Modeling Market Downside Volatility

Bruno Feunou∗ Roméo Tédongap†

Duke University Stockholm School of Economics

Previous Version: November 2007This Version: March 2009

Abstract

Downside risks are important in more volatile and negatively asymmetric marketsrelative to upside risks, and an increase in the relative downside risk should be re-munerated by appropriate returns. Motivated by these statements, we present a newmethodology for modeling and estimating time-varying downside and upside variancesin stock returns. The assumed binormal distribution explicitly relates downside andupside risks to conditional heteroscedasticity and conditional asymmetry. Moreover,we estimate the relation between the mode of conditional return distribution anddownside and upside volatilities. Empirical results and tests of the model stronglysuggest that relative downside risks are compensated by an increase in the conditionalmode, while expected returns show a negative relation with risk.

Keywords: Option Valuation, Value-at-Risk, GARCH, Risk-Return Tradeoff, Binormal

JEL Classification: G12, C01, C22, C51

∗Duke University, Economics Department, 213 Social Sciences Building, Durham, NC 27708, USA.Email: Bruno.Feunou@duke.edu.

†Corresponding Author: Stockholm School of Economics, Finance Department, Sveavägen 65, 6th floor,Box 6501, SE-113 83 Stockholm, Sweden. Email: Romeo.Tedongap@hhs.se.

1 Introduction

Investors do not face in a rising stock market the same amount of risk they face in a

declining market, and precisely if the market is asymmetric. Given available information,

if investors evaluate that they will be facing more risk if the market falls below a certain

threshold (downside risk) than if it realizes above the threshold (upside risk), then, they will

require an additional compensation to bear the relative downside risk (difference between

downside and upside risks). Downside risks measure the likelihood that a security or other

investment will suffer a decline in price, or the amount of loss that could result from that

potential decline, if market conditions turn bad. In opposite, upside risks measure the

potential dollar or percentage amount by which the market or a stock could fall given

market conditions are good.

Measuring downside or upside risk is basically making an educated guess on how low

or how high a stock could go in the near future depending on market conditions. Variance

of returns has been widely used as a proxy for risk in financial returns. Natural measures

for downside and upside risks are variances of returns conditional on returns falling be-

low or realizing above a wisely chosen threshold, or simply downside and upside variances

with respect to this threshold. Keeping this classical view, we introduce a new general-

ized autoregressive conditional heteroscedasticity (GARCH) model for downside and upside

volatilities. Since downside risks are important in more volatile and negatively asymmetric

markets relative to upside risks, the model explicitly relates downside and upside volatilities

to conditional heteroscedasticity and conditional asymmetry in asset returns. We rely on

useful properties of the binormal distribution to build our model (Bin-GARCH). The bi-

normal distribution has three parameters: the mode, the downside and the upside variances

with respect to the mode. If downside and upside variances are equal, then the distribution

is normal and the mode is equivalent to the mean. Downside and upside variances are also

explicitly related to total variance and asymmetry, the latter measured by the Pearson’s

first coefficient of asymmetry. The binormal distribution is then alternately parameterized

by the mode, the variance and the asymmetry.

In the i.i.d. case with binormally distributed returns, we derive closed-form formulas

for option prices, value-at-risk and expected shortfalls. This option pricing formula is a

single parameter generalization of the Black-Scholes-Merton option valuation and offers the

same simplicity and tractability. This is also similar to the formula provided in Feunou,

Fontaine and Tédongap (2009) and which assumes a homoscedastic gamma distribution.

Next, we assume that conditional on available information, returns are binormally dis-

tributed with time-varying conditional mode, variance and asymmetry. This specification

1

nests the NGARCH model of Engle and Ng (1993) as we borrow our volatility dynam-

ics from the NGARCH. This heteroscedasticity specification features both persistence and

asymmetry in conditional variance, the so-called leverage effect. Conditional asymmetry

in our model follows a nonlinear GARCH-type dynamics and features asymmetry in asym-

metry. Since the binormal distribution explicitly relates downside and upside volatilities to

heteroscedasticity and conditional asymmetry, our model is equivalent to modeling down-

side and upside risks. In the spirit of the ARCH-M model of Engle, Lilien and Robins

(1987), we allow the relative downside volatility to be a determinant of the conditional

mode. In consequence, expected returns are also determined by the relative downside risk.

