A, m1A, ΨA, Ψz) = E1

t

(

e−ρ(T−t)

(A(t)

A(T )

1 + λ(T )1/γ

1 + λ(t)1/γ

)γ

(S (T ) − Ke)+

)

.

From the formulas in Proposition 4, we obtain a semi-explicit description for the dependence of the prices of

corporate bonds, stock options, and equity on the degree of disagreement about cash flows and the signal. Therefore,

these formulas are very useful in studying how disagreement influences the prices of these assets and credit spreads

in our economy.

II. Model Predictions

In this section, we use the solutions of the model in Proposition 4 to investigate the joint behavior of credit

spreads, option prices, and stock returns as a function of the difference in beliefs ΨA(t) and the difference in agents

perceived volatilities σ1µA

− σ2µA

for the cash flows expected growth rate µA. We compute all equilibrium quantities

with respect to the steady state distribution of beliefs, which is non–trivial when agent disagree about σµAin our

model, and specify with γ1A − γ2

A the difference in the steady–state posterior variances of firm cash flows.13 We

13For computational tractability, we set β = 0. In this case, the disagreement about the signal, Ψz , converges to zero when a1z isnegative. It follows that at the steady state, disagreement about the signal’s growth rate is zero, even if the steady-state posterior varianceof the signal is different across agents when σ1

µz6= σ2

µz.

14

calibrate the model to the cash–flow dynamics of a representative firm in our sample. Table 1 summarizes the set

of calibrated parameters. We assume a level of risk aversion equal to 2 and a firm cash-flows volatility equal to

7% (Zhang, 2006).14 The calibrated values of the learning parameters are consistent with the estimates obtained in

Xia (2001) and Brennan and Xia (2001). The median difference in beliefs of firms’ future earnings in our I/B/E/S

forecast data is 0.22. Consequently, we consider belief dispersion ΨA(t) between zero and 0.2 and differences in

steady state cash flow growth rate posterior volatility between zero and 0.02.

Insert Table 1 approximately here.

A. Firm Value and Firm Value Volatility

When ΨA(t) = Ψz(t) = 0 and γ1A = γ2

A the model implications are those of Merton’s (1974): At the calibrated

parameters the firm value is 161 and the constant firm value volatility is 7%. Disagreement across investors affects

the equilibrium stochastic discount factor and thus prices of all financial assets. Figure 1 (left panel) shows that the

firm value decreases with disagreement and the difference in growth rate volatility between agent one and two: An

increase in ΨA(t) and γ1A − γ2

A from zero to 0.2 and 0.02, respectively, reduces the equilibrium (asset) value of the

firm by approximately 1.5%.

Insert Figure 1 approximately here.

To understand the intuition for the negative effect of belief disagreement on the firm value, it is helpful to compare

the discount factor of the optimistic agent in the model with the stochastic discount factor in Merton’s (1974) model.

The stochastic discount factor of the optimist can be written as:

ξi(t) =1

yie−ρtA(t)−γsi(t)

−γ , (14)

where si(t) = ci(t)/A(t) is this investors share of total consumption A(t): ξi(t) is proportional to the marginal

utility of a stochastic share si(t) of total consumption A(t). In the economy with homogenous beliefs, this share is

a constant that depends only on the tightness of the individual budget constraints (i.e., the ratio y1/y2). Thus, the

discount factor is simply proportional to the marginal utility A(t)−γ of aggregate consumption. In the economy with

disagreement, the random share si(t) is greater (lower) when A(t) is higher (lower). Therefore, in good (bad) cash

flows states the marginal utility of the optimist has a larger (lower) impact on the stochastic discount factor. Since

in good (bad) states the marginal utility of the optimist (pessimist) is lower, the present value of future cash-flows

is lower, implying a lower equilibrium firm value than in the economy with homogeneous beliefs. To implement

the optimal ex-ante consumption plan, the optimist (pessimist) buys financial assets that finance the higher future

consumption share in good (bad) cash flow states. Therefore, in the competitive equilibrium the optimist sells

financial protection against low cash flow states to the pessimist. It follows that the additional risk created by

14Brennan and Xia (2001), Aıt-Sahalia, Parker, and Yogo (2004) and Chen, Collin-Dufresne, and Goldstein (2006) use risk aversionparameters between six and ten. The main focus of these studies, however, is to match the equity premium.

15

the stochastic consumption share in the economy is compensated to investors in an asymmetric way because the

individual state prices are proportional to different marginal utilities. Since the optimist (pessimist) perceives a lower

(higher) state price for large cash flow states, the market price of risk of cash flow shocks is higher (lower) for the

optimist (pessimist).

A further important feature of the stochastic discount factor (14) is that its volatility is stochastic when the

consumption share is stochastic. When disagreement is zero, si(t) is constant and the volatility of dξi(t)/ξi(t) is

proportional to the volatility of firm cash flows: The firm value process features a constant volatility as in Merton’s

(1974) model. When beliefs are heterogenous, the volatility of si(t) is stochastic and proportional to Ψ(t) because

it depends on the volatility of the ratio of individual marginal utilities of optimal consumption. It follows that the

volatility of the firm value is stochastic as well and increasing with the degree of belief disagreement: A higher

uncertainty about future state prices increases the volatility of the discounted value of future cash flows. Figure 1

(middle panel) shows a plot of the firm value volatility (obtained from Ito’s Lemma): At the calibrated parameters

an increase in ΨA(t) and γ1A − γ2

A to 0.2 and 0.02, respectively, increases the equilibrium firm value volatility from

7% to about 12.5%.

B. Firm Value Risk-Neutral Skewness and Probability of Default

As disagreement increases, the model generates a negative endogenous co–movement between the value of the firm

and the firm value volatility, even if the local volatility of the firm cash flows is constant. This feature implies

a moderate negative skewness of the physical distribution of the firm value, which generates a moderate increase

of the physical probability of default from 3.2% (5.6%) to 4% (7.1%) for low (high) leverage firms. This positive

relation between default probabilities and volatility is consistent with the recent evidence documented in Bharath

and Shumway (2007).15

It is well-known that negative skewness can also be obtained in partial equilibrium models with jumps; see Pan

(2002), in option pricing, and Zhang, Zhou, and Zhu (2006), Cremers, Driessen, and Maenhout (2007) and Tauchen

and Zhou (2006), in credit risk. In our model, negative skewness arises endogenously, even if cash flows and security

prices do not include a jump component. This follows from the form of the equilibrium stochastic discount factor

in the economy with heterogeneous beliefs. To understand this point, consider for brevity an economy, in which the

signal is not related to cash flows (α = β = 0). From Proposition 1 and Ito’s Lemma, the diffusion term of the

dynamics of the individual stochastic discount factors is given by:

dξi(t)/ξi(t) − Et[dξi(t)/ξi(t)] = − (γσA + (1 − si(t))ΨA(t)) dW iA(t), (15)

where si(t) = ci(t)/A(t) is the share of agent i of total consumption. It follows that the volatility of the individual

state prices is asymmetric and systematically related to the consumption share of agents in the economy: A positive

15Further recent studies on default probabilities include Vassalou and Xing (2004), Duffie, Saita, and Wang (2006), and Campbell,Hilscher, and Szilagyi (2007).

16

cash flow shock lowers the volatility of the individual stochastic discount factors, and vice versa. This feature is

due to the decreasing marginal utility of consumption of the two investors and generates endogenously the negative

skewness in the economy.

Consistent with the feature for the physical measure, the model also generates endogenously a pronounced neg-

ative risk–neutral skewness. Figure 1, right panel, illustrates the link between difference in beliefs and risk–neutral

skewness: An increase in ΨA(t) (γ1A−γ2

A) from zero to 0.2 (0.02) decreases the risk-neutral skewness of the firm value

from zero to -0.5 (-0.3).16 This feature is important because it directly affects corporate bonds and equity prices,

given that these are non-linear claims on firm’s asset values. It implies a large disagreement-induced risk premium

for the default event, which can have a substantial impact on the price of default-dependent securities, even if the

physical probability of default at the calibrated parameters is less than 4%.17

C. Credit Spreads and the Volatility of Stock Returns

It is well–known that the Merton (1974) model implies corporate credit spreads which are on average too low.

Structural credit risk models fail to explain credit spreads especially for firms with high credit ratings; see, e.g.,

Eom, Helwege, and Huang (2004), Huang and Huang (2004) and David (2007) for an alternative explanation. This is

known as the “credit spread puzzle”. Bhamra, Kuhn, and Strebulaev (2007) propose a model with recursive utility,

in which they try to fit both the credit spread and the equity premium puzzle. A potentially interesting topic for

future research is the analysis of the joint impact of disagreement on credit spreads and equity premia in a setting

with Epstein-Zin preferences. We find that belief disagreement can help to explain this puzzle: An increase in the

difference in beliefs reduces the firm value and increases the risk premium for the default event, by lowering the

risk-neutral skewness of the firm. These features generate the higher credit spreads, even if the probability of default

is held below 4% for low leverage firms. Figure 2 illustrates this effect: In the right panel, as ΨA(t) and γ1A − γ2

A

increase from zero to 0.2 and 0.02, respectively, the senior credit spread increases by 29%, from 123 to 159 basis

points, even for moderate levels of risk aversion. This increase is larger than the standard deviation of credit spreads

for senior debt in our data set. For a larger relative risk aversion γ = 3, the increase in credit spreads is as large as 45

basis points. Given the time separable preferences in our economy, a too large risk aversion implies a low elasticity

of intertemporal substitution and a higher risk-less interest rate.

Another main implication of our model is that disagreement increases the volatility of equity.18 This feature is

important because Campbell and Taksler (2003) have documented empirically a positive co-movement between credit

spreads and the volatility of stock returns, which even exceeds the co–movement predicted by standard structural

16To compute the risk-neutral skewness, we follow Bakshi and Madan (2000), who show that any payoff function can be spanned by acontinuum of out-of-the-money calls and puts: When the risk-neutral distribution is left skewed the combined cost of the positioning inputs is larger than the one of the combined positions in calls. The expression for the risk-neutral skewness of the firm value returns isprovided in the technical Appendix.

17For comparison, the average default probability of senior secured bonds between 1980 and 2006 is 4.3%, according to Moody’sCorporate Default and Recovery Rates Report 2007.

18The detailed specification of the equity volatility is derived in the technical Appendix.

17

models.19 Our model offers a structural explanation for these findings, in which the co–movement between credit

spreads and equity volatility follows endogenously from the time-variation of the difference in beliefs. For instance,

in the same scenario as above, the volatility of equity increases from 7% to 29% as disagreement increases; see the

left panel of Figure 2. For firms with lower leverage, not shown in Figure 2, the volatility increases to 20%. These

volatilities are consistent with the average volatility of stock returns in our data set.

Insert Figure 2 approximately here.

D. Price of Equity and the Skewness of Stock Returns

In Merton’s (1974) model, firm value volatility is constant and the risk–neutral skewness is zero. It follows that the

price of equity is increasing in the firm value volatility parameter and co-moves positively with the value of the firm.

In an economy with disagreement, markets are incomplete and the volatility and risk-neutral skewness of the firm

value are stochastic. For firms with different degrees of leverage, we find that the price of equity can either increase

or decrease with disagreement. This feature has important implications for the co–movement of credit spreads, stock

prices, and the volatility of stock returns. Moreover, it has key implications for the smile of individual stock options.

Figure 3 illustrates the different effects of an increase in the difference in beliefs index ΨA(t) and the difference

in agents’ posterior cash flow growth rate volatility, γ1A− γ2

A, conditional on a low or a high firm leverage.

Insert Figure 3 approximately here.

For high leverage firms, an increase in ΨA(t) and γ1A − γ2

A to 0.2 and 0.02, respectively, lowers the price of equity

by 1 percent. For low leverage firms, the price increases by 3.1 percent. From the results in the previous sections, it

follows that in the first case the price and the volatility of equity co–move negatively. A positive co–movement arises

in the second case. This is an important departure from Merton’s (1974) model – with immediate implications for

capital structure arbitrage strategies – because it implies that the standard hedge ratio might even change sign.

To understand why these features exist, note that the price of equity can be represented as a portfolio consisting

of a long position in the firm value V (t), a short position in K1 risk–less zero bonds with price ZCB(t), and a long

position in an out-of-the-money put on the firm value, with strike K1 and price P (t):20

S(t) = V (t) − K1 · ZCB(t) + P (t, K1) .

The first term, V , is independent of leverage and is decreasing in disagreement. The price of the zero coupon bond

can be shown to be decreasing in disagreement for a relative risk aversion parameter greater than one. Thus, the

19This finding has been documented also by Zhang, Zhou, and Zhu (2006), and Avramov, Jostova, and Philipov (2007), using reducedform models. Chen, Collin-Dufresne, and Goldstein (2006) and Bhamra, Kuhn, and Strebulaev (2007) obtain similar effects from twoconsumption-based equilibrium models.

20We consider for brevity of exposition a firm without junior debt.

18

effects of the first two components of the price of equity tend to offset each other, with the second component

increasing proportionally to firm leverage. The last term – i.e. the price of the put option P (t, K1) – has a positive

impact on the price of equity, but the size of the effect depends significantly and in a non monotonic way on firm’s

leverage. For some regions of leverage, we find that this effect can be large enough to reverse the negative impact of

the change in the value of the firm:

dS

dΨ+/−

=dV

dΨ−

− K1 ·dZCB

dΨ−

+[

Delta: +︷ ︸︸ ︷

dP

dV−

·dV

dΨ−

+

Vega: +︷ ︸︸ ︷

dP

dσV+

·dσV

dΨ+

+

Skewness: +︷ ︸︸ ︷

dP

dSkV−

·dSkV

dΨ−

]

. (16)

When leverage is high, the dominating effect on the price of equity comes from the first two terms in (16), as the

Delta, Vega, and Skewness effects on the put price are all small in relative terms. For very low leverage values, the

values of the put option and the position in the zero bond are a small fraction of firm value. Therefore, the price

of equity is dominated by the first term in (16). It follows that for high and very low leverage the value of equity

is decreasing with disagreement at the calibrated model parameters. For the intermediate leverage region, however,

the price of the embedded out-of-the-money put option can be a non–negligible fraction of the firm value, and its

sensitivity to increases in negative skewness (the last term in square brackets) is high. We find that the last effect can

be high enough to compensate the negative change of the firm value and make the price of equity increase. Figure

4 illustrates the trade-off between these effects as ΨA(t) changes from 0 to 0.20, dependent on firm leverage: For

levels of leverage between approximately 0.01 and 0.03 the effect of the higher negative skewness is large enough to

increase the price of equity as beliefs dispersion increases. The leverage region, in which disagreemet and stock price

have a positive co–movement depends on the calibrated parameters in the model. For instance, for a relative risk

aversion parameter γ = 4 this region is broader and contains leverage ratios between 0.01 and 0.06.

Insert Figure 4 approximately here.

Similar to the findings for the firm value, the endogenous stochastic co–movement between the price and the volatility

of equity generates an asymmetric physical stock price density. However, in contrast to the unambiguously negative

sign of the skewness of firm value, the skewness of stock returns can be both positive and negative in our model:

The positive (negative) co–movement between the price and the volatility of equity tends to generate stock returns

that are positively (negatively) skewed. This is important because skewness has been found by several authors to be

a key determinant of stock returns; see, among others, Kraus and Litzenberger (1976), Harvey and Siddique (2000)

and Dittmar (2002).21 Moreover, these features have important implications for the co-movement between stock and

corporate bond prices and for the relation between disagreement and firm-specific measures of distance to default.

As disagreement increases, the price of the corporate bond decreases, but the price of equity can increase if the firm

21Kraus and Litzenberger (1976) study a CAPM with investors that have preferences for skewness. Harvey and Siddique (2000) showthat stocks with increasing prices when volatility spikes up have a positive skewness. Moreover, investors with preferences for skewnessbid up the prices of assets with positive (co)skewness. Dittmar (2002) studies the impact of a non-linear pricing kernel in an economy, inwhich agents are averse to kurtosis and prefer positive skewness. Barberis and Huang (2007) study the impact of a preference for skewnessin a Prospect Theory–type model with exogenously distorted beliefs. Brunnermeier, Gollier, and Parker (2007) analyze preferences forskewness in general equilibrium and find that positively skewed assets have lower expected returns. Recently, Conrad, Dittmar, andGhysels (2007) link the higher prices of assets with positive skewness to the existence of stock market bubbles.

19

is moderately levered. It follows that in our model the higher credit spreads do not have to be coupled with, for

example a lower distance to default.

E. The Implied–Volatility Smile

In addition to the physical skewness of stock returns, the co–movement between the price and the volatility of equity

in our model also implies an endogenous risk-neutral skewness, which is more likely to be positive (negative) for firms

with low (high) leverage. This feature of the model is important because it is related to the empirical behavior of

stock options. While index options mostly exhibit a left skewed smile, Bakshi, Kapadia, and Madan (2003) document

that individual options behave differently and have, in some cases, a positively sloped implied–volatility smile. They

discuss in detail the exact link between implied volatility and risk-neutral skewness. Figure 5 illustrates the effect of

the difference in beliefs on the risk-neutral skewness of stock returns in our model, conditional on leverage.

Insert Figure 5 approximately here.

In the Merton (1974) model (ΨA(t) = 0 and γ1A = γ2

A), volatility is constant and the risk-neutral skewness of stock

returns is zero. As ΨA(t) and γ1A − γ2

A increase to 0.2 and 0.02, respectively, the risk-neutral skewness increases

to 0.9 for the firm with low leverage, but it decreases to -0.99 for the firm with high leverage. Figure 6 illustrates

the implications of this effect for the implied volatility smiles of individual options. We consider three firms, with

decreasing levels of leverage, in the left, middle, and right Panels of Figure 6, respectively.

Insert Figure 6 approximately here.

In the Merton (1974) model, the implied volatility smile is flat, independent of leverage. As disagreement increases,

at–the–money implied volatilities increase. The slope of the smile is positive and negative for the firms with lowest

and highest leverage, respectively. The smile is approximately symmetric for the firm with average leverage and its

convexity typically increases as disagreement increases. The at-the-money implied volatility is 0.11 (0.17) for the

firm with highest (lowest) leverage. Out-of-the-money puts are more expensive when leverage is high: A 10% out-

of-the-money put has an implied volatility of 0.16, but a 10% out-of-the-money call has an implied volatility of 0.08.

For the firm with average leverage, the pattern is more symmetric: Out-of-the-money calls have an implied volatility

of 0.14 and out-of-the-money puts an implied volatility of 0.16. We can quantify the slope of the smile by regressing

the implied volatility on moneyness, as in Bakshi, Kapadia, and Madan (2003). From these regressions, we obtain a

slope coefficient of -0.36, -0.26, and 0.6 for the settings with highest, average and lowest leverage, respectively.

20

III. The Data Sets

To empirically test the main implications of our model, we merge four data sets and match, for each firm,

information on professional earning forecasts, balance-sheet data, corporate bond spreads, stock returns, and stock

option prices. The merged data set contains monthly information on 337 firms for the period 1996 – 2005.

A. Bond Data

The bond data is obtained from the Fixed Income Securities Database (FISD) on corporate bond characteristics and

the National Association of Insurance Commissioners (NAIC) database on bond transactions. The FISD database

contains issue and issuer-specific information for all U.S. corporate bonds. The NAIC data set contains all transactions

on these bonds by life insurance, property and casualty insurance, and health maintenance companies, as distributed

by Warga (2000). This database is an alternative to the no longer available database used by Duffee (1998), Elton,

Gruber, Agrawal, and Mann (2001), and Collin-Dufresne, Goldstein, and Martin (2001). U.S. regulations stipulate

that insurance companies must report all changes in their fixed income portfolios, including prices at which fixed

income instruments were bought and sold. Insurance companies are major investors in the fixed income market and,

according to Campbell and Taksler (2003), they hold about one-third of outstanding corporate bonds. These data

represent actual transaction data and not trader quotes or matrix prices.22

Initially, we eliminate all bonds with embedded optionalities, such as callable, putable, exchangeable, convertible

securities, bonds with sinking fund provisions, non-fixed coupon bonds, and asset-backed issues. The data set contains

information on the seniority level of the bonds. We are thus able to divide our data sample into senior secured and

junior subordinated bonds. We manually delete all data entry errors. Moreover, to control for the possibility of

residual errors, we windsorize our database at the 1% and 99% level. We are then left with a final database of 337

firms with senior secured bonds and junior subordinated bonds. Finally, to compute corporate bond credit spreads,

we use zero-coupon yields available from the Center for Research in Security Prices (CRSP).

B. Difference in Beliefs Index Data

To obtain a proxy of belief disagreement, we use analyst forecasts of earnings per share, from the Institutional

Brokers Estimate System (I/B/E/S) database. This database contains individual analyst’s forecasts organized by

forecast date and the last date the forecast was revised and confirmed as accurate. To circumvent the problem of

using stock-split adjusted data, as described in Diether, Malloy, and Scherbina (2002), we use unadjusted data. In

an initial step, we match analysts forecast data with the bond data. We extend each forecast date to its revision

date.23 If an analyst makes more than one forecast per month, we take the last forecast that was confirmed.

22Earlier data sets offered only indirect information about actual market prices since values of non-traded bonds were estimated basedon matrix algorithms.

23E.g., if a forecast is made in July and last confirmed in September, then we use this information for the months July, August, andSeptember.

21

In our model, belief disagreement is defined as the difference between the subjective expected growth of firm

cash flows and signals, scaled by the volatility of dividends and signals, respectively. We proxy disagreement by

computing for each firm the mean absolute difference in the available earnings forecasts, scaled by an indicator of

earnings uncertainty. Since the data on subjective earnings uncertainty of analyst forecasts is not available in our

data set, we proxy earnings uncertainty by the cross-sectional standard deviation of earning forecasts. Therefore, our

measure of disagreement is the ratio of the mean absolute difference and the standard deviation of earning forecasts.

In the robustness section, we compare the explanatory power of our disagreement proxy with the one of other proxies

proposed in the literature and find it to be highest and more robust in terms of both statistical significance and

explanatory power.

C. Option Data

The option data is taken from OptionMetrics, LLC. This database covers all exchange listed call and put options

on US equities. With each trade, OptionMetrics reports the option’s implied volatility. Implied volatilities are

calculated using LIBOR and Eurodollar rates, taking into account European and American exercise styles. We apply

the following data filters to eliminate possible data errors. First, we exclude options which mature in the given

month, since it is known (see, e.g., Bondarenko, 2003) that these suffer from illiquidity. Second, we eliminate all

observations for which the ask is lower than the bid, for which the bid price is equal to zero, or for which the bid

ask spread is lower than the minimum ticksize.24 In a first step, we take implied volatilities of single-stock options

which are closest to at-the-money since these are known to be the most liquid ones. The implied volatility skew

is calculated as the difference between the implied volatility of a put option with moneyness 0.92 and the implied

volatility of an at-the-money put, scaled by the difference 0.92 − 1 in strike to spot ratios. We also calculate the

right skew of the implied volatility smile, which is defined as the difference between the implied volatility of a call

option with moneyness 1.08 and the implied volatility of a call option which is nearest to at-the-money, scaled by

the difference 1 − 0.8 of the moneyness ratios of these options.

D. Stock Returns Data

Stock returns serve as a dependent variable in the regressions for stock returns and as a control variable in the

regressions for credit spreads. They are taken from the CRSP database.

E. Control Variables

A large amount of empirical literature has studied the potential determining factors of credit spreads, option implied

volatility and skewness, and stock returns. This literature has suggested several economic variables having explana-

tory power for credit spreads, option prices, or stock returns. To focus on the additional explanatory power of belief

disagreement, we include in our regressions several of these variables as controls.

24The minimum ticksize equals USD 0.05 for options trading below USD 3 and USD 0.1 in any other case.

22

A first obvious control variable in all our regressions is firm leverage, which is defined as total debt divided by

the sum of total debt and the book value of shareholders’ equity. In the regression for credit spreads, we additionally

control for firm size, defined as the log total book value of assets. Leverage and firm size data are retrieved from the

COMPUSTAT database.

Since trading is generated endogenously in our model, a second set of natural control variables includes proxies

related to trading activity, either in bond, stock, or option markets. In the regressions for credit spreads, we take

stock returns and stock trading volume from the CRSP database as control variables, together with the 10 year swap

rate. This last variable is used as a further proxy of liquidity in stock and bond markets, similarly to Campbell and

Taksler (2003). In the robustness checks, we additionally control for idiosyncratic volatility, measured by the time-

series standard deviation of daily market adjusted stock returns 180 days preceding every observation, as in Campbell

and Taksler (2003). As control variables for trading activity and liquidity on option markets, in all regressions we

use option open interest and volume.25 In the regressions for stock returns, we additionally control for trading

pressure, defined as the ratio of put and call trading volume (see Dennis and Mayhew, 2002). In the regressions

for the option implied volatility and skewness, we also account for volatility mean reversion,26 which is proxied by

the difference between the twelve months moving averages of the option implied volatilities and the stock returns

realized volatilities.

A third set of natural control variables captures business–cycle and term structure effects, as well as further

systematic pricing factors. To this end, we include in the regressions for credit spreads and stock returns the

monthly S&P500 returns and non-farm payroll, which is available from FRED. Further systematic risk effects are

captured by including in these regressions a market excess return and the two Fama and French factors. This data is

available from Kenneth French’s web page. In the regressions for credit spreads, we also control for the level and the

slope of the term structure, by including as explanatory variables the yield of a 2 year Treasury bond and the slope

with respect to the 10 year Treasury bond yield. This data is obtained from the CRSP database. Table 2 provides

summary statistics for the main variables used in our empirical analysis.

Insert Table 2 approximately here.

IV. Empirical Analysis

In this section, we test the main empirical predictions of our model. We analyze in a set of panel regressions the

impact of disagreement on corporate credit spreads, option-implied volatility and skewness, and stock returns.27 In

25Since stock and option trading volume contain a deterministic time trend, we de-trend these series using a quadratic deterministiccomponent; see Gallant, Rossi, and Tauchen (1992) and Chordia, Sarkar, and Subrahmanyam (2006).

26See, e.g., Granger and Poon (2003) and Andersen, Bollerslev, Christoffersen, and Diebold (2006) for an extensive review of theempirical evidence on volatility mean reversion.

27Regression coefficients are estimated with Ordinary Least Squares (OLS). The standard errors are corrected for autocorrelation andheteroscedasticity. We also estimated the models with Instrumental Variables estimators to control for the potential endogeneity of someof the explanatory variables in the regressions. The results, however, were the same.

23

addition, we investigate the relation between disagreement and no–arbitrage violations of single factor models for

credit and option markets.

In our model, belief disagreement unambiguously increases credit spreads and the option implied volatility.

Therefore, we expect a positive sign for the coefficient of disagreement in these regressions. Disagreement implies a

potentially ambiguous sign for the relation between disagreement and stock returns, depending on the leverage of

the firm. Moreover, it can increase or decrease the slope of the smile, depending on its sign. To account for these

features, we introduce dummy variables for the coefficient of disagreement in our panel regressions, according to three

different, equally weighted, leverage bins. The low bin corresponds to leverage ratios below 0.045, the average bin to

leverage ratios between 0.045 and 0.35, and the high bin to leverage ratios above 0.35. According to the predictions

of our model at the calibrated parameters, a positive relation between disagreement and future returns is more likely

for higher levels of leverage because in this case the skewness effect of the firm value tends to vanish. Similarly, a

negative relation is more likely for regions of low leverage. We also use, where needed, some dummy variables to

differentiate firms with positive and negative slope of the smile. In this case, we expect disagreement to lower the

slope of the smile when it is negative, and vice versa.

A. Corporate Credit Spreads

In the regressions for credit spreads, we investigate the relevance of disagreement with respect to several empirical

models studied in the credit risk literature. Accordingly, we define the following empirical specifications, which

embody specific sets of control variables investigated by the previous literature. Given these specifications and

control variables, we study the economic importance and statistical significance of differences in beliefs:

(1) Option-related variables,

(2) Macro-financial variables,

(3) Firm-specific variables,

(4) Liquidity proxies,

(5) Fama-French factors and option implied-volatility,

(6) Full model with disagreement proxy,

(7) Full model without disagreement proxy.

In model (1), we account for option-related control variables. Cremers, Driessen, Maenhout, and Weinbaum (2006)

and Cao, Yu, and Zhong (2007) find that the option-implied volatility and skewness are important explanatory factors

of corporate credit spreads, both in the cross-section and in the time-series. They interpret these variables as proxies

for a volatility and jump risk premium component hidden in credit spreads. In our model, the option implied volatility

and skewness are endogenously driven by heterogeneity in beliefs. Therefore, we expect a significant relation between

option-related variables and credit spreads when heterogeneity in beliefs is omitted as an explanatory variable in

the credit spreads regressions. Table 3, second column, presents the results for model (1). Beliefs disagreement

and option-related variables alone explain 77% of the variation in senior credit spreads and 76% of the variation in

24

the full sample. Our belief disagreement proxy is highly statistically significant and the estimated coefficient has

the expected sign, but the option-implied volatility and skewness are not significant at standard significance levels.

In a regression with beliefs dispersion as the only explanatory variable, a one-standard deviation monthly shock in

disagreement widens credit by more that 12 basis points, which accounts for approximately one third the sample

standard deviation of credit spreads. The basic findings are unchanged in the results for the full model (6), in column

seven of Table 3, which includes all control variables available in our data set. Finally, note that the endogenous role

of option implied variables with respect to disagreement is supported by the results seen in the last column (model

(7)), since these variables are highly significant when disagreement is omitted as a regressor.

In model (2), we include macro-financial variables as controls. First, we follow Collin-Dufresne, Goldstein, and

Martin (2001) in using as explanatory variables the index and individual stock returns, the risk-free rate level, and

a proxy for the slope of the yield curve. Second, Huang and Kong (2007) show that macroeconomic announcements

have a significant effect on corporate credit spreads. Among many possible announcement variables, we use non-

farm pay-roll as a further control for the uncertainty about the future state of the economy. Non-farm pay-roll

is labeled by Andersen and Bollerslev (1998) the “king” of announcements. Beber and Brandt (2007) document

that it is the most influential macro announcement variable. The results in Table 3 (third column) show that, in

addition to the disagreement proxy, the risk-free rate and non-farm pay-roll are highly significant.28 In the full

model (6), these macro-financial factors maintain the expected sign, but only non-farm payroll has a significant

coefficient, as in Huang and Kong (2007). The sign of the coefficient for the other macro-financial variables is as in

Collin-Dufresne, Goldstein, and Martin (2001), but the return of the S&P500 is not statistically significant in our

regressions. Anderson, Ghysels, and Juergens (2005) regress an index of belief disagreement on the returns of S&P500

stocks and find a highly significant coefficient. In our economy, the price of equity is endogenous, and so should the

index be. From this perspective, the non-significance of stock and index returns in model (2) is not surprising.

Model (3) controls for firm-specific features. Leverage is an obvious control variable, positively related to the

probability of default in structural models of credit risk, and is an important explanatory variable of credit spreads,

as documented – among others – by Avramov, Jostova, and Philipov (2007). We also use firm size as a control

variable, to measure the higher sensitivity of smaller firms to business–cycle factors (Fama and French, 1993). In the

third column of Table 3, the regression results show that belief disagreement is highly significant, despite the high

significance of leverage and firm size. Consistent with economic intuition, these variables have a positive estimated

coefficient. Moreover, their significance is robust to the inclusion of additional control variables in model (6).

In model (4), we control for liquidity factors. Longstaff, Mithal, and Neis (2005), Driessen (2005), Ericsson

and Renault (2006), and Chen, Lesmond, and Wei (2007) investigate whether the non-default component of credit

spreads reflects cross-sectional differences in bond liquidity, and regress it on a number of liquidity proxies. The

significance and the sign of the estimated coefficients in these regressions can be very different.29 Direct measures

28The negative coefficient of the risk-less rate is consistent with the intuition that a higher risk-neutral drift of the firm value processreduces the risk–neutral probability of default; see, e.g., Longstaff and Schwartz (1995).

29For instance, Tang and Yan (2007) find a negative and insignificant coefficient for bond age, but Driessen (2005) finds a positive andsignificant one.

25

of bond liquidity are not available in our data set. Therefore, we use option volume and open interest as proxies of

option market liquidity, and stock volume as a measure of stock market liquidity. In this way, we attempt to capture

possible spill-over effects between option, stock, and corporate bond markets due, for example to capital arbitrage

trading.30 Since trading in credit, option, and credit markets is endogenous to belief disagreement in our model,

we test if disagreement impacts on credit spreads, beyond its influence on trading portfolios. The fifth column of

Table 3 shows that disagreement has a significant impact on credit spreads also when we control for liquidity effects.

Moreover, option–related liquidity proxies impact significantly on credit spreads, in a way that is robust to the

inclusion of further control variables in model (6).

In model (5), we control for Fama-French and volatility risk factors. Schaefer and Strebulaev (2005) find that

corporate bond prices are significantly influenced by two Fama-French factors and the VIX implied volatility index.

In our regressions, we use individual stock option rather than index option implied volatility to proxy for volatility

risk. The sixth column of Table 3 shows that disagreement is again highly significant. Additionally, only the SMB

factor additionally is. However, this last result is not robust to the inclusion of further control variables in model

(6).31

Overall, our results show that the explanatory power of belief disagreement for credit spreads is high and robust

with respect to several common control variables. The regression results for the full model (6) yields an adjusted R2

of approximately 0.9. The adjusted R2 for the model including only the significant variables is 0.86 for the regression

with senior credit spreads and 0.89 for the regression with the full data set. These findings are remarkably robust also

with respect to a stratification of the sample with respect to firm leverage. Table 5, second column, shows that the

significance and the size of the estimated coefficients for disagreement are very stable when we allow the coefficients

to be leverage-dependent in a panel regression with dummy variables.

Insert Table 3 approximately here.

B. Implied Volatility Smile

In this section, we study the empirical relation between belief disagreement and the implied volatility smile of

individual stocks.

(i) At-the-money implied volatility: In our economy, the at-the-money implied volatility of individual options un-

ambiguously increases with the degree of disagreement among investors. Therefore, we expect a positive empirical

relation between disagreement and the average implied volatility level. We consider several control variables to isolate

the marginal effect of disagreement on the smile of individual options. As proxies of trading activity, we use calls

and puts trading volume. In incomplete markets, the prices of individual options might reflect a liquidity premium

that depends on the direction of the trading pressure from public order flow. Therefore, we additionally control for

30Tang and Yan (2006) find spill-over effects between option and corporate bond markets. Hotchkiss, Warga, and Jostova (2002) findevidence that stock trading volume is a determinant of corporate bond trading volume.

31Avramov, Jostova, and Philipov (2007) also find that Fama and French factors loose their significance for credit spreads whencombined with other control variables.

26

trading pressure using the ratio of call and put trading volume.32 Another important control variable in our setting is

leverage because it is crucially related to the level and the slope of the smile of individual stocks. Finally, to account

for volatility mean reversion, we control also for the spread between at-the-money implied volatility and realized

historical volatility of individual stock returns. The results of Table 4, third column, show that the relation between

disagreement and at-the-money implied volatility of individual stocks is positive and significant. The results of the

panel regression with dummy variables for the three leverage bins in Table 5, fourth column, emphasize that this

positive relation is strongest for average and high leverage firms, but it is not significant for low levels of leverage.

With the exception of variables measuring trading volume on option markets, all other control variables in Table 4

are also highly statistically significant.