We estimate the Bin-GARCH model by maximum likelihood using daily S&P500 index

returns from January 2, 1980 to December 30, 2005. We find significant presence of high

downside risks relative to upside risks, a high correlation between downside and upside

volatilities (0.82), an annualized daily relative downside volatility of 2%, and a significant

impact of relative downside volatility on the conditional mode. Allowing for time-varying

conditional mode significantly increases asymmetry in asymmetry by more than twice, rela-

tive to a constant conditional mode. We perform several diagnostics of the binormal model

for asset returns, using daily S&P500 index option data. These experiments assess the per-

formance of the Bin-GARCH model in different dimensions. In particular, the Bin-GARCH

model generates enough unconditional skewness to restore the observed implied volatility

curve. Also, back testing and density forecast evaluation of the model show its accuracy

in measuring conditional value-at-risk and in fitting conditional return distribution. News

impact curves reveal that downside volatility strictly declines with return shocks while up-

side volatility has a convex shape, and in consequence, market asymmetry strictly increases

with return shocks.

This paper touches on the literature that studies downside risks. Ang, Chen and Xing

(2006) show that the cross-section of stock returns reflects a premium for downside risk, and

provide a methodology for estimating this downside risk-premium using daily data. Instead

we study downside risks in the time series dimension, modeling and estimating downside

volatility, and studying its relation to the central tendency in asset returns. Barndorff-

Nielsen, Kinnebrock and Shephard (2008) introduce measures of downside risks, termed

downside realized semivariances, and based entirely on downward moves measured using

high frequency data. We rely on a GARCH framework to measure and estimate downside

risks by maximum likelihood using daily data.

We also touch on the literature on the risk-return tradeoff. The traditional way of

assessing the risk-return tradeoff is measuring and testing the relation between expected

return and variance. This has been an important topic in asset pricing which led to con-

2

flicting findings. Campbell (1987), Harvey (1989), Nelson (1991), Campbell and Hentschel

(1992), Hentschel (1995), Glosten, Jagannathan, and Runkle (1993), and Wu (1998) have

focused on the intertemporal relation between return and risk where risk is measured in the

form of variance or covariance. An important concern has been the sign and magnitude of

this tradeoff. Ghysels, Santa-Clara and Valkanov (2005) find a significantly positive rela-

tion between risk and return in the stock market. French, Schwert and Stambaugh (1987)

and Campbell and Hentschel (1992) find that the relation is positive and insignificant. In

contrast, Campbell (1987) and Nelson (1991) find a significantly negative relation.

The relation between the conditional mode and the relative downside risk, explicitly

specified and tested in our setting, is appealing for the risk-return tradeoff. In contrast

to previous literature, we argue that expected returns might not be a critical or even

reliable measure for risk compensation, especially if markets are negatively asymmetric. If

the return distribution is unimodal and negatively skewed (with a negative Pearson’s first

coefficient of asymmetry), we argue that an increase in the mode of the return distribution

may provide a better compensation for risk than focusing on the mean. First, the mode is

greater than the mean since returns are negatively skewed with a negative Pearson’s first

coefficient. Second, the mode has the highest probability to realize. Then, investors would

like to see the mode increasing with risk as they know it is the most likely outcome, that

would compensate their risk anyway, even if the mean decreases with risk. Third, compared

to the mean, the mode is less sensitive to outliers that may affect return prediction. We

find a significantly positive relation between the relative downside risk and the mode of

returns in the stock market, and the magnitudes of our estimates also suggest that expected

returns are negatively related to relative downside risk.

We finally build on the literature that explicitly models conditional asymmetry and fat-

tailedness in stock returns. Following the work of Hansen (1994), many authors have shown

interest in modeling conditional skewness and kurtosis. Harvey and Siddique (1999) intro-

duce a methodology for estimating time-varying conditional skewness, using a maximum

likelihood framework with instruments, and assuming a non-central t distribution. They

reparameterize the standardized residual’s conditional density in terms of skewness, and

model the mean, the volatility and the skewness independently. Brooks et al. (2005) use

a modified version of the Student’s distribution which allows for independent modeling of

volatility and kurtosis, assuming that the skewness is zero. Leon, Rubio and Serna (2004)

use a Gram-Charlier series expansion, and perform parameterizations which yield indepen-

dent modeling of volatility, skewness and kurtosis. However, as they use the Gallant and

Tauchen (1989)’s transformation, the interpretation of their parameters as volatility, skew-

ness and kurtosis is lost. In the GARCH framework, an efficient estimation of conditional

3

mean and variance requires a full description of the conditional distribution, and a good

predictive density relies on the knowledge of a distribution that better approximates the

true distribution of standardized returns. We do not explicitly model conditional skewness,

but the Pearson’s first coefficient of asymmetry, which is a more robust measure of asym-

metry (see Kim and White (2004)), and explicitly relates to the relative downside volatility

which is a focus of this article. We also depart from previous literature by modeling the

central tendency in returns through the conditional mode instead of expected returns.