Insert Table 4 and 5 approximately here.

(ii) Slope of the implied volatility smile: In our model, disagreement can higher or lower the slope of the smile,

depending on whether the slope itself is positive or negative. For high leverage firms, the slope tends to be negative.

In this case, a higher disagreement across investors further lowers the slope. For low leverage firms, which tend to

produce a smile with a positive slope, a higher disagreement increases the slope.

Insert Table 6 approximately here.

Table 6 presents the results for panel regressions, in which the endogenous variable is the left and right slope of the

option implied volatility smile, respectively. To account for the potentially ambiguous impact of disagreement on

the slope of the smile in our model, we introduce for each firm an additional dummy variable, indicating whether

the firm had a smile with a positive or a negative slope. We find that in all regressions the estimated coefficient

for the disagreement proxy is statistically significant. As suggested by our theory, it is negative for firms having a

negatively sloped smile, and positive in the other cases. With the exception of control variables measuring option

trading volume and leverage, all other variables are also statistically significant. In particular, a larger at-the-money

implied volatility decreases the absolute slope of the smile. Table 7 summarizes the findings for the panel regression

including dummy variables for the three leverage bins. It shows that the impact of disagreement on the absolute

slope of the smile is larger when leverage is higher.

Insert Table 7 approximately here.

Overall, the results of this section support the hypothesis that disagreement among investors is a key driver of the

level and slope of the implied volatility smile of individual stock options.

32This is motivated also by the empirical evidence, e.g., in Bollen and Whaley (2004), that option implied volatility smiles of individualstock options are significantly linked to trading pressure.

27

C. No–Arbitrage Violations of Single–Factor Models

The simplifying structure of single-factor credit risk and option pricing models, such as the Merton (1974) and

Black and Scholes (1973) models, implies a negative relation between stock prices and credit spreads, and a positive

(negative) relation between the prices of stocks and those of their European call (put) options: A call (put) option’s

delta should be positive (negative). For index option markets, it is well-known that these relations are often violated

in the data. Bakshi, Cao, and Chen (2000) and Perignon (2006) document these violations empirically. Buraschi

and Jiltsov (2006) show that belief disagreement increases the probability of such violations. In this section, we first

analyze the empirical regularities of these violations for credit and individual option markets, in dependence of belief

disagreement and the leverage structure of the firm. In a second step, we study if they can be explained by our

model.

C.1. Model Predictions

At the calibrated parameters and for low leverage, our model implies that credit spreads and stock returns can be

positively linked. Therefore, we expect the probability of a violation on the credit market to be higher for such firms.

The probability of a no arbitrage violation on option markets in our model depends, in addition to leverage, on the

type and the moneyness of the option. This feature is due to the stochastic volatility and skewness of stock returns,

which has an impact on the prices of options with different Delta, Vega, and Skewness sensitivities in potentially

many different ways. For firms with low leverage, an increase in the stock price is linked to an increase in both

the volatility and the positive skewness of stock returns. Therefore, the prices of European call options on these

stocks increase with the stock price. At the same time, some out-of-the-money put options on the same stock can

have a low Delta and Skewness sensitivity, but a positive Vega. Therefore, their price can increase as the stock

price increases, leading to a no-arbitrage violation in the market for put options. It follows that the probability of a

violation in the put option market in our model is higher for out-of-the-money options on stocks with low leverage.

The same intuition applies, but with opposite effects, for the prices of individual options of firms with high leverage.

In this case, the prices of all put options decrease with the stock prices and the probability of a violation is higher

for out-of-the-money call options.

Figure 7 and Figure 8 illustrate the above intuition. At the calibrated parameters, Figure 7 (Figure 8) plots

the prices of in-, at-, and out-of-the-money put (call) options on the stock of a firm with low (high) leverage, as a

function of the belief dispersion index ΨA(t) and the difference in agents’ perceived growth rate volatility γ1A − γ2

A.

Insert Figures 7 and 8 approximately here.

In our model, we know that the stock price increases (decreases) with disagreement under the leverage structure

assumed in Figure 7 (Figure 8). It follows that the prices of in-the-money and at-the-money put (call) options in

Figure 7 (Figure 8) decrease with disagreement, which is consistent with a (negative) positive Delta for these options.

28

However, the prices of out-of-the-money put and call options in the two Figures are decreasing with disagreement.

This feature generates a no–arbitrage violation relative to single–factor models.

These findings have important implications for the usefulness of hedging strategies based on single-factor option

pricing models and imply that simple Delta-Vega hedging might perform poorly in states with a high dispersion

of beliefs. They also suggest that it might be useful to condition hedging strategies of individual stock options

on firm’s leverage. During the hedge fund crisis of 1998, market participants were given a revealing glimpse into

the proprietary trading strategies used by a number of large hedge funds, such as Long Term Capital Management

(LTCM). Among these strategies, capital structure arbitrage has become increasingly popular. These strategies,

however, are based on some key assumptions about the joint behavior of the value of debt and equity, such as the

sensitivities of corporate bond prices to changes in the price of equity. Schaefer and Strebulaev (2005) study the

hedge ratios implied by Merton’s (1974) model and find that the hedging quality is quite poor at the individual

bond level. Our model provides a rationale for this finding. The hedge ratio of standard single-factor models takes

the form (1/∆S − 1)S/D, where ∆S is the sensitivity of the price of equity to the underlying price of the corporate

bond and S/D is the inverse leverage ratio. In contrast to single-factor models, our results imply that the sign of

∆S might be both positive or negative, depending on the leverage of the firm.

Next, we study the frequency at which arbitrage violations of single-factor models can arise in our setting and

compare it to the one observed in the data. Next, we systematically differentiate between violations on credit markets,

on option markets, and joint violations.

Definition 2. (i) A Type 1 Violation [Credit Spread] is observed when ∆CS∆S > 0, where ∆CS is the change in

the credit spread and ∆S the change in the corresponding individual stock price. (ii) A Type 2 Violation [Options]

is observed on call (put) markets when ∆S∆C < 0 (∆S∆P > 0), where ∆C (∆P ) is the change in the price of the

call (put) option. (iii) A Type 3 Violation [Joint Behavior] is observed when a Type 1 and a Type 2 violation occur

at the same time.

Using the calibrated parameters in Table 1, we can simulate our model and calculate the occurrence frequencies

of Type 1, Type 2, and Type 3 violations. The results are reported in Table 8.

Insert Table 8 approximately here.

The unconditional frequency of Type 1 violations is approximately 13.9 percent. As expected, these violations are

more frequent for low leverage firms (15.3 percent) than for high leverage firms (12.2 percent). The frequency of Type

2 violations ranges from 1.7 percent to 10.9 percent. The highest number of violations is observed for out-of-the-

money call options of firms with high leverage. For puts, the frequency of Type 2 violations ranges from 1.1 percent

to 12.7 percent. In contrast to calls, the highest number of violations is observed for out-of-the-money put options of

firms with low leverage. The general patterns for Type 3 violations are similar to those for Type 2 violations, even

if Type 3 violations are, by construction, less frequent.

29

C.2. Empirical Analysis

Table 12 summarizes the occurrence frequencies of Type 1, Type 2, and Type 3 violations in our data set. The

structure and the patterns of the results are remarkably consistent with those obtained in Table 8 for the violations

simulated from the calibrated model.

Insert Table 12 approximately here.

The unconditional frequency of Type 1 violations is approximately 16.2 percent, which is quite high. As in the

simulated model, these violations are more frequent for low leverage firms (18.9 percent) than for high leverage firms

(14.3 percent). The frequency of Type 2 violations for call option markets ranges from 3 to 10.3 percent. The

highest number of violations is observed for out-of-the-money call options of firms with high leverage. For puts, the

frequency of Type 2 violations ranges from 2 percent to 10.2 percent and the highest number of violations is observed

for out-of-the-money options of firms with low leverage. These patterns are remarkably stable across different times

to maturity. For instance, the empirical frequency of Type 2 violations for out-of-the-money put (call) options of

low (high) leverage firms ranges between 10.1 (10.1) and 10.2 (10.3) percent. Moreover, the frequency of Type 2

violations for out-of the money calls and puts of firms with high and low leverage, respectively, are quite comparable.

Overall, the average number of violations in our data is quite comparable with the one reported by Bakshi, Cao, and

Chen (2000) for index options.

To study how belief disagreement influences the conditional probability of a violation in credit or option markets,

we estimate a set of panel Logit regressions, in which the binary variable y(it)j, denoting the occurrence of a violation

of type j at time t for firm i, is regressed onto a set of variables that include our proxy for disagreement. More

precisely, the probability that a Type 1 violation event occurs at time t for firm j is specified as:

P(y(it)j = 1

)= F (β0 + β1 log Ψ(t)),

where F is the cumulative distribution function of a logistic distribution and β0, β1 are parameters that have

to be estimated. For Type 2 and Type 3 violations, we additionally include as control variables option-specific

characteristics, such as moneyness and time to maturity. These models are estimated by Maximum-Likelihood. The

results for credit markets are given in in Table 9. Those for Type 2 violations in option markets are presented in

Table 10.

Insert Table 9 and 10 approximately here.

Overall, disagreement increases the conditional probability of a violation of Type 1 and Type 2, respectively, with an

estimated coefficient that is highly significant and very stable across the different leverage, moneyness, and maturity

regions. In the comparison between call and put violations, the marginal impact of disagreement on the conditional

probability of a violation is slightly higher for puts. At the same time, moneyness and maturity have a larger positive

30

impact on the likelihood of a violation for call options. The Logit regressions for the joint violations in Table 11

yield quite similar findings.

Insert Table 11 approximately here.

Disagreement has consistently a positive significant impact on the likelihood of a Type 3 violation, across different

leverage, moneyness, and maturity regions. In contrast to the results for Type 2 violations, maturity has a larger

positive impact on the likelihood of a violation for put options.

Overall, we obtain supporting evidence that belief disagreement is a potentially important missing factor in

structural models, explaining the empirical no-arbitrage violations of single-factor models in credit and individual

stock option markets.

D. Stock Returns

We conclude our analysis by studying the relation between stock returns and our belief disagreement proxy. At

the calibrated parameters, disagreement and stock returns tend to be positively related. For low levels of leverage,

however, it is also possible to observe a negative relation. It follows that the sign of the regression coefficient for

disagreement can be both positive and negative, whereby the last situation is more likely, according to our findings,

for firms with a low leverage.

The sign of the relation between disagreement and stock returns has been investigated in a series of papers,

obtaining mixed findings. Diether, Malloy, and Scherbina (2002), for example estimate a negative relation, but

Anderson, Ghysels, and Juergens (2005) find, using the same proxy for disagreement and a model, in which the

pricing kernel is affected by traders’ sentiment, a positive relation. To motivate the negative relation estimated in

their regressions, Diether, Malloy, and Scherbina (2002) invoke Miller’s (1977) argument that short selling constraints

cause optimistic investors to drive market prices above fundamental values. Our model can generate this feature for

firms with low leverage without introducing short selling constraints or other market frictions.

The second column of Table 4 summarizes the aggregate results of our benchmark regression for stock returns.

The estimated coefficient of our proxy for belief disagreement is positive and highly significant, which is at odds with

Miller’s (1977) theory. Among the control variables used, the trading volume on calls, the S&P500 returns and the

Fama and French factors are also significant at standard levels.33 As in Beber and Brandt (2007), the coefficient

estimated for non-farm pay-roll is not significant.34

Table 5, third column, presents the disaggregated results using the regression model with dummy variables for

the three leverage bins. The estimated coefficient of disagreement is statistically significant across all leverage bins,

33We obtain a different sign for the estimated coefficient of call and put trading volumes. This finding is not irrealistic. Easley, O’Hara,and Srinivas (1998), e.g., show that positive option trades – buying calls and selling puts – provide positive signals to the market makers,who increase bid and ask prices. In a similar vein, negative option trades – buying puts or selling calls – reduce the prices.

34Beber and Brandt (2007) attribute this lacking explanatory power to the fact that macro-economic news are swamped away byfirm-specific news on cash flows.

31

but the sign of the estimated coefficient for the low leverage bin is negative. The significance and the sign of the

other control variables is similar to those obtained for the benchmark model without dummies. These mixed signs

of the estimated coefficients are consistent with our main model implications and suggest that leverage might indeed

be a relevant variable for explaining the direction of the impact of disagreement on stock returns.

V. Robustness

Our results support the hypothesis that belief disagreement is an important priced risk factor for credit spreads,

the implied volatilities and skewness of single-stock options, and stock returns. In this section, we assess the robustness

of our results by studying (i) the extent to which our disagreement measure captures other sources of risk and (ii)

how our results compare to those of studies using other proxies of divergence of opinions.

A. Idiosyncratic Volatility

Idiosyncratic volatility is a potentially important risk factor for stock returns and credit spreads. Johnson (2004),

e.g., studies a model, in which stock returns of a levered firm are decreasing in the asset’s idiosyncratic risk. Mei,

Scheinkman, and Xiong (2005) use the idiosyncratic volatility of stock returns as a proxy for firm uncertainty in the

Chinese stock market. Chen, Collin-Dufresne, and Goldstein (2006) emphasize that the credit spread puzzle is closely

related to the ratio of idiosyncratic and total volatility of stock returns. Campbell and Taksler (2003) empirically

document that idiosyncratic stock return volatility is an important explanatory factor for the cross-section of credit

spreads.

Even if belief disagreement is not itself a measure of pure idiosyncratic risk, it is a natural robustness check

to investigate the extent to which it proxies for idiosyncratic volatility. To this end, we calculate the time-series

standard deviation of daily market adjusted stock returns 180 days preceding each observation and use it as a further

explanatory variable in our regressions, leading to the results in Table 13.

Insert Table 13 approximately here.

As in Campbell and Taksler (2003), idiosyncratic risk significantly and positively affects corporate credit spreads.

However, it does not have an impact on either the economic or statistical significance of our belief disagreement

proxy, which is the most significant variable in the model. The adjusted R2 of the regressions with idiosyncratic

risk without disagreement are slightly lower than those of our benchmark regressions without idiosyncratic risk. The

estimated coefficient for disagreement in the aggregated regression for stock returns is highly significant. The one

of idiosyncratic volatility is negative and significant, but only at the 10 percent significance level. This last finding

is consistent with the results in Ang, Hodrick, Xing, and Zhang (2006) and Guo and Savickas (2006), even if the

empirical literature also finds opposite results in some cases.35 Interestingly, in the regression for stock returns with

35In a time-series analysis Goyal and Santa-Clara (2003) find that the average stock variance has explanatory power for market returns,where the average stock variance is mainly idiosyncratic risk. Malkiel and Xu (2006) find a positive link between stock returns andidiosyncratic risk. Bali, Cakici, Yan, and Zhang (2005) find no significant link between idiosyncratic risk and market returns.

32

dummy variables for leverage, our proxy for disagreement consistently maintains a significant explanatory power,

but the idiosyncratic volatility does not.

B. Measures of Divergence of Opinions

In the literature, the mixed sign of the empirical relation between disagreement and stock returns can partially be

due to the different proxies used to measure the divergence of opinions. Diether, Malloy, and Scherbina (2002) find a

negative relation when measuring disagreement by the cross-sectional standard deviation of analysts’ forecasts about

future earnings. Using the same proxy and a different data set, Anderson, Ghysels, and Juergens (2005) obtain a

positive relation. Barron, Kim, Lim, and Stevens (1998) emphasize that the standard deviation of analysts’ forecast

is contaminated by uncertainty. Thus, it can be a poor proxy for divergence of opinions. Doukas, Kim, and Pantzalis

(2006) follow this hint and find a positive relation after adjusting their proxy for the effect of uncertainty.

To compare our findings with those of these other studies, we re-run all of our regressions for credit spreads

and stock returns using both the proxy of divergence of opinions used in Diether, Malloy, and Scherbina (2002) and

the one in Doukas, Kim, and Pantzalis (2006). Diether, Malloy, and Scherbina (2002) proxy disagreement by the

cross-sectional standard deviation of analysts’ forecasts, scaled by the lagged stock price, where the lagged stock price

is calculated on the previous month to avoid any look-ahead bias. The proxy used in Doukas, Kim, and Pantzalis

(2006) is computed using the procedure described in Barron, Kim, Lim, and Stevens (1998).

Table 14 presents the corresponding panel regression results, where column “Dispersion 1” uses the Diether,

Malloy, and Scherbina (2002) proxy and column “Dispersion 2” the Doukas, Kim, and Pantzalis (2006) proxy,

respectively.

Insert Table 14 approximately here.

The estimated coefficient for the proxy “Dispersion 1” is negative in the regressions for stock returns, as in Diether,

Malloy, and Scherbina (2002). It is positive in the regressions for credit spreads, consistent with the findings in

Guntay and Hackbarth (2006). Estimated coefficients in the regressions without dummies are significant only at the

10 percent significance level. When we stratify for leverage, the proxy in the regression for credit spreads is significant

only for the high leverage bin at the five percent significance level. The proxy “Dispersion 2” is significant at the

10 percent and 5 percent level, respectively, in the credit spreads and stock returns regression with aggregated data.

Moreover, the estimated sign for stock returns is positive.

In contrast to the results obtained for our disagreement proxy, the statistical significance of the estimated coeffi-

cients for the proxies “Dispersion 1” and “Dispersion 2” is clearly lower and less consistent. Adjusted R2 implied by

all regressions, both for credit spreads and beliefs dispersion, are also lower. Finally, it is interesting to note that in

the credit spreads regression of Table 14, the implied volatility of individual stocks is much more significant than the

corresponding disagreement proxies. It follows that the high significance of the estimated disagreement coefficient in

the regressions of Guntay and Hackbarth (2006), which use the index-option implied volatility as a control variable,

is not robust to the inclusion of control variables measuring the implied volatility smile of individual stock options.

33

VI. Conclusion

In this paper, we study theoretically and empirically the joint behavior of credit, option, and stock markets for

an economy, in which disagreement is a priced risk factor in equilibrium. Our model extends in a parsimonious way

the standard (partial) equilibrium Merton (1974) model of credit risk by considering an incomplete–market economy,

in which investors have heterogeneous beliefs about firm cash flows and another signal. These features generate an

additional risk factor due to disagreement, which is not priced by standard structural models of credit risk but which

determines the co–movement of credit spreads, option prices, their implied volatility smile, and stock returns. We

use the model solutions to study the joint behavior of these variables and to derive testable model predictions for

our empirical analysis. We build a comprehensive panel data set consisting of data on professional earning forecasts,

firm balance-sheets, corporate bond spreads, stock returns, and option–implied volatilities, and test the predictions

of the model. Our findings are as follows.

First, disagreement unambiguously widens corporate credit spreads. It is a well-known empirical fact that stan-

dard structural models tend to predict credit spreads which are on average too low, especially for highly rated firms.

In our theoretical model, disagreement generates trading patterns, in which risk is transferred from pessimistic to

optimistic investors. As disagreement increases, firm value decreases, but the firm value volatility and negative risk-

neutral skewness increase. Since defaultable bonds are proportional to a short put on the firm value, credit spreads

widen. The empirical analysis shows that the coefficient of disagreement in the regressions for credit spreads is highly

significant, both economically and statistically, in a way that is very robust to the inclusion of a large number of

control variables. Moreover, since disagreement drives the implied volatility and skewness of single-stock options,

these last variables have explanatory power for spreads when disagreement is not included as a right-hand variable

in the regressions.

Second, disagreement implies a positive co–movement between credit spreads, the volatility of equity, and the

implied volatility of single stock options: As disagreement increases, the volatility of stock returns and the option

implied volatility of single stocks also increase. The positive co–movement of these variables in our structural model

is supported by the empirical analysis, where we find that disagreement is the main driver of the implied volatility

smile and that both disagreement and idiosyncratic volatility are positively linked to credit spreads.

Third, disagreement is a crucial driving factor for the shape of the implied volatility smile of single stock options.

As disagreement increases, we find that the smile can be either more negatively and positively sloped, or even

symmetric and convex, depending on firm leverage. To understand this finding, note that in our model the co-

movement of stock returns and the volatility of equity can be both positive or negative because the relation between

disagreement and stock returns can be inverted for some regions of leverage. As in stochastic volatility option pricing

models, a negative (positive) correlation between return and volatility shocks generates a negative (positive) risk-

neutral skewness and the negatively (positively) sloped implied volatility smile. Bakshi, Kapadia, and Madan (2003)

have documented empirically the puzzling inverted implied volatility smirk of some stock options, which is difficult

to explain with standard single-factor option pricing models. Our model gives a structural explanation for their

34

findings. In our empirical analysis, we find that disagreement increases the at-the-money implied volatility of single

stocks and that it has explanatory power for both the left and the right slope of the smile.

Fourth, disagreement helps to explain the no-arbitrage violations implied by single–factor models for credit

spreads and single-stock options. Single–factor structural models of credit risk imply a negative relation between

credit spreads and the price of equity. Single factor option pricing models imply a positive (negative) relation between

the prices of a call (put) option and the price of the underlying stock. These relations can be violated in the data.

For instance, Bakshi, Cao, and Chen (2000) and Perignon (2006) find that the deltas of call (put) index options can

be not between zero and +1 (-1) for a good fraction of the prices observed empirically. In our model, the monotonic

relation between credit spreads, option prices, and the prices of the underlying stock can be violated for some regions

of leverage because of the important role of time–varying endogenous risk-neutral skewness. The percentage of

no–arbitrage violations in our data for individual options and credit spreads is substantial and is comparable to

the one predicted by our model at the calibrated parameters. We investigate empirically the extent to which belief

disagreement can explain no-arbitrage violations of single–factor models with a set of Logit regressions, and find that

the slope coefficient of disagreement is always positive and highly statistically significant.

Finally, disagreement tends to increase stock returns, but can be negatively related to them for certain, typically

low, regions of leverage. The recent empirical asset pricing literature has debated on the sign of the relation between

stock returns and divergence of opinions. Our model with disagreement and credit risk offers a structural explanation

for these mixed results. An increase in disagreement can raise the price of equity in some cases. This feature emerges

because when leverage is sufficiently low the higher negative skewness of the firm value can have a dominating impact

on the price of equity, which contains a long put option on the firm value. This result is obtained in a frictionless

economy and is completely due to the interaction of disagreement and credit risk. The empirical analysis yields a

very significant positive relation between stock returns and disagreement using aggregated data. This relation is

reversed and significant for low leverage firms when we stratify our panel with respect to leverage.

35

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39

Appendix

Appendix A. Beliefs Disagreement Dynamics and Learning

Given the parametric structure of the dynamics for dividends and signals in our model, it is possible to solve the filteringproblem of a Bayesian investor that believes in this dynamics. Since the relevant state variables define a multivariate Gaussianprocess, this problem is standard. In particular, with Gaussian initial conditions the implied conditional beliefs are Gaussian;see Rogers and Williams (1987) Sec. 6.9, and Liptser and Shiryaev (2000), Theorem 12.7.

Lemma 1. Let m(t) = E(µ(t)|FY

t

)and γ(t) = E

((µ(t) − m(t)) (µ(t) − m(t))′ |FY

t

). It then follows that m(t) and γ(t) are

continuous FYt -measurable for any t solutions of the system of equations:

dm(t) = (a0 + a1m(t))dt + γ(t)A′(BB′)−1(dY (t) − Am(t)dt), (A-1)

dγ(t)/dt = a1γ(t) + γ(t)a′1 + bb′ − γ(t)A′(BB′)−1Aγ(t), (A-2)

subject to the initial conditions m(0) = E(µ0|FY0 ) and γ(0) = E

((µ0 − m(0)) (µ0 − m(0))′ |FY

0

). Moreover, if the matrix γ(0)

is positive definite, then the matrices γ(t), 0 ≤ t ≤ T , have the same property.

Closed form solutions of the matrix Riccati equation36 for γ(t) in Lemma 1 are obtained, via Radon’s Lemma, by linearizingthe flow of the differential equation (A-2).

Lemma 2. Let (g11(t) g12(t)g21(t) g22(t)

)

= exp

(

t

(a1 A′(BB′)−1Abb′ −a′

1

))

.

Then the solution of equation (A-2) can be written as:

γ(t) = (γ(0)g12(t) + g22(t))−1 (γ(0)g11(t) + g21(t)) .

Proof: Let γ(t) = F (t)−1G(t), for two differentiable functions F (t) and G(t) such that G(t) is invertible and F (0) = id2×2.It then follows:

d

dt[F (t)γ(t)]− d

dt[F (t)] γ(t) = F (t)

d

dtγ(t),

andd

dtG(t) − d

dt[F (t)]γ(t) = bb′F (t) + G(t)a1 +

(F (t)a′

1 + G(t)A′(BB′)−1A)γ(t). (A-3)

The last ordinary differential equation leads to the following system of linear equations:

d

dtG(t) = F (t)bb′ + G(t)a′

1,

d

dtF (t) = −F (t)a1 + G(t)A′(BB′)−1A .

The solution of this system of differential equations is:

(G(t) F (t)) = (G(0) F (0)) exp

(

t

(a1 A′(BB′)−1Abb′ −a′

1

))

,

= (γ(0)g11(t) + g21(t) γ(0)g12(t) + g22(t)) .

It then follows that the closed form solution of the matrix Riccati equation (A-2) is:

γ(t) = (γ(0)g12(t) + g22(t))−1 (γ(0)g11(t) + g21(t)).

This concludes the proof.

�

From Lemma 1, the disagreement dynamics in our economy follows in a straightforward way, by applying the above resultsindividually to the filtering problems of the two investors in our economy. The dynamics of the individual beliefs are:

dm1(t) = (a0 + a1m1(t))dt + γ1(t)A′B−1dW 1

Y (t),

dm2(t) = (a0 + a1m2(t))dt + γ2(t)A′(BB′)−1(m1(t) − m2(t))dt + γ2(t)A′B−1dW 1

Y (t) ,

36For a review of Riccati equations, see Freiling (2002).

40

using the symmetry of B. The dynamics of Ψ(t) = B−1(m1(t) − m2(t)) follows, as:

dΨ(t) = B−1 (a1 + γ2(t)A′(BB′)−1) (m1(t) − m2(t))dt + B−1 (γ1(t) − γ2(t))A′B−1dW 1

Y (t)

= B−1(a1B + γ2(t)A′B−1

)Ψ(t)dt + B−1

(γ1(t) − γ2(t)

)A′B−1dW 1

Y (t),

with initial condition Ψ(0) = B−1(m1(0) − m2(0)). The solution of this stochastic differential condition is:

Ψ(t) = exp

{

−∫ t

0

M(s)ds

}

Ψ(0) +

∫ t

0

exp

{

−∫ s

0

M(u)du

}

B−1(γ1(s) − γ2(s)

)A′B−1dW 1

Y (s), (A-4)

where M(s) = B−1(a1B + γ2(s)A′B−1

). It follows that Ψ(t) is normally distributed as:

Ψ(t) ∼ N(

e−∫

t

0M(s)dsΨ(0),

∫ t

0

e−∫

s

0M(u)duB−1(γ1(s) − γ2(s))A′(BB′)−1A

(γ1(s) − γ2(s)

)′B−1e−

∫s

0M(u)duds

)

.

The parameter b = diag(σµA, σµz

) in the dynamics for γ(t) in Lemma 1 impacts the distribution of m(t) only indirectly, byinfluencing the Riccati differential equation for γ(t). Therefore, when we assume that this parameter is perceived identicallyby all investors, we can model a setting of rational Bayesian investors that can disagree because of different priors at time zero.If we assume that this parameter is perceived differently by some investor, we can model parsimoniously an economy withoverconfidence, in which, e.g., some investor perceives a lower variance for the expected consumption growth or the expectedchange in the signal. To this end, we just need to use an investor–dependent parameter bi = diag(σi

µA, σi

µz) in the matrix

Riccati differential equation for γi(t), where i = 1, 2.

For i = 1, 2, define:

γi(t) =

(γi

A(t) γiAz(t)

γiAz(t) γi

z(t)

)

. (A-5)

From the vector dynamics for Ψ(t), the dynamics for ΨA(t) and Ψz(t) read explicitly:

dΨA(t) =

((

a1A +γ2

A(t)

σ2A

)

ΨA(t) +αγ2

A(t) + βγ2Az(t)

σAσzΨz(t)

)

dt

+γ1

A(t) − γ2A(t)

σ2A

dW 1A(t) +

α(γ1A(t) − γ2

A(t)) + β(γ1Az(t) − γ2

Az(t))

σAσzdW 1

z (t) ,

and

dΨz(t) =

(γ2

Az(t)

σAσzΨA(t) +

(

a1z +αγ2

Az(t) + βγ2z (t)

σ2z

)

Ψz(t)

)

dt

+γ1

Az(t) − γ2Az(t)

σAσzdW 1

A(t) +α(γ1

Az(t) − γ2Az(t)) + β(γ1

z(t) − γ2z(t))

σ2z

dW 1z (t) .

Similarly, the dynamics for m1A(t) and m1

z(t) read explicitly:

dm1A(t) =

(a0A + a1Am1

A(t))dt +

γ1A(t)

σAdW 1

A(t) +αγ1

A(t) + βγ1Az(t)

σzdW 1

z (t) ,

and

dm1z(t) =

(a0z + a1zm1

z(t))dt +

γ1Az(t)

σAdW 1

A(t) +αγ1

Az(t) + βγ1z (t)

σzdW 1

z (t) .

The steady-state solution of the matrix Riccati differential equation (A-2) satisfies the symmetric algebraic matrix Riccatiequation:

0 = a1γ + γa′1 + bb′ − γA′(BB′)−1Aγ , (A-6)

41

which is equivalent to the following system of equations:

0 = 2a1AγA + σ2µA

− γ2A

σ2A

− (αγA + βγAz)2

σ2z

,

0 = 2a1zγz + σ2µz

− γ2Az

σ2A

− (αγAz + βγz)2

σ2z

,

0 = (a1A + a1z) γAz − γAzγA

σ2A

− αγAz (αγA + βγAz) + βγz (αγA + βγAz)

σ2z

.

In the special case β = 0, the solution to these equations takes the particularly simple form γAz = 0 and:

γA =

a1A +

√

a21A + σ2

µA

(σ2

z+α2σ2

A

σ2

Aσ2

z

)

(σ2

z+α2σ2

A

σ2

Aσz

) ,

γz = − σ2µz

2a1z.

This concludes the discussion of the disagreement dynamics in our model.

�

Appendix B. Proof of Proposition 1 (Equilibrium)

(i) Dynamics of the stochastic weighting process λ: Ito’s Lemma applied to η(t) = ξ1(t)/ξ2(t) gives:

dη(t) =dξ1(t)

ξ2(t)− ξ1(t)

(ξ1(t))2dξ2(t) +

1

2

2ξ1(t)

(ξ2(t))3(dξ2(t)

)2 − 1

(ξ2(t))2dξ2(t)dξ1(t).

Since markets are complete, there exists a unique stochastic discount factor for each agent. Absence of arbitrage implies fori = 1, 2:

dξi(t)

ξi(t)= −r(t)dt − θi(A(t), z(t))′dW i

Y ,

where θi = (θiA(t), θi

z(t))′ is the vector of market prices of risk perceived by agent i. It then follows,

dη(t) =ξ1(t)

ξ2(t)

dξ1(t)

ξ1(t)− ξ1(t)

ξ2(t)

dξ2(t)

ξ2(t)+

ξ1(t)

ξ2(t)

(dξ2(t)

ξ2(t)

)2

− 1

(ξ2(t))2dξ2(t)dξ1(t),

= η(t)

(

− r(t)dt − θ1A(t)dWA(t) − θ1

z(t)dWz(t) −(−r(t)dt − θ2

A(t)dWA(t) − θ2z(t)dWz(t)

)

+((

θ2A(t)

)2+(θ2

z(t))2 − θ1

A(t)θ2A(t) − θ1

z(t)θ2z(t)

)

dt

)

. (A-7)

The prices of the stock and the senior bond in our economy follow the dynamics:

dS(t) = S(t) (µS(t)dt + σSAdWA(t) + σSzdWz(t)) , (A-8)

dBS(t) = BS(t) (µBS (t)dt + σBSAdWA(t) + σBSzdWz(t)) , (A-9)

where S(t) is the price of equity and Bs(t) the price of the senior bond, and the expected growth rates µS(t) and µBs (t) andthe volatility coefficients σSA, σBSA, σSz and σBSz are determined in equilibrium. It is easily shown that the difference in theperceived rates of return have to satisfy the consistency condition:

µ1n(t) − µ2

n(t) = σn

(

ΨA(t), αΨA(t)σA

σz+ βΨz(t)

)′

,

where n denotes security n. The definition of market price of risk yields:

σnAθiA(t) + σnzθi

z(t) = µin(t) − r(t).

After some simple algebra, we obtain:

σnA(t)(θ1

A(t) − θ2A

)+ σnz(t)

(θ1

z(t) − θ2z(t)

)= σnA(t)ΨA(t) + σnz(t)

(

αΨA(t)σA

σz+ βΨz(t)

)

.

42

Since this equation has to hold for any σnA(t) and σnz(t), it follows:

θ1A(t) − θ2

A(t) = ΨA(t),

θ1z(t) − θ2

z(t) =

(

αΨA(t)σA

σz+ βΨz(t)

)

.

By construction, we also have:

dWA(t) =mi

A(t) − µA(t)

σAdt + dW i

A(t), dWz(t) =

(

αmi

A(t) − µA(t)

σz+ β

miz(t) − µz(t)

σz+ dW i

z(t)

)

.

Therefore, after substituting in equation (A-7), we get:

dη(t)

η(t)= −dW 1

A(t)ΨA(t) − θ1z(t)dW 1

z (t) + θ2z(t)

(

dW 1z (t) + αΨA(t)

σA

σz+ βΨz(t)

)

+((

θ1A(t) − ΨA(t)

)2+ θ2

z(t)(θ2

z(t) − θ1z(t)

)− θ1

A(t)(θ1

A(t) − ΨA(t)))

dt,

= −dW 1A(t)ΨA(t) − dW 1

z (t)

(

αΨA(t)σA

σz+ βΨz(t)

)

.

(ii) Representative investor optimization and optimal consumption policies: The representative agent in the econ-omy faces the following optimization problem:

supc1(t)+c2(t)=A(t)

U(c1(t), c2(t), λ(t)) =c1(t)

1−γ

1 − γ+ λ(t)

c2(t)1−γ

1 − γ, (A-10)

where λ(t) > 0. Optimality of individual consumption plans implies that the stochastic weight takes the following form:

λ(t) = u′(c1(t))/u′(c2(t)) = y1ξ1(t)/y2ξ

2(t),

where u′(c(t)) = c(t)−1/γ is the marginal utility function, which is assumed identical across agents. The first order conditionfor agent one is:

e−ρtc1(t)−γ = y1ξ

1(t).

The first order condition for agent two is:η(t)e−ρtc2(t)

−γ = y2ξ1(t).

The aggregate resource constraint can now be easily derived as:

(y2ξ

1(t)eρt

η(t)

)−1/γ

+(y1ξ

1(t)eρt)−1/γ

= A(t).

Thus, the solutions for the individual state price densities are:

ξ1(t) = e−ρt 1

y1A(t)−γ

(

1 + λ(t)1/γ)γ

, ξ2(t) = e−ρt 1

y2A(t)−γ

(

1 + λ(t)1/γ)γ

λ(t)−1.

To solve for the optimal consumption policy of each agent, we plug in the functional forms for the individual state pricedensities:

c1(t) = (y1ξ1(t)eρt)−1/γ = A(t)

(

1 + λ(t)1/γ)−1

.