The remaining of the paper is organized as follows. Section 2 discusses properties of

the binormal distribution that are useful for financial modeling. Section 3 introduces the

i.i.d. binormal model of asset returns and derive closed-form formulas of option prices and

value-at-risk consistent with the i.i.d. model. Section 4 introduces the binormal GARCH

model and Section 5 presents empirical results and various model diagnostics. Section 6

concludes.

2 The Binormal Distribution

We model innovations in asset prices through the binormal distribution. The Binormal

distribution is introduced in Gibbbons and Mylroie (1973) as an analytically tractable dis-

tribution which accommodates practical values of skewness and kurtosis, and strictly nests

the normal distribution.1 In a static setting, log returns R follow a binormal distribution

of parameter (m, σ1, σ2)⊤ if their density is given by:

f(x) = A exp

(

−1

2

(

x − m

σ1

)2)

1{x<m} + A exp

(

−1

2

(

x − m

σ2

)2)

1{x≥m} (2.1)

where m is the mode of the distribution and A =√

2 /π /(σ1 + σ2) . The attractiveness of

the binormal distribution is the fact that it is directly parameterized by its mode. For this

distribution, the mean µ, the variance σ2, the Pearson’s first coefficient of asymmetry p,

that is the difference between the mean and the mode divided by the standard deviation,

and the skewness s are given by:

µ = m + σp

σ2 = (1 − 2 /π ) (σ2 − σ1)2 + σ1σ2

s = p (1 − (π − 3) p2)

p =√

2 /π (σ2 − σ1) /σ .

(2.2)

1Other related works show that the binormal distribution can be useful for data modeling, statisticalanalysis and robustness studies of normal theory methods. These works include Bangert et al. (1986),Kimber and Jeynes (1987), Toth and Szentimrey (1990).

4

We notice that σ21 and σ2

2 are, up to a multiplicative constant, interpretable as variances

of returns conditional on returns being less than the mode (downside variance), and con-

ditional on returns being greater than the mode (upside variance), respectively. More

specifically,

V ar [R | R < m] =

(

1 −2

π

)

σ21 and V ar [R | R ≥ m] =

(

1 −2

π

)

σ22. (2.3)

This is a particularly important characteristic of the binormal distribution that we will

discuss in details later in subsequent sections when modeling asset prices in dynamic setting.

We show that the initial parameters σ1 and σ2 are expressed in terms of the variance

and the Pearson’s coefficient as:

σ1 = σ(

−√

π /8p +√

1 − (3π /8 − 1) p2)

σ2 = σ(

√

π /8p +√

1 − (3π /8 − 1) p2)

,(2.4)

and it also implies that the Pearson’s coefficient is bounded: |p| ≤ 1/

√

π /2 − 1 ≈ 1.3236.

Since return skewness is related to its Pearson’s coefficient through the third equation

in (2.2), then, bounds on the Pearson’s coefficient also imply bounds on the skewness:

|s| ≤ 0.9953. Also, excess kurtosis is positive and less or equal to 3.8692. We alternately

parameterize the binormal distribution by the vector (m, σ, p)⊤ which is also informative

about the distribution as it represents the central tendency, the standard deviation and

the asymmetry, as well as the vector (m, σ1, σ2)⊤, as we now know how to recover the

latter from the former.2 In a dynamic setting with changing parameters over time, given

equation (2.4), modeling the variance and the Pearson’s coefficient of asymmetry would be

equivalent to modeling downside and upside variances.

Financial data are subject to many outliers and contrarily to the mean and the skewness,

the mode and the Pearson’s coefficient are more robust to outliers (see Kim and White

(2004) for a more detailed discussion on that issue). In a dynamic setting, we will exploit

this empirical advantage of the mode and the Person’s first coefficient over the mean and

the skewness, and, instead of modeling the mean, the variance and the skewness of returns

as in the common approach in previous literature (see for example Harvey and Siddique

(1999, 2000) and Feunou and Tédongap (2009)), we directly model the mode, the variance

and the Pearson’s first coefficient.