Good’s market clearing, finally implies:

c2(t) = A(t) − c1(t) = A(t)λ(t)1/γ(

1 + λ(t)1/γ)−1

.

This concludes the proof of the Proposition.

�

43

Remark 1 (Equilibrium risk-less interest rate). The expression for the equilibrium interest rate can be now easily obtainedfrom the above results, using the same arguments as in Basak (2005):

r(t) = ρ + γ

(1

1 + λ(t)1/γµ1

A(t) +λ(t)1/γ

1 + λ(t)1/γµ2

A(t)

)

− 1

2(1 + γ)γσ2

A

+γ − 1

2γ

λ(t)1/γ

(1 + λ(t)1/γ)2

(

(αΨA(t)σA

σz+ βΨz(t))

2 + ΨA(t)2)

. (A-11)

In particular, one can see that for a relative risk aversion γ > 1 the contribution of disagreement to the equilibrium interestrate is positive.

Appendix C. Proof of Proposition 2 (Market Price of Risk)

The optimal consumption policy of agent i is:ci(t) = I(yiξ

i(t)),

where I(h) = h−1/γ is the inverse marginal utility function of consumption, which is identical across agents. Applying Ito’sLemma to this equation yields:

dci(t) =∂I(yiξ

i(t))

∂ξi(t)dξi(t) +

1

2

∂2I(yiξi(t))

∂(ξi(t))2

(

dξi(t))2

,

= µci(t)dt + σciA(t)dWA(t) + σciz(t)dWz(t).

Good market clearing and equation (A-12) imply:

c1(t)

γθ1

A(t) +c2(t)

γθ2

A(t) = σAA(t).

and

θ1A(t) =

(c1(t)

γ+

c2(t)

γ

)−1 (

σAA(t) + ΨA(t)c2(t)

γ

)

= γσA +c2(t)

A(t)ΨA(t).

The market price of risk of cash flow for the second agent follows easily as:

θ2A(t) = γσA − c1(t)

A(t)ΨA(t).

By inserting in these formulas the optimal consumption policies, the individual market prices of risk for cash flow risk are:

θ1A(t) = γσA + ΨA(t)

(

1 + λ(t)1/γ)−1

λ(t)1/γ , θ2A(t) = γσA − ΨA(t)

(

1 + λ(t)1/γ)−1

.

For the market prices of signal risk we first obtain, using good market clearing:

c1(t)

γθ1

z(t) +c2(t)

γθ2

z(t) = 0 .

Since, by construction:

θ1z(t) − θ2

z(t) =

(

αΨA(t)σA

σz+ βΨz(t)

)

,

we immediately obtain:

θ1z(t) =

c2(t)

A(t)

(

αΨA(t)σA

σz+ βΨz(t)

)

, θ2z(t) = − c1(t)

A(t)

(

αΨA(t)σA

σz+ βΨz(t)

)

.

By inserting in these formulas the form of the optimal consumption policies in Proposition 1, the market prices of signal riskfollow in closed form:

θ1z(t) =

(

1 + λ(t)1/γ)−1

λ(t)1/γ

(

αΨA(t)σA

σz+ βΨz(t)

)

, θ2z(t) = −

(

1 + λ(t)1/γ)−1

(

αΨA(t)σA

σz+ βΨz(t)

)

.

This concludes the proof.

�

44

Appendix D. Proof of Proposition 3 (Joint Laplace Transform of A(t) and η(t))

We want to compute the following moment-generating function

F(A, η, m1

A, ΨA, Ψz, t, u; ǫ, χ)

= EA,η,m1

A,ΨA,Ψz

(A(u)ǫη(u)χ) .

This function satisfies the following partial differential equation (PDE):

0 ≡ DF(A, η, m1

A, ΨA, Ψz, t, u; ǫ, χ)

+∂F

∂t

(A, η, m1

A, ΨA, Ψz, t, u; ǫ, χ), (A-12)

with the initial condition F(A, η, m1

A, ΨA, Ψz, t, t; ǫ, χ)

= Aǫηχ, and where D is the differential generator of the multivariateprocess

(A(t), η(t), m1

A(t),ΨA(t), Ψz(t))

under the probability measure of agent 1. Spelling out Feynman-Kac (A-13), we get

0 =∂F

∂AAm1

A +∂F

∂m1A

(a0A + a1Am1

A

)+

∂F

∂ΨA

((

a1A +γ2

A

σ2A

)

ΨA +

(αγ2

A + βγ2Az

σAσz

)

Ψz

)

+∂F

∂Ψz

((

a1z +αγ2

Az + βγ2z

σ2z

)

Ψz +γ2

Az

σAσzΨA

)

+1

2

∂2F

∂A2(AσA)2 +

1

2

∂2F

∂ (m1A)2

((γ1

A

σA

)2

+

(αγ1

A + βγ1Az

σz

)2)

+1

2

∂2F

(∂ΨA)2

((γ1

A − γ2A

σ2A

)2

+

(

α(γ1

A − γ2A

)+ β

(γ1

Az − γ2Az

)

σAσz

)2)

+1

2

∂2F

(∂Ψz)2

((γ1

Az − γ2Az

σAσz

)2

+

(

α(γ1

Az − γ2Az

)+ β

(γ1

z − γ2z

)

σ2z

)2)

+1

2

∂2F

(∂η)2η2

(

Ψ2A +

(

ασA

σzΨA + βΨz

)2)

+∂2F

∂A∂m1A

γ1AA +

∂2F

∂A∂ΨA

(γ1

A − γ2A

σA

)

A +∂2F

∂A∂Ψz

(γ1

Az − γ2Az

σz

)

A − ∂2F

∂A∂ηAηΨAσA

+∂2F

∂m1A∂ΨA

(

γ1A

(γ1

A − γ2A

)

σ3A

+

(αγ1

A + βγ1Az

σz

)(

α(γ1

A − γ2A

)+ β

(γ1

Az − γ2Az

)

σAσz

))

+∂2F

∂m1A∂Ψz

(

γ1A

(γ1

Az − γ2Az

)

σ2Aσz

+

(αγ1

A + βγ1Az

σz

)(

α(γ1

Az − γ2Az

)+ β

(γ1

z − γ2z

)

σ2z

))

− ∂2F

∂m1A∂η

η

(

ΨAγ1

A

σA+

(αγ1

A + βγ1Az

σz

)(

αΨAσA

σz+ βΨz

))

− ∂2F

∂ΨA∂ηη

(

ΨAγ1

A − γ2A

σ2A

+

(

α(γ1

A − γ2A

)+ β

(γ1

Az − γ2Az

)

σAσz

)(

αΨAσA

σz+ βΨz

))

− ∂2F

∂Ψz∂ηη

(

ΨAγ1

Az − γ2Az

σzσA+

(

α(γ1

Az − γ2Az

)+ β

(γ1

z − γ2z

)

σ2z

)(

αΨAσA

σz+ βΨz

))

+∂2F

∂ΨA∂Ψz

((γ1

A − γ2A

σ2A

)(γ1

Az − γ2Az

σAσz

)

+

(

α(γ1

A − γ2A

)+ β

(γ1

Az − γ2Az

)

σAσz

)(

α(γ1

Az − γ2Az

)+ β

(γ1

z − γ2z

)

σ2z

))

+∂F

∂t.

The solution to this PDE takes the functional form

F (A, η, m1A, ΨA, Ψz, t, u; ǫ, χ) = AǫηχFm1

A

(m1

A, t, u; ǫ)FΨA,Ψz

(ΨA, Ψz, t, u; ǫ, χ) =: AǫηχF(m1

A, ΨA, Ψz, t, u; , ǫ, χ).

45

Plugging in this expression into equation (A-13), yields:

0 = F ǫm1A +

∂F

∂m1A

(a0A + a1Am1

A

)+

∂F

∂ΨA

((

a1A +γ2

A

σ2A

)

ΨA +

(αγ2

A + βγ2Az

σAσz

)

Ψz

)

+∂F

∂Ψz

((

a1z +αγ2

Az + βγ2z

σ2z

)

Ψz +γ2

Az

σAσzΨA

)

+1

2ǫ(ǫ − 1)F σ2

A +1

2

∂2F

∂ (m1A)2

((γ1

A

σA

)2

+

(αγ1

A + βγ1Az

σz

)2)

+1

2

∂2F

(∂ΨA)2

((γ1

A − γ2A

σ2A

)2

+

(

α(γ1

A − γ2A

)+ β

(γ1

Az − γ2Az

)

σAσz

)2)

+1

2

∂2F

(∂Ψz)2

((γ1

Az − γ2Az

σAσz

)2

+

(

α(γ1

Az − γ2Az

)+ β

(γ1

z − γ2z

)

σ2z

)2)

+1

2χ(χ − 1)F

(

Ψ2A +

(

ασA

σzΨA + βΨz

)2)

+∂F

∂m1A

ǫγ1A +

∂F

∂ΨA

(γ1

A − γ2A

σA

)

ǫ +∂F

∂Ψz

(γ1

Az − γ2Az

σz

)

ǫ − ǫχFΨAσA

+∂2F

∂m1A∂ΨA

(

γ1A

(γ1

A − γ2A

)

σ3A

+

(αγ1

A + βγ1Az

σz

)(

α(γ1

A − γ2A

)+ β

(γ1

Az − γ2Az

)

σAσz

))

+∂2F

∂m1A∂Ψz

(

γ1A

(γ1

Az − γ2Az

)

σ2Aσz

+

(αγ1

A + βγ1Az

σz

)(

α(γ1

Az − γ2Az

)+ β

(γ1

z − γ2z

)

σ2z

))

− ∂F

∂m1A

χ

(

ΨAγ1

A

σA+

(αγ1

A + βγ1Az

σz

)(

αΨAσA

σz+ βΨz

))

− ∂F

∂ΨAχ

(

ΨAγ1

A − γ2A

σ2A

+

(

α(γ1

A − γ2A

)+ β

(γ1

Az − γ2Az

)

σAσz

)(

αΨAσA

σz+ βΨz

))

− ∂F

∂Ψzχ

(

ΨAγ1

Az − γ2Az

σzσA+

(

α(γ1

Az − γ2Az

)+ β

(γ1

z − γ2z

)

σ2z

)(

αΨAσA

σz+ βΨz

))

+∂2F

∂ΨA∂Ψz

((γ1

A − γ2A

σ2A

)(γ1

Az − γ2Az

σAσz

)

+

(

α(γ1

A − γ2A

)+ β

(γ1

Az − γ2Az

)

σAσz

)(

α(γ1

Az − γ2Az

)+ β

(γ1

z − γ2z

)

σ2z

))

+∂F

∂t.

We can first factor out the expressions that do not involve η and Ψ (given in blue) and solve for Fm1

A

by direct integration.To this end, we guess the following functional form:

Fm1

A

(m1A, t, u, ǫ) = exp(A(ǫ, u − t)m1

A + C(ǫ, u − t)),

with the explicit solutions for A(ǫ, τ ) and C(ǫ, τ ) given by:

A(ǫ, u − t) =ǫ(

ea1A(u−t) − 1)

a1A,

C(ǫ, u − t) =1

2ǫ (ǫ − 1) σA (u − t) +

1

a1A

(a0A + ǫγ1

A

) (

e−a1A(u−t) + u − t)

+1

a1A

((γ1

A

σA

)2

+

(αγ1

A + βγ1Az

σz

)2)(

3

2ea1A(u−t) − a1A (u − t)

)

.

Next, we guess the following functional form:

FΨA,Ψz(ΨA, Ψz, t, ǫ, χ, u) = exp

(

A0(ǫ, χ, u − t) + B1(ǫ, χ, u − t)ΨA + B2(ǫ, χ, u − t)Ψz

+C1(ǫ, χ, u − t)Ψ2A + C2(ǫ, χ, u − t)Ψ2

z + D0(ǫ, χ, u − t)ΨA(t)Ψz

)

.

46

From this guess, we obtain the derivatives:

∂F

∂ΨA= F (B1(u − t) + 2C1(u − t)ΨA + D0(u − t)Ψz) ,

∂2F

∂Ψ2A

= F((B1(u − t) + 2C1(u − t)ΨA + D0(u − t)Ψz)

2 + 2C1(u − t)),

∂F

∂Ψz= F (B2(u − t) + 2C2(u − t)Ψz + D0(u − t)ΨA) ,

∂2F

∂Ψ2z

= F((B2(u − t) + 2C2(u − t)Ψz + D0(u − t)ΨA)2 + 2C2(u − t)

),

∂F

∂ΨAΨz= F ((B1(u − t) + 2C1(u − t)ΨA + D0(u − t)Ψz) (B2(u − t) + 2C2(u − t)Ψz + D0(u − t)ΨA) + D0(u − t)) ,

∂F

∂t= −F

(A′

0(u − t) + B′1(u − t)ΨA + B′

2(u − t)Ψz + C′1(u − t)Ψ2

A + C′2(u − t)Ψ2

z + D′0(u − t)ΨAΨz

),

which, plugged–in into the initial differential equation imply:

0 = (B1(τ ) + 2C1(τ )ΨA + D0(τ )Ψz)

((

a1A +γ2

A

σ2A

)

ΨA +

(αγ2

A + βγ2Az

σAσz

)

Ψz

)

+ (B2(τ ) + 2C2(τ )Ψz + D0(τ )ΨA)

((

a1z +αγ2

Az + βγ2z

σ2z

)

Ψz +γ2

Az

σAσzΨA

)

+

+1

2

((B1(τ ) + 2C1(τ )ΨA + D0(τ )Ψz)

2 + 2C1(τ ))

((γ1

A − γ2A

σ2A

)2

+

(

α(γ1

A − γ2A

)+ β

(γ1

Az − γ2Az

)

σAσz

)2)

+1

2

((B2(τ ) + 2C2(τ )Ψz + D0(τ )ΨA)2 + 2C2(τ )

)

((γ1

Az − γ2Az

σAσz

)2

+

(

α(γ1

Az − γ2Az

)+ β

(γ1

z − γ2z

)

σ2z

)2)

+1

2χ(χ − 1)

(

Ψ2A +

(

ασA

σzΨA + βΨz

)2)

+ (B1(τ ) + 2C1(τ )ΨA + D0(τ )Ψz)

(γ1

A − γ2A

σA

)

ǫ + (B2(τ ) + 2C2(τ )Ψz + D0(τ )ΨA)

(γ1

Az − γ2Az

σz

)

ǫ − ǫχΨAσA

+A(ǫ, τ ) (B1(τ ) + 2C1(τ )ΨA + D0(τ )Ψz)

(

γ1A

(γ1

A − γ2A

)

σ3A

+

(αγ1

A + βγ1Az

σz

)(

α(γ1

A − γ2A

)+ β

(γ1

Az − γ2Az

)

σAσz

))

+A(ǫ, τ ) (B2(τ ) + 2C2(τ )Ψz + D0(τ )ΨA)

(

γ1A

(γ1

Az − γ2Az

)

σ2Aσz

+

(αγ1

A + βγ1Az

σz

)(

α(γ1

Az − γ2Az

)+ β

(γ1

z − γ2z

)

σ2z

))

−A(ǫ, τ )χ

(

ΨAγ1

A

σA+

(αγ1

A + βγ1Az

σz

)(

αΨAσA

σz+ βΨz

))

− (B1(τ ) + 2C1(τ )ΨA + D0(τ )Ψz)χ

(

ΨAγ1

A − γ2A

σ2A

+

(

α(γ1

A − γ2A

)+ β

(γ1

Az − γ2Az

)

σAσz

)(

αΨAσA

σz+ βΨz

))

−((B2(τ ) + 2C2(τ )Ψz + D0(τ )ΨA)χ

(

ΨAγ1

Az − γ2Az

σzσA+

(

α(γ1

Az − γ2Az

)+ β

(γ1

z − γ2z

)

σ2z

)(

αΨAσA

σz+ βΨz

))

+ ((B1(τ ) + 2C1(τ )ΨA + D0(τ )Ψz) (B2(τ ) + 2C2(τ )Ψz + D0(τ )ΨA) + D0(τ ))

×((

γ1A − γ2

A

σ2A

)(γ1

Az − γ2Az

σAσz

)

+

(

α(γ1

A − γ2A

)+ β

(γ1

Az − γ2Az

)

σAσz

)(

α(γ1

Az − γ2Az

)+ β

(γ1

z − γ2z

)

σ2z

))

−(A′

0(τ ) + B′1(τ )ΨA + B′

2(τ )Ψz + C′1(τ )Ψ2

A + C′2(τ )Ψ2

z + D′0(τ )ΨAΨz

).

47

It follows that functions A0, B1, B2, C1, C2 and D0 must solve the following system of ODEs

C′1(τ ) = 2a1C

21 (τ ) + 2b1C1(τ ) +

1

2a1D

20(τ ) + b1D0(τ ) + 2c1C1(τ )D0(τ ) + d1, (A-13)

C′2(τ ) = 2a1C

22 (τ ) + 2b2C2(τ ) +

1

2a1D

20(τ ) + b2D0(τ ) + 2c1C2(τ )D0(τ ) + d2, (A-14)

D′0(τ ) = c1D

20(τ ) +

1

2(b1 + b2) D0(τ ) + 2b1C2(τ ) + 2b2C1(τ ) + 2a1C1(τ )D0(τ ) + 2a1C2(τ )D0(τ )

+4c1C1(τ )C2(τ ) + d5, (A-15)

B′1(τ ) = b1B1(τ ) + b1B2(τ ) + a1B1(τ )C1(τ ) + a1B2D0(τ ) + c1B1(τ )D0(τ ) + 2c1C1(τ )B2(τ )

+d3 − ea1Aτ d3 + C1(τ )(

b3ea1Aτ − b3

)

+ D0(τ ) (c3ea1Aτ − c3) + d3, (A-16)

B′2(τ ) = b2B2(τ ) + b1B1(τ ) + a1B1(τ )D0(τ ) + a1B2C2(τ ) + (2B1(τ )C2(τ ) + B2(τ )D0(τ )) c1

+C2(τ )(

b3ea1Aτ − b3

)

+ D0(τ ) (c3ea1Aτ − c3) + d4 (ea1Aτ − 1) , (A-17)

A′0(τ ) =

1

2a1B

21(τ ) +

1

2a1B

22(τ ) + a1C2(τ ) + b3B1(τ ) +

1

2c3B2(τ ) + c1B1(τ )B2(τ ), (A-18)

subject to the initial condition:

C1(0) = C2(0) = B1(0) = B2(0) = D0(0) = A0(0) = 0.

48

In these equations, the coefficients are given explicitly by:

a1 =

(γ1

A − γ2A

σ2A

)2

+

(

α(γ1

A − γ2A

)+ β

(γ1

Az − γ2Az

)

σAσz

)2

,

b1 = a1A +γ2

A

σ2A

− χ

(

γ1A − γ2

A

σA+

ασA

σz

(

α(γ1

A − γ2A

)+ β

(γ1

Az − γ2Az

)

σAσz

))

,

a1 =

(γ1

Az − γ2Az

σAσz

)2

+

(

α(γ1

Az − γ2Az

)+ β

(γ1

z − γ2z

)

σ2z

)2

,

b1 =γ2

Az

σAσz− χ

(

γ1Az − γ2

Az

σAσz+

(ασA

σz

)(

α(γ1

Az − γ2Az

)+ β

(γ1

z − γ2z

)

σ2z

))

,

c1 =

(γ1

A − γ2A

σ2A

)(γ1

Az − γ2Az

σAσz

)

+

(

α(γ1

A − γ2A

)+ β

(γ1

Az − γ2Az

)

σAσz

)(

α(γ1

Az − γ2Az

)+ β

(γ1

z − γ2z

)

σ2z

)

,

d1 =1

2χ (χ − 1)

(

1 +

(ασA

σz

)2)

,

b2 = a1z +αγ2

Az − χβγ2Az

σ2z

+ β

(

α(γ1

Az − γ2Az

)+ β

(γ1

z − γ2z

)

σ2z

)

,

b2 =αγ2

A + βγ2Az

σAσz− χβ

(

α(γ1

A − γ2A

)+ β

(γ1

Az − γ2Az

)

σAσz

)

,

d2 =1

2χ (χ − 1) β2,

b3 = ǫ

(γ1

A − γ2A

σA

)

− ǫ

a1A

(

γ1A

(γ1

A − γ2A

)

γ3A

+

(αγ1

A + βγ1Az

σz

)(

α(γ1

A − γ2A

)+ β

(γ1

Az − γ2Az

)

σAσz

))

,

b3 =ǫ

a1A

(

γ1A

(γ1

A − γ2A

)

σ3A

+

(αγ1

A + βγ1Az

σz

)(

α(γ1

A − γ2A

)+ β

(γ1

Az − γ2Az

)

σAσz

))

,

c3 = 2ǫ

(γ1

Az − γ2Az

σz

)

− 2ǫ

a1A

(

γ1A

(γ1

Az − γ2Az

)

σ2Aσz

+

(αγ1

A + βγ1Az

σz

)(

α(γ1

Az − γ2Az

)+ β

(γ1

z − γ2z

)

σ2z

))

,

c3 =2ǫ

a1A

(

γ1A

(γ1

Az − γ2Az

)

σ2Aσz

+

(αγ1

A + βγ1Az

σz

)(

α(γ1

Az − γ2Az

)+ β

(γ1

z − γ2z

)

σz

))

,

d3 = ǫχ

(

σA − 1

a1A

(γ1

A

σA+

αγ1A + βγ1

Az

σz

)(ασA

σz

))

,

d3 = − ǫχ

a1a

(γ1

A

σA+

(αγ1

A + βγ1Az

σz

)(ασA

σz

))

,

d5 =ασA

σzχ (χ − 1) .

To solve the system of equations (A-13)-(A-18), we first solve equations (A-13)-(A-15). First, we observe that the system ofdifferential equations (A-13)-(A-15) can be written as the following matrix Riccati equation:

dA

dτ= AM ′MA + AP + P ′A + D (A-19)

with coefficient matrices defined by:

A =

(C1 D0

D0 C2

)

, M =

(2√

a1 0

c1√a1

√

2a1 − c21

a1

)

, P =

(2b1 b2

b1 2b2

)

D =

(d1 d5

d5 d1

)

.

The matrix Riccati equation (A-19) can be solved in closed form by transforming it into a locally equivalent linear system ofordinary differential equations by a homogenization procedure (Radon’s Lemma). Let

(C11(t) C12(t)C21(t) C22(t)

)

= exp

(

t

(P M ′MD −P ′

))

.

49

Then, the solution of differential equation (A-19) is:

A(t) = C22(t)−1C21(t), (A-20)

using the fact that A(0) = 0. The solutions for B1, B2, and A0 follow by direct integration. This concludes the proof.

�

Appendix E. Proof of Proposition 4 (Security Prices)

By definition, the risk-less zero coupon bond price is given by:

B(t, T ) =1

ξ1(t)E1

t

(

e−ρ(T−t)ξ1(T ))

.

Using the expression for ξ1(t) in Proposition 1, we get:

B(t, T ) = E1t

(

e−ρ(T−t)

(A(T )

A(t)

)−γ (1 + λ(T )1/γ

1 + λ(t)1/γ

)γ)

. (A-21)

Let

G(t, T, x;ΨA, Ψz) ≡∫ ∞

0

(1 + λ(T )1/γ

1 + λ(t)1/γ

)γ[

1

2π

∫ +∞

−∞

(λ(T )

λ(t)

)−iχ

FΨA,Ψz(ΨA, Ψz, t, T ;−γ, iχ)dχ

]

dλ(T )

λ(T ).

By Fourier inversion, it then follows:

B(t, T ) = e−ρ(T−t)Fm1

A

(m1A, t, T ;−γ)G(t, T,−γ;ΨA, Ψz).

In a similar way, the firm value is:

V (t) = E1t

(∫ ∞

t

e−ρ(u−t) ξ1(u)

ξ1(t)A(u)du

)

,

= A(t)E1t

(∫ ∞

t

e−ρ(u−t)

(1 + λ(t)1/γ

1 + λ(t)1/γ

)γ (A(u)

A(t)

)1−γ

du

)

,

= A(t)

∫ ∞

t

(

e−ρ(u−t)Fm1

A

(m1A, t, u; 1 − γ)G(u, T, 1 − γ; ΨA, Ψz)

)

du.

The price of the senior bond is:

Bs(t, T ) = K1B(t, T ) − E1t

(

e−ρ(T−t) ξ1(T )

ξ1(t)(K1 − V (T ))+

)

,

= K1B(t, T ) − E1t

(

e−ρ(T−t)

(A(T )

A(t)

)−γ (1 + λ(T )1/γ

1 + λ(t)1/γ

)γ

(K1 − V (T ))+)

,

= K1B(t, T ) − P (t, T, K1),

where P (t, T, K1) is the price of the put option on the firm value. The price of the junior bond is:

Bj(t, T ) = E1t

(

e−ρ(T−t) ξ1(T )

ξ1(t)(V (T ) − K1)

+

)

− E1t

(

e−ρ(T−t) ξ1(T )

ξ1(t)(V (T ) − (K1 + K2))

+

)

,

= C(t, T, K1) − C(t, T, K1 + K2),

where C(t, T, K1) and C(t, T, K1 + K2) are call options on the firm value with strikes K1 and K1 + K2, respectively. Equityin our economy is a call option on the firm value with strike price K1 + K2. Therefore:

S(t) = E1t

(

e−ρ(T−t) ξ1(T )

ξ1(t)(V (T ) − (K1 + K2))

+

)

= C(t, T, K1 + K2).

A European call option on the equity value is derived in the following way:

O(t, T ) = E1t

(

e−ρ(T−t) ξ1(T )

ξ1(t)(S(T ) − Ke)

+

)

.

50

This concludes the proof.

�

Appendix F. Risk-neutral Skewness and Volatility Formulas

It follows from Bakshi and Madan (2000) that the entire collection of twice-continuously differentiable payoff functions withbounded expectation can be spanned algebraically. Applying this result to the firm value V (t) (or equivalently to the stockvalue S(t)) yields:

G(V ) = G(V ) + (V − V )GV (V ) +

∫ ∞

V

GV V (K)(V − K)+dK +

∫ V

0

GV V (K)(K − V )+dK,

where GV is the partial derivative of the payoff function G(V ) with respect to V and GV V the corresponding second-orderpartial derivative. By setting V = V (t) we obtain the final formula for the firm value risk-neutral skewness, after applyingthe same steps as in Bakshi, Kapadia, and Madan (2003) (Theorem 1, p. 137).

Proposition 5. Let v(t, T ) = ln (V (t + T )) − ln (V (t)) be the firm value return between time t and T . The risk-neutralskewness of v(t, T ) is given by:

skew(t, T ) =Et

((v(t, T ) − Et (v(t, T )))3

)

(Et (v(t, T ) − Et (v(t, T )))2

)3/2=

erT W (t, T ) − 2µ(t, T )erT R(t, T ) + 2µ(t, T )3

(erT R(t, T ) − µ(t, T )2)3/2,

where

R(t, T ) =

∫ ∞

V (t)

2(

1 − ln(

KV (t)

))

K2(V (T ) − K)+ dK +

∫ V (t)

0

2(

1 + ln(

V (t)K

))

K2(K − V (T ))+ dK,

W (t, T ) =

∫ ∞

V (t)

6 ln(

KV (t)

)

− 3(

ln(

KV (t)

))2

K2(V (T ) − K)+ dK

−∫ V (t)

0

6 ln(

V (t)K

)

− 3(

ln(

V (t)K

))2

K2(K − V (T ))+ dK ,

and

X(t, T ) =

∫ ∞

V (t)

12(

ln(

KV (t)

))2

− 4(

ln(

KV (t)

))3

K2(V (T ) − K)+ dK

−∫ V (t)

0

12(

ln(

V (t)K

))2

− 4(

ln(

V (t)K

))3

K2(K − V (T ))+ dK,

µ(t, T ) = Et

(

ln

(V (t + T )

V (t)

))

≈ erT − 1 − erT

2R(t, T ) − erT

6W (t, T ) − erT

24X(t, T ).

The expression for the stock returns volatility in the paper follows from a straight-forward application of Ito’s Lemma.37

The price of the stock, defined in equation (A-8), satisfies a diffusion process given by:

dS

S= µ1

S(t)dt + σSA(t)dW 1A(t) + σSz(t)dW 1

z (t),

37With the same procedure one also obtains the volatility of firm value returns.

51

Therefore, the diffusion term in this dynamics is characterized by:

dS(t) − S(t)µ1S(t)dt =

∂S

∂A(dA(t) − E1

t (dA(t))) +∂S

∂m1A

(dm1A(t) − E1

t (dm1A(t)))

+∂S

∂ΨA(dΨA(t) − E1

t (dΨA(t))) +∂S

∂Ψz(dΨz(t) − E1

t (dΨz(t)))

=∂S

∂AA(t)σAdW 1

A(t) +∂S

∂m1A

(γ1

A(t)

σAdW 1

A(t) +

(αγ1

A(t) + βγ1Az(t)

σz

)

dW 1z (t)

)

+∂S

∂ΨA

(γ1

A(t) − γ2A(t)

σ2A

dW 1A(t) +

(α(γ1

A(t) − γ2A(t)) + β(γ1

Az(t) − γ2Az(t))

σAσz

)

dW 1z (t)

)

+∂S

∂Ψz

(γ1

Az(t) − γ2Az(t)

σAσzdW 1

A(t) +

(α(γ1

Az(t) − γ2Az(t)) + β(γ1

z(t) − γ2z (t))

σ2z

)

dW 1z (t)

)

.

By matching coefficients, we obtain:

σSA(t) =1

S(t)

(∂S

∂AA(t)σA +

∂S

∂m1A

γ1A(t)

σA+

∂S

∂ΨA

γ1A(t) − γ2

A(t)

σ2A

+∂S

∂Ψz

γ1Az(t) − γ2

Az(t)

σAσz

)

and

σSz(t) =1

S(t)

(∂S

∂m1A

αγ1A(t) + βγ1

Az(t)

σz+

∂S

∂ΨA

α(γ1A(t) − γ2

A(t)) + β(γ1Az(t) − γ2

Az(t))

σAσz

)

+1

S(t)

∂S

∂Ψz

α(γ1Az(t) − γ2

Az(t)) + β(γ1z (t) − γ2

z (t))

σ2z

Thus, the volatility of stock returns at time t is (σ2SA(t) + σ2

Sz(t))1/2, with σSA and σSz given above. For the special case, in

which β = 0, the second of these expressions simplifies:

σSz(t) =1

S(t)

(∂S

∂m1A

αγ1A(t)

σz+

∂S

∂ΨA

α(γ1A(t) − γ2

A(t))

σAσz+

∂S

∂Ψz

α(γ1Az(t) − γ2

Az(t))

σ2z

)

.

This concludes the discussion of the risk-neutral skewness and the volatility of stock returns in our model.

52

Table 1

Choice of Parameter Values and Benchmark Values of State Variables

This table lists the parameter values used for all figures in the paper. We calibrate to the mean and volatility of the time-series averageof operating cash flow for all firms present in our database. Operating cash flow is earnings before extraordinary items (Compustat item18) minus total accruals, scaled by average total assets (Compustat item 6), where total accruals are equal to changes in current assets(Compustat item 4) minus changes in cash (Compustat item 1), changes in current liabilities (Compustat item 5), and depreciationexpense (Compustat item 14) plus changes in short-term debt (Compustat item 34). The initial values for the conditional variances areset to their steady-state variances. Agent specific values are consistent with estimated values from Brennan and Xia (2001). In our plotswe set α equal to 1 and β equal to zero.

Name Symbol Value

Parameters for Cash FlowLong-term growth rate of cash flow growth a0A 0.01

Mean-reversion parameter of cash flow growth a1A -0.01

Volatility of cash flow σA 0.07

Initial level of cash flow A 1.00

Initial level of cash flow growth m1A 0.01

Parameters for Signal

Long-term growth rate of signal a0z 0.01

Mean-reversion parameter of signal a1z -0.03

Volatility of signal σz 0.06Agent specific Parameters

Relative risk aversion for both agents γ 2

Time Preference Parameter ρ 0.02Firm specific Parameters

Face Value of Senior Bond K1 ∈ [5, 50]

Strike Price of Equity Option Ke ∈ [40, 100]

53

Table 2

Summary Statistics

We report summary statistics of the main variables used in the analysis. The data runs from January 1996 to December 2004, withmonthly frequency. Credit spreads are reported in basis points and taken from the FISD Mergent database. Dispersion is defined as theratio of the average absolute difference of analysts’ forecasts and the standard deviation of these forecasts, retrieved from the I/B/E/Sdatabase. The option’s implied volatility is for at-the-money options taken from the OptionMetrics database. The implied volatilityskew is calculated as the left slope of the implied volatility smile, which is defined as the difference of the implied volatility of a putoption with a 0.92 strike to spot ratio and the implied volatility of an at-the-money put, divided by the difference in strike to spotratio. Leverage is defined as total debt divided by the sum of total debt and the book value of shareholders’ equity and data is fromCOMPUSTAT. Reported numbers are time-series means of cross-sectional averages. Correlations are computed as time-series correlationof cross-sectional averages.

Variable Mean Max Min Median StDev Corr Senior Corr Junior

Senior Spread 124.939 177.640 74.404 131.640 34.398 1.00 0.85Junior Spread 134.432 186.060 67.290 142.945 34.237 0.85 1.00Dispersion 0.246 0.821 0.014 0.224 0.143 0.18 0.20Implied Vola 0.368 0.527 0.250 0.352 0.082 0.45 0.33Implied Vola Skew 0.202 0.633 -0.155 0.215 0.159 -0.46 -0.62Stock Returns 0.010 0.051 -0.090 0.003 0.024 -0.01 0.09Leverage Senior 0.041 0.780 0.010 0.039 0.080 0.78 0.78

54

Table 3

OLS Panel Regression Results for Credit Spreads

Using data running from January 1996 to December 2004, we regress credit spreads on corporate bonds on a set of variables listed below.⋆ denotes significance at the 10% level, ⋆⋆ denotes significants at the 5% level and ⋆ ⋆ ⋆ denotes significance at the 1% level. Model(1) corresponds to the regression model with option-implied determinants (see Cremers, Driessen, and Maenhout (2006)). Model (2)corresponds to determinants used in Collin-Dufresne, Goldstein, and Martin (2001). Model (3) corresponds to firm specific variables,Model (4) takes into account liquidity concerns, Model (5) corresponds to the model of Schaefer and Strebulaev (2005) and Model(6) includes all determinants. Model (7) is the full model without the disagreement proxy. All estimations use autocorrelation andheteroscedasticity-consistent t-statistics.

Model (1) (2) (3) (4) (5) (6) (7)

Senior Credit SpreadConstant 140.57⋆⋆⋆ 281.03⋆⋆⋆ −288.22⋆⋆⋆ 217.70⋆⋆⋆ 183.21⋆⋆⋆ −148.51⋆⋆⋆ 111.21⋆⋆

Dispersion 22.72⋆⋆ 16.69⋆⋆ 23.44⋆⋆⋆ 13.51⋆⋆ 28.91⋆⋆⋆ 15.68⋆⋆⋆

Implied Volatility 49.49 168.45 14.12 26.43⋆⋆⋆

Implied Volatility Skew −13.94 -6.16 −15.09⋆⋆

Open Interest −0.17⋆⋆⋆ −0.13⋆⋆⋆ −0.10⋆⋆⋆ −0.15⋆⋆⋆

Volume 0.02⋆⋆⋆ 0.01⋆⋆⋆ 0.03⋆⋆⋆ 0.03⋆⋆⋆

Slope of Term Structure −6.06⋆ 3.65 -5.98 13.68Risk-free Rate −25.42⋆⋆⋆ −22.85⋆⋆⋆ -7.95 -10.64S&P 500 Returns -25.00 -3.91 -7.75Non-Farm Payroll (/1000) −9.12⋆⋆⋆ −3.74⋆⋆⋆ −3.40⋆⋆⋆

Stock Returns −24.73 18.43 −30.12 -33.78Stock Volume 10.35 8.85 5.25Leverage (/1000) 3.06⋆⋆⋆ 1.83⋆⋆⋆ 2.06⋆⋆⋆

Firm Size (/100) 1.06⋆⋆⋆ 0.89⋆⋆⋆ 0.57⋆⋆⋆

Swap Rate (/100) −8.98⋆⋆⋆ −0.38 -0.37Rm − Rf -0.57 -0.07 0.00SMB 0.84⋆ 0.14 0.05HML 0.00 0.18 0.04Adjusted R2 0.77 0.69 0.80 0.81 0.79 0.91 0.87

Full SampleConstant 170.18⋆⋆⋆ 226.20⋆⋆⋆ −277.15⋆⋆⋆ 219.82⋆⋆⋆ 129.96⋆⋆⋆ 54.35 69.45⋆

Dispersion 25.06⋆⋆⋆ 16.08⋆⋆ 25.35⋆⋆⋆ 20.45⋆⋆⋆ 27.15⋆⋆⋆ 16.27⋆⋆⋆

Implied Volatility 46.21 163.97 12.63 26.80⋆⋆

Implied Volatility Skew −25.57⋆ −26.53 −30.72⋆⋆

Open Interest −0.11⋆⋆⋆ −0.10⋆⋆⋆ −0.04⋆⋆⋆ −0.04⋆⋆⋆

Volume 0.00 0.01⋆⋆⋆ 0.02⋆⋆⋆ 21.85⋆⋆

Slope of Term Structure −2.38 12.38⋆⋆⋆ −1.24 −1.74Risk-free Rate −17.31⋆⋆⋆ −14.89⋆⋆ -6.89 -0.09S&P 500 Returns -23.49 -10.11 -9.78Non-Farm Payroll (/1000) −9.09⋆⋆⋆ −3.92⋆⋆⋆ −4.07⋆⋆⋆

Stock Returns −23.54 25.54 −16.19 -30.42Stock Volume 9.29 9.78⋆ 9.57⋆

Leverage (/ 1000) 3.06⋆⋆⋆ 0.97⋆⋆ 1.01⋆⋆

Firm Size (/ 100) 1.27⋆⋆⋆ 0.35 0.32⋆⋆⋆

Swap Rate (/ 100) −0.96⋆⋆⋆ -0.14 -2.06Rm − Rf -0.63 -0.01 -0.04SMB 1.10⋆⋆⋆ 0.33 0.32HML 0.53⋆⋆⋆ 0.49⋆⋆ 0.49⋆

Adjusted R2 0.76 0.72 0.49 0.84 0.78 0.92 0.92

55

Table 4

OLS Panel Regression Results for Stock Returns and Implied Volatility

Using data running from January 1996 to December 2004, we regress stock returns and implied volatility on a set of variables listedbelow. ⋆ denotes significance at the 10% level, ⋆⋆ denotes significants at the 5% level and ⋆ ⋆ ⋆ denotes significance at the 1% level. Allestimations use autocorrelation and heteroskedasticity-consistent t-statistics.

Stock Returns Implied Volatility

Constant 0.01⋆⋆⋆ 0.45⋆⋆⋆

Dispersion 0.01⋆⋆⋆ 0.50⋆⋆⋆

Implied Volatility −0.01Implied Volatility Skew (left) −0.02 0.12⋆⋆⋆

Implied Volatility Skew (right) −0.03⋆⋆⋆

Call Volume −0.01⋆⋆ -0.07Put Volume 0.00 0.08Open Interest -0.00 −0.00⋆⋆⋆

S&P 500 Returns −0.03⋆⋆

Non-Farm Payroll 1.08Leverage 0.42 1.14⋆⋆⋆

Rm − Rf 0.00⋆⋆⋆

SMB 0.00⋆⋆⋆

HML 0.00⋆⋆⋆

IV - RV 0.16⋆⋆⋆

Trading Pressure −0.14⋆⋆⋆

Adjusted R2 0.06 0.58

56

Table 5

OLS Panel Regressions with Dummies

Using data running from January 1996 to December 2004, we regress credit spreads on corporate bonds, firm stock returns, and impliedvolatilities of single-stocks on a set of variables listed below. The coefficients for Dispersion (HL), Dispersion (AL), and Dispersion (LL)are obtained by multiplying the coefficient with a dummy variable that takes the value 1 if the firm is in the high, average, and lowleverage bin and zero otherwise. The same applies to the variables Implied Volatility, Implied Volatility Skewness, and Leverage. ⋆

denotes significance at the 10% level, ⋆⋆ denotes significants at the 5% level and ⋆⋆⋆ denotes significance at the 1% level. All estimationsuse autocorrelation and heteroskedasticity-consistent t-statistics.

Dependant Credit Spreads Stock Returns Implied Volatility

Constant 94.78⋆⋆⋆ 0.01⋆⋆⋆ 0.65⋆⋆⋆

Dispersion (LL) 35.68⋆⋆⋆ −0.01⋆⋆ 0.24Dispersion (AL) 36.06⋆⋆ 0.00⋆ 0.33⋆⋆⋆

Dispersion (HL) 38.48⋆⋆ 0.01⋆⋆ 0.44⋆⋆⋆

Implied Volatility (LL) 16.27⋆ −0.00Implied Volatility (AL) 15.92 −0.01Implied Volatility (HL) 15.46 −0.02Implied Volatility Skew (LL) −2.84 −0.02 0.06⋆⋆

Implied Volatility Skew (AL) −6.84 −0.02 0.02Implied Volatility Skew (HL) −7.37 −0.01 0.01Implied Volatility Skew (LL) (right) −0.06⋆⋆⋆

Implied Volatility Skew (AL) (right) −0.04Implied Volatility Skew (HL) (right) −0.02⋆⋆⋆

Open Interest −0.03⋆⋆⋆ -0.00 −0.00Volume 25.28⋆⋆

Call Option Volume −0.01⋆⋆ −0.07⋆

Put Option Volume 0.01⋆ 0.08⋆⋆

Slope of Term Structure −8.87Risk-free Rate −5.96S&P 500 Returns −1.13 −0.02⋆⋆

Non-Farm Payroll (/1000) −3.52⋆⋆ 0.91Stock Returns −10.55Stock Volume 7.19Leverage (LL) 6.73⋆⋆⋆ 0.59 1.64⋆⋆

Leverage (AL) 2.66⋆⋆⋆ 0.47 1.78⋆

Leverage (HL) 2.03⋆⋆⋆ 0.31 1.98Firm Size 0.04Swap Rate (/100) −4.10Rm − Rf −0.29 0.00⋆⋆⋆

SMB 0.42 0.00⋆⋆⋆

HML 0.18 0.00⋆⋆⋆

IV - RV 0.15⋆⋆⋆

Treading Pressure −0.11⋆

Adjusted R2 0.89 0.07 0.69

57

Table 6

OLS Panel Regression Results for Implied Volatility Skewness

Using data running from January 1996 to December 2004, we regress left and right implied volatility skewness on a set of variables listedbelow. Dispersion Positive (Negative) denotes a dummy for positive (negative) implied volatility slope. ⋆ denotes significance at the10% level, ⋆⋆ denotes significants at the 5% level and ⋆ ⋆ ⋆ denotes significance at the 1% level. All estimations use autocorrelation andheteroskedasticity-consistent t-statistics.

Skewness (left) Skewness (right)

Constant −0.31⋆⋆⋆ 3.57⋆⋆⋆

Dispersion Positive 0.11⋆ 0.62⋆⋆

Dispersion Negative −0.17⋆⋆ −0.90⋆⋆

Implied Volatility 1.08⋆⋆⋆ −5.36⋆⋆⋆

Call Volume 0.03 -0.92Put Volume −0.05 0.83Open Interest 0.00⋆⋆⋆ −0.00⋆⋆⋆

Leverage 0.76 1.32IV - RV −0.23⋆⋆⋆ 1.27⋆⋆⋆

Trading Pressure −0.02⋆⋆⋆ −2.01⋆⋆⋆

Adjusted R2 0.42 0.44

58

Table 7

OLS Panel Regressions for Implied Volatility Skewness with Dummies

Using data running from January 1996 to December 2004, we regress implied volatilities skewness of single-stocks on a set of variableslisted below. The coefficients for Dispersion (LL), Dispersion (AL), and Dispersion (HL) are obtained by multiplying the coefficient witha dummy variable that takes the value 1 if the firm is in the low, average, or high leverage bin and zero otherwise. The same applies tothe variables Implied Volatility and Leverage. Dispersion Positive (Negative) denotes a dummy for positive (negative) implied volatilityslope. ⋆ denotes significance at the 10% level, ⋆⋆ denotes significants at the 5% level and ⋆ ⋆ ⋆ denotes significance at the 1% level. Allestimations use autocorrelation and heteroskedasticity-consistent t-statistics.

Dependant Skewness (left) Skewness (right)

Constant −0.12⋆ 2.12Dispersion (LL) Positive 0.14⋆⋆ 0.65⋆⋆

Dispersion (LL) Negative −0.18⋆⋆ −0.53⋆⋆

Dispersion (AL) Positive 0.15⋆⋆ 0.84⋆⋆

Dispersion (AL) Negative −0.22⋆⋆ −0.65⋆

Dispersion (HL) Positive 0.17⋆ 1.17⋆

Dispersion (HL) Negative −0.30⋆ −1.11⋆⋆

Implied Volatility (LL) 1.25⋆⋆⋆ −5.09⋆⋆⋆

Implied Volatility (AL) 1.20⋆⋆⋆ −5.18⋆⋆⋆

Implied Volatility (HL) 1.15⋆⋆ −5.56⋆⋆⋆

Open Interest 0.00⋆⋆⋆ −0.00⋆⋆

Call Option Volume 0.03 0.70⋆

Put Option Volume −0.03 0.73Low Leverage 0.20 1.45Average Leverage 0.15 1.68High Leverage 0.11 1.69IV - RV −0.21⋆⋆⋆ 1.14⋆⋆⋆

Treading Pressure −0.10 −1.70⋆⋆

Adjusted R2 0.44 0.39

59

Table 8

Simulated Occurrence of Violations

This table shows the violation frequencies implied by the Monte Carlo simulation of our model. Low, Average, and High refer to theleverage ratios. OTM, ATM, and ITM stand for out-of-the-money, at-the-money, and in-the-money, respectively. Short, medium, andlong-term refer to options with time to maturity less than 60 days, between 60 and 180 days, and higher than 180 days, respectively. Thereported numbers are the simulated fractions of the violation occurrence across 10,000 simulation trials of the model.

Violation 1Low Average High

15.3 14.2 12.2

Violation 2Low Average High

Short Medium Long Short Medium Long Short Medium Long

OTMCall 5.8 6.0 6.2 5.5 5.7 5.6 10.0 10.2 10.9

Put 12.7 12.2 12.5 8.2 8.0 8.1 7.2 7.0 6.5

ATMCall 3.8 3.0 3.1 2.7 2.6 2.5 2.0 2.1 2.0

Put 3.5 3.0 2.0 2.0 2.1 2.1 2.0 2.1 2.2

ITMCall 3.0 3.0 3.2 1.8 1.7 1.9 1.7 1.8 2.0

Put 1.2 1.1 1.3 1.5 1.6 1.3 1.2 1.3 1.5

Violation 3Low Average High

Short Medium Long Short Medium Long Short Medium Long

OTMCall 4.2 4.1 3.7 4.2 3.7 3.6 7.2 6.5 7.8

Put 8.4 7.6 7.3 6.5 6.1 5.4 5.2 5.4 5.7

ATMCall 2.7 2.5 2.3 2.2 2.1 2.1 1.8 1.6 1.4

Put 2.5 2.2 1.2 0.9 0.8 0.8 0.9 0.8 0.8

ITMCall 1.8 1.6 1.6 1.5 1.4 1.5 1.5 1.6 1.4

Put 0.8 0.6 0.8 0.8 0.8 0.7 0.7 0.6 0.7

60

Table 9

Logit Regression of Arbitrage Violations on Credit Markets

This table summarizes the Logit regression results for the violation frequency according to Type 1 violations in Definition 2. ⋆ denotessignificance at the 10% level, ⋆⋆ denotes significants at the 5% level and ⋆ ⋆ ⋆ denotes significance at the 1% level, respectively.

Dependant LeverageLow Average High

Constant −0.89⋆⋆⋆ −0.86⋆⋆⋆ −0.84⋆⋆⋆

Dispersion 0.23⋆⋆⋆ 0.20⋆⋆⋆ 0.18⋆⋆⋆

Adjusted R2 0.08 0.05 0.07

Table 10

Logit Regression of Arbitrage Violations on Option Markets

This table summarizes the Logit regression results for Type 2 violations according to Definition 2, for different leverage and moneynessregions. Panel A reports the estimates for call options. Panel B those for put options. ⋆ denotes significance at the 10% level, ⋆⋆ denotessignificants at the 5% level and ⋆ ⋆ ⋆ denotes significance at the 1% level, respectively.

Panel A: CallsDependant Low Average High

OTM ATM ITM OTM ATM ITM OTM ATM ITM

Constant −0.48⋆⋆⋆ −0.45⋆⋆⋆ −0.46⋆⋆⋆ −0.49⋆⋆⋆ −0.50⋆⋆⋆ −0.51⋆⋆⋆ −0.50⋆⋆⋆ −0.52⋆⋆⋆ −0.52⋆⋆⋆

Moneyness 1.18⋆⋆⋆ 1.20⋆⋆ 1.22⋆⋆ 1.25⋆ 1.21⋆ 1.24⋆⋆ 1.21⋆⋆ 1.25⋆⋆ 1.30⋆⋆

Maturity 0.50⋆⋆⋆ 0.51⋆⋆⋆ 0.50⋆⋆⋆ 0.52⋆⋆⋆ 0.54⋆⋆⋆ 0.52⋆⋆⋆ 0.56⋆⋆⋆ 0.54⋆⋆⋆ 0.57⋆⋆⋆

Dispersion 0.65⋆⋆⋆ 0.60⋆⋆⋆ 0.62⋆⋆⋆ 0.61⋆⋆⋆ 0.63⋆⋆⋆ 0.61⋆⋆⋆ 0.62⋆⋆⋆ 0.65⋆⋆⋆ 0.66⋆⋆⋆

Adjusted R2 0.11 0.12 0.12 0.13 0.11 0.12 0.10 0.12 0.11

Panel B: PutsDependant Low Average High

OTM ATM ITM OTM ATM ITM OTM ATM ITM

Constant −0.62⋆⋆⋆ −0.60⋆⋆⋆ −0.61⋆⋆⋆ −0.63⋆⋆⋆ −0.61⋆⋆⋆ −0.64⋆⋆⋆ −0.65⋆⋆⋆ −0.65⋆⋆⋆ −0.67⋆⋆⋆

Moneyness 1.02⋆⋆ 1.03⋆⋆ 1.04⋆⋆ 1.07⋆ 1.02⋆⋆ 1.03⋆⋆ 1.01⋆⋆ 1.05⋆⋆ 1.04⋆⋆⋆

Maturity 0.24⋆⋆⋆ 0.21⋆⋆⋆ 0.26⋆⋆⋆ 0.23⋆⋆⋆ 0.24⋆⋆⋆ 0.21⋆⋆⋆ 0.20⋆⋆⋆ 0.24⋆⋆⋆ 0.23⋆⋆⋆

Dispersion 0.70⋆⋆⋆ 0.71⋆⋆⋆ 0.72⋆⋆⋆ 0.73⋆⋆⋆ 0.74⋆⋆⋆ 0.75⋆⋆⋆ 0.72⋆⋆⋆ 0.76⋆⋆⋆ 0.68⋆⋆⋆

Adjusted R2 0.10 0.11 0.12 0.12 0.12 0.13 0.09 0.12 0.12

61

Table 11

Logit Regression of Joint Arbitrage Violations

This table summarizes the Logit regression results for Type 3 violations according to Definition 2, for different leverage and moneynessregions. Panel A reports the estimates for call options. Panel B those for put options. ⋆ denotes significance at the 10% level, ⋆⋆ denotessignificants at the 5% level and ⋆ ⋆ ⋆ denotes significance at the 1% level, respectively.

Panel A: CallsDependant Low Average High

OTM ATM ITM OTM ATM ITM OTM ATM ITM

Constant −0.52⋆⋆⋆ −0.51⋆⋆⋆ −0.50⋆⋆⋆ −0.47⋆⋆⋆ −0.43⋆⋆⋆ −0.47⋆⋆⋆ −0.45⋆⋆⋆ −0.47⋆⋆⋆ −0.50⋆⋆⋆

Moneyness 1.20⋆⋆ 1.24⋆⋆ 1.20⋆⋆ 1.24⋆ 1.24⋆ 1.26⋆⋆ 1.27⋆⋆ 1.28⋆⋆ 1.21⋆⋆

Maturity 0.24⋆⋆⋆ 0.26⋆⋆⋆ 0.25⋆⋆⋆ 0.23⋆⋆⋆ 0.25⋆⋆⋆ 0.23⋆⋆⋆ 0.21⋆⋆⋆ 0.25⋆⋆⋆ 0.23⋆⋆⋆

Dispersion 0.51⋆⋆⋆ 0.50⋆⋆⋆ 0.52⋆⋆⋆ 0.53⋆⋆⋆ 0.57⋆⋆⋆ 0.54⋆⋆⋆ 0.52⋆⋆⋆ 0.56⋆⋆ 0.58⋆⋆

Adjusted R2 0.10 0.11 0.08 0.10 0.05 0.05 0.06 0.05 0.07

Panel B: PutsDependant Low Average High

OTM ATM ITM OTM ATM ITM OTM ATM ITM

Constant −0.48⋆⋆ −0.50⋆⋆⋆ −0.51⋆⋆⋆ −0.52⋆⋆ −0.58⋆⋆⋆ −0.54⋆⋆⋆ −0.60⋆⋆⋆ −0.61⋆⋆⋆ −0.68⋆⋆⋆

Moneyness 1.00⋆ 0.98⋆⋆ 0.87⋆⋆ 0.98⋆ 0.85⋆⋆ 0.81⋆⋆ 0.89⋆⋆ 0.87⋆⋆ 0.90⋆⋆⋆

Maturity 0.31⋆⋆⋆ 0.32⋆⋆⋆ 0.30⋆⋆⋆ 0.28⋆⋆⋆ 0.29⋆⋆⋆ 0.27⋆⋆⋆ 0.28⋆⋆⋆ 0.27⋆⋆⋆ 0.25⋆⋆⋆

Dispersion 0.34⋆⋆⋆ 0.37⋆⋆ 0.38⋆⋆⋆ 0.33⋆⋆ 0.32⋆⋆⋆ 0.37⋆⋆⋆ 0.32⋆⋆⋆ 0.41⋆⋆⋆ 0.48⋆⋆⋆

Adjusted R2 0.08 0.07 0.05 0.07 0.08 0.06 0.07 0.04 0.08

62

Table 12

Frequency of Arbitrage Violations

This table summarizes the empirical frequency arbitrage violations of Type 1, 2, and 3, each as a percentage of total observations at agiven leverage, moneyness, or time to maturity region. Low, Average, and High refer to the leverage ratios. OTM, ATM, and ITM standfor out-of-the-money, at-the-money, and in-the-money, respectively. Short, medium, and long-term refer to options with time to maturityless than 60 days, between 60 and 180 days, and higher than 180 days, respectively.

Violation 1Low Average High

18.9 15.4 14.3

Violation 2Low Average High

Short Medium Long Short Medium Long Short Medium Long

OTMCall 5.3 5.1 5.8 5.1 5.7 5.9 10.3 10.1 10.1

Put 10.2 10.1 10.1 7.1 7.3 7.2 6.0 6.1 6.3

ATMCall 4.0 4.0 4.3 4.3 4.5 4.0 3.8 3.2 3.7

Put 4.2 4.1 4.0 4.5 4.6 4.5 3.6 3.3 3.1

ITMCall 3.0 3.2 3.8 3.5 3.7 3.2 3.2 3.0 3.1

Put 2.1 2.0 2.2 2.5 2.6 2.0 2.5 2.3 2.1

Violation 3Low Average High

Short Medium Long Short Medium Long Short Medium Long

OTMCall 4.1 4.2 4.3 4.3 4.2 4.8 7.1 7.2 7.7

Put 5.2 3.3 4.1 2.1 1.2 2.2 3.2 2.1 1.2

ATMCall 3.2 2.4 1.8 2.5 2.0 1.8 2.1 1.0 1.2

Put 2.0 1.0 1.2 1.2 1.0 1.1 1.1 1.0 1.1

ITMCall 2.3 2.4 2.8 2.8 2.9 2.1 3.0 1.2 1.8

Put 1.0 1.1 1.2 1.2 1.0 1.0 1.1 1.2 1.0

63

Table 13

OLS Panel Regressions with Idiosyncratic Volatility

Using data running from January 1996 to December 2004, we regress credit spreads on corporate bonds and firm stock returns on a set ofvariables listed below. The variable Idiosyncratic Volatility is the time-series sample standard deviation of daily stock returns 180 dayspreceding the observation. ⋆ denotes significance at the 10% level, ⋆⋆ denotes significants at the 5% level and ⋆ ⋆ ⋆ denotes significanceat the 1% level. All estimations use autocorrelation and heteroskedasticity-consistent t-statistics.

Dependant Credit StockSpreads Returns

Constant −24.18 −1.44 0.02 0.02 0.02Dispersion 22.25⋆⋆⋆ 0.01⋆⋆⋆

Dispersion (LL) −0.01⋆

Dispersion (AL) 0.01⋆⋆

Dispersion (HL) 0.01⋆⋆

Idiosyncratic Volatility 1781.54⋆⋆ 1535.20⋆⋆ −0.44⋆ −0.40⋆

Idiosyncratic Volatility (LL) -0.35Idiosyncratic Volatility (AL) -0.40Idiosyncratic Volatility (HL) -0.48Implied Volatility 23.33 30.61 −0.04 −0.03Implied Volatility (LL) 0.07Implied Volatility (AL) -0.02Implied Volatility (HL) -0.05Implied Volatility Skew (left) −15.63 −20.49 0.00 −0.00 0.00Implied Volatility Skew (right)Open Interest −0.02⋆⋆ −0.03⋆⋆⋆ −0.00 −0.00⋆ −0.00⋆

Volume 23.75 19.96Call Option Volume 0.01 0.01 −0.02⋆

Put Option Volume 0.01 0.01 0.01⋆

Slope of Term Structure −12.00⋆⋆⋆ −12.82⋆⋆

Risk-free Rate −14.05⋆⋆ −12.63⋆

S&P 500 Returns 11.43 11.30 −0.07⋆ −0.07⋆⋆ -0.04Non-Farm Payroll (/100) −2.30⋆ −2.63⋆⋆ 1.10 1.05 1.04Stock Returns −31.34 −52.65Stock Volume 8.27 7.56Leverage (/1000) 2.12⋆⋆⋆ 2.16⋆⋆⋆ 0.38 0.40Leverage (LL) (/1000) 0.38Leverage (AL) (/1000) 0.35Leverage (HL) (/1000) 0.28Firm Size (/100) 60.69⋆⋆⋆ 56.90Swap Rate (/100) 1.14 −52.80Rm − Rf −0.07 −0.08 −0.00 −0.00⋆⋆⋆ 0.00⋆⋆⋆

SMB 0.11 0.09 0.00⋆⋆⋆ 0.00⋆⋆⋆ −0.00⋆⋆⋆

HML −0.03 −0.04 0.00 0.00⋆⋆⋆ −0.00⋆⋆⋆

Adjusted R2 0.88 0.87 0.06 0.05 0.06

64

Table 14

OLS Panel Regressions with Other Measures of Beliefs Disagreement

Using data running from January 1996 to December 2004, we regress credit spreads on corporate bonds and firm stock returns on a set ofvariables listed below. Dispersion 1 is the dispersion measure of Diether, Malloy, and Scherbina (2002) and Dispersion 2 is the divergencemeasure of Doukas, Kim, and Pantzalis (2006). ⋆ denotes significance at the 10% level, ⋆⋆ denotes significants at the 5% level and ⋆ ⋆ ⋆

denotes significance at the 1% level. All estimations use autocorrelation and heteroskedasticity-consistent t-statistics.

Dependant Credit Spreads Stock ReturnsDispersion 1 Dispersion 2 Dispersion 1 Dispersion 2

Constant 94.83 94.76 101.02 98.92 -0.01 0.00 0.01 0.00Dispersion 29.09⋆ 22.02⋆ −0.01⋆ 0.01⋆⋆

Low Dispersion 28.45 20.10⋆ −0.01⋆⋆ 0.01⋆⋆

Average Dispersion 29.15 21.15⋆ −0.01⋆ 0.01⋆⋆

High Dispersion 30.41⋆⋆ 22.05⋆ −0.01⋆ 0.01⋆

Implied Volatility 24.83⋆⋆ 25.45⋆⋆ 0.02 0.01⋆

Low Implied Volatility 29.23⋆ 25.48⋆⋆ 0.01 0.01⋆⋆

Average Implied Volatility 26.55⋆ 25.14⋆⋆ 0.02 0.02⋆

High Implied Volatility 23.55⋆ 24.14⋆ 0.01 0.01⋆

Implied Volatility Skew (left) −12.32 −15.14⋆ 0.01 0.00Low Implied Volatility Skew −12.35 −14.12 0.02 0.01Average Implied Volatility Skew −7.00 −15.17 0.01 0.01High Implied Volatility Skew −4.69 −16.25 0.02 0.01Open Interest −0.03⋆⋆ −0.02⋆⋆ −0.01⋆⋆⋆ −0.03⋆⋆ 0.00 0.00⋆ 0.00 0.00Volume 19.69 24.59⋆⋆ 20.14 17.75Call Option Volume −0.01⋆ −0.01⋆ 0.00⋆⋆ 0.00⋆⋆

Put Option Volume 0.01⋆⋆ 0.00 0.03⋆ 0.01⋆

Slope of Term Structure −13.78⋆⋆⋆ −10.45⋆⋆ −13.89⋆⋆⋆ 9.21⋆

Risk-free Rate −11.38⋆ −6.96 -10.47 −10.45S&P 500 Returns 8.09 1.84 10.98 10.78 −0.03⋆ −0.02⋆⋆ −0.03⋆⋆ −0.01⋆⋆

Non-Farm Payroll (/100) −3.30⋆ −4.25⋆⋆⋆ −4.21⋆⋆ −4.14⋆⋆⋆ -0.37 -0.52 0.90 0.80Stock Returns −56.78 −9.04 -25.45 −14.47Stock Volume 5.25 6.14 4.58 5.12Leverage (/1000) 2.00⋆⋆⋆ 1.19⋆⋆⋆ -0.51 −1.40⋆

Low Leverage (/1000) 3.08⋆⋆⋆ 1.18⋆⋆⋆ −0.47 −1.40⋆

Average Leverage (/1000) 2.98⋆⋆⋆ 1.18⋆⋆⋆ −0.52 −1.50High Leverage (/1000) 2.80⋆⋆⋆ 1.17⋆⋆⋆ −0.52 −1.52⋆

Firm Size (/100) 5.82⋆⋆⋆ 3.69 5.41⋆⋆⋆ 4.78⋆

Swap Rate (/100) −2.19 −3.12 -3.78 −3.98Rm − Rf 0.03 −0.38 −0.01 −0.12 0.00⋆⋆⋆ 0.00⋆⋆⋆ 0.00⋆⋆⋆ 0.00⋆⋆⋆

SMB 0.06 0.42 −0.01 −0.01 0.00⋆⋆⋆ 0.00⋆⋆⋆ 0.00⋆⋆⋆ 0.00⋆⋆⋆

HML 0.03 0.14 0.00 0.01 0.00⋆⋆⋆ 0.00⋆⋆⋆ 0.00⋆⋆⋆ 0.00⋆⋆⋆

Adjusted R2 0.80 0.82 0.82 0.83 0.04 0.04 0.05 0.05

65

0

0.005

0.01

0.015

0.02 0

0.05

0.1

0.15

0.2

158

159

160

161

ΨA

Firm Value

γ1A − γ2

A

Firm

Valu

e

0.2

0.15

0.1

0.05

0

0

0.005

0.01

0.015

0.02

0.07

0.10

0.13

ΨA

Firm Value Volatility

γ1A − γ2

A

Firm

Valu

eV

ola

tility

0

0.005

0.01

0.015

0.02 0

0.05

0.1

0.15

0.2

−0.5

−0.4

−0.3

−0.2

−0.1

0

ΨA

Risk-Neutral Skewness (Firm Value Returns)

γ1A − γ2

A

Risk-n

eutr

alSkew

ness

Figure 1. Firm Value, Firm Value Volatility, and Risk-Neutral SkewnessThese figures plot the firm value, the firm value volatility, and the firm value risk-neutral skewness as a function of belief disagreement ΨA(t) and the difference between the agents’volatility of cash flow growth rate. The parameter values used are given in Table 1.

66

0.2

0.15

0.1

0.05

0

0

0.005

0.01

0.015

0.02

0

0.15

0.30

ΨA

Equity Volatility (high Leverage)

γ1A − γ2

A

Equity

Vola

tility

0.02

0.015

0.01

0.005

00.2

0.15

0.1

0.05

0

1.2

1.3

1.4

1.5

1.6

1.2

1.3

1.4

1.5

ΨA

Senior Bond Credit Spread (high Leverage)

γ1A − γ2

A

Sen

ior

Bond

Cre

dit

Spre

ad

(in

%)

Figure 2. Equity Volatility and Senior Bond Credit Spreads for High Leverage RatioThese figures plot equity volatility (left panel) and senior bond credit spreads (right panel) for a high leverage ratio ratio as a functionof difference in beliefs ΨA(t) and the difference between the agents’ volatility of cash flow growth rate. The parameter values used aregiven in Table 1.

67

0

0.005

0.01

0.015

0.02 0

0.05

0.1

0.15

0.2

82

82.25

82.5

82.75

83

ΨA

Equity Price (high Leverage)

γ1A − γ2

A

Equity

Price

0

0.005

0.01

0.015

0.02 0

0.05

0.1

0.15

0.2116

117

118

119

120

ΨA

Equity Price (low Leverage)

γ1A − γ2

A

Equity

Price

Figure 3. Firm Equity Price for High and Low Leverage RatioThe price of equity with high leverage ratio (left panel) and low leverage ratio (right panel) is plotted as a function of the difference inbeliefs ΨA(t) and the difference between the agents’ volatility of cash flow growth rate. The parameter values used are given in Table 1.

68

0 0.0625 0.125 0.1875 0.25 0.3125 0.375 0.4375 0.5−2

−1

0

1

2

3

4

5

Leverage

∆S/∆

Ψ

ZCBFirm ValueDeltaVegaEquitySkewness

Figure 4. Change in the Equity PriceThis figure plots the change in the equity price as a function of the bonds face values. We split up the total variation of equity, into fourmain effects: The Delta effect, which is due to a change in the underlying, the Vega effect, which is due to a change in the firm valuevolatility, a Skew effect, which is due to a change in the risk-neutral skewness and a bond effect.

69

0

0.005

0.01

0.015

0.02 0

0.05

0.1

0.15

0.2

−1

−0.75

−0.5

−0.25

0

ΨA

Risk-neutral Skewness (Equity Returns), high Leverage

γ1A − γ2

A

Risk-n

eutr

alSke

wnes

s

0.2

0.15

0.1

0.05

0

0

0.005

0.01

0.015

0.02

0

0.25

0.5

0.75

1

ΨA

Risk-Neutral Skewness (Equity Returns), low Leverage

γ1A − γ2

A

Risk-n

eutr

alSke

wnes

s

Figure 5. Risk-neutral Skewness of Equity Returns for High and Low Leverage RatiosThese figures depict risk-neutral skewness of equity returns with high leverage ratio (left panel) and low leverage ratio (right panel).They are plotted as a function of the difference in beliefs ΨA(t) and the difference between the agents’ volatility of cash flow growth rate.The parameter values used are given in Table 1.

70

0.8 0.85 0.9 0.95 1 1.05 1.10

0.1

0.2

0.3Volatility Smile as Function of ΨA (high Leverage)

Moneyness

Implied

Vola

tility

ΨA = 0.2ΨA = 0

0.8 0.85 0.9 0.95 1 1.05 1.10

0.1

0.2

0.3Volatility Smile as a Function of ΨA (average Leverage)

Moneyness

Implied

Vola

tility

ΨA = 0.2ΨA = 0

0.8 0.85 0.9 0.95 1 1.05 1.10

0.1

0.2

0.3Volatility Smile as Function of ΨA (low Leverage)

Moneyness

Implied

Vola

tility

ΨA = 0.2ΨA = 0

Figure 6. Implied Volatility Smile of the European Option on Firm Equity for High, Average, and Low LeverageImplied volatilities of equity options are plotted as a function of the difference in beliefs ΨA(t). The parameter values used are given in Table 1.

71

0

0.005

0.01

0.015

0.02 0

0.05

0.1

0.15

0.2

0.0025

0.003

0.0035

0.004

ΨA

10% ITM Put Option Price, low Leverage

γ1A − γ2

A

Put

Option

Price

0

0.005

0.01

0.015

0.02 0

0.05

0.1

0.15

0.2

0.025

0.03

0.035

0.04

ΨA

ATM Put Option Price, low Leverage

γ1A − γ2

A

Put

Option

Price

0

0.005

0.01

0.015

0.02 0

0.05

0.1

0.15

0.2

0.30

0.35

0.40

0.45

0.50

ΨA

10% OTM Put Option Price, low Leverage

γ1A − γ2

A

Put

Option

Price

Figure 7. Put Option Prices for Low Leverage FirmIn-the-money (left panel), at-the-money (middle panel), and out-of-the-money (right panel) put option prices are depicted as a function of belief disagreement ΨA(t) and thedifference between the agents’ volatility of cash flow growth rate. The parameter values used are given in Table 1.

0

0.005

0.01

0.015

0.02 0

0.05

0.1

0.15

0.2

13

13.5

14

14.5

ΨA

10% ITM Call Option Price, high Leverage

γ1A − γ2

A

Call

Option

Price

0

0.005

0.01

0.015

0.02 0

0.05

0.1

0.15

0.2

25.5

26

26.5

27

ΨA

ATM Call Option Price, high Leverage

γ1A − γ2

A

Call

Option

Price

0

0.005

0.01

0.015

0.02 0

0.05

0.1

0.15

0.239

39.5

40

40.5

41

ΨA

10% OTM Call Option Price, high Leverage

γ1A − γ2

A

Call

Option

Price

Figure 8. Call Option Prices for High Leverage FirmIn-the-money (left panel), at-the-money (middle panel), and out-of-the-money (right panel) call option prices are depicted as a function of belief disagreement ΨA(t) and thedifference between the agents’ volatility of cash flow growth rate. The parameter values used are given in Table 1.

72