To derive a valuation formula for a European option when changes in log asset prices

are i.i.d. binormal, and to derive value-at-risk (VaR) and expected shortfall (ES) formulas,

2Parameterizing by the mode, the variance and the Pearson’s first coefficient of asymmetry is equivalentto the epsilon-skew-normal distribution (ESN) of Mudholkar and Hutson (2000). They provide statisti-cal properties of GMM, MLE and Bayesian estimator of ESN parameters and conclude that the ESNdistribution is very accurate for analyzing near-normal data.

5

one needs the moment generating function M (u) = E [exp (uR)] as well as the truncated

moment generating function M (u; x) = E[

exp (uR) 1{R≥x}

]

of returns. These moments

are given by:

M (u) =2σ1

σ1 + σ2exp

(

mu +σ2

1u2

2

)

Φ (−σ1u) +2σ2

σ1 + σ2exp

(

mu +σ2

2u2

2

)

Φ (σ2u) (2.5)

and

M (u; x) = M (u) −2σ1

σ1 + σ2exp

(

mu +σ2

1u2

2

)

Φ

(

x − m

σ1− σ1u

)

if x < m (2.6)

=2σ2

σ1 + σ2

exp

(

mu +σ2

2u2

2

)

Φ

(

−x − m

σ2

+ σ2u

)

if x ≥ m, (2.7)

where Φ is the cumulative distribution function of a standard normal random variable. In

particular, one can show that the quantile function needed for VaR calculation is given by:

F−1 (α) = m + σ1Φ−1

(

ασ1 + σ2

2σ1

)

1{α<

σ1σ1+σ2

}

+ σ2Φ−1

(

ασ1 + σ2

2σ2

+σ2 − σ1

2σ2

)

1{α≥

σ1σ1+σ2

} (2.8)

where F is the cumulative distribution function of returns.

We finally notice that the two terms adding up to M (u) in equation (2.5) have particu-

lar meanings. The first term is up to the multiplicative constant σ1 /(σ1 + σ2) , the moment

generating function of m − σ1 |Z1| where Z1 is a standard normal random variable. Simi-

larly, the second term term is up to the multiplicative constant σ2 /(σ1 + σ2) , the moment

generating function of m + σ2 |Z2| where Z2 is a standard normal random variable, and

Z1 and Z2 are independent. Since the two multiplicative constants sum up to one, this

suggests that the distribution of returns can be viewed as a mixture of two distributions

and that the mix parameter is σ1 /(σ1 + σ2) . Interestingly, returns can be written:

R = m − σ1U |Z1| + σ2 (1 − U) |Z2| (2.9)

where U is a Bernoulli random variable of parameter σ1 /(σ1 + σ2) , independent of Z1 and

Z2. This suggests a procedure for simulating returns, a technique particularly useful where

closed-form formulas for expressions depending on moments of nonlinear functions of asset

prices are not available. We are now armed with necessary tools to explore financial models

assuming a binormal distribution of asset returns.

3 The IID Binormal Model: Option Valuation and Value-at-Risk

3.1 Generalizing the Black-Scholes-Merton Option Valuation

We assume that under the risk-neutral measure, log returns R (t) = ln (S (t) /S (t − 1)) are

i.i.d. binormal with parameter (m, σ1, σ2)⊤ or equivalently (m, σ, p)⊤ as discussed previ-

6

ously, as (2.2) and (2.4) hold. Since expected stock returns under the risk-neutral measure

are equal to the risk-free return, that is

E [exp (R (t + 1))] = exp (r) (3.1)

where r is the risk free rate, then the restriction (3.1) implies that the risk neutral mode is

given by:

m = r + ln

(

σ1 + σ2

2

)

− ln

(

σ1 exp

(

σ21

2

)

Φ (−σ1) + σ2 exp

(

σ22

2

)

Φ (σ2)

)

. (3.2)

Garvin and McClean (1997) discuss aggregation issues, and show that the sum of two inde-

pendent binormal distributions is well approximated by another binormal distribution. Fol-

lowing their work, the risk-neutral distribution of aggregate returns R (t, τ) =τ∑

i=1

R (t + i)

is approximately a binormal distribution of parameter (mτ , σ1τ , σ2τ )⊤, where:

mτ = τm + σ (τp − s)

σ1τ = σ(

−√

π /8s +√

τ − (3π /8 − 1) s2)

σ2τ = σ(

√

π /8s +√

τ − (3π /8 − 1) s2)

.

(3.3)

Equations (2.5), (2.6) and (2.7) express the moment generating function and truncated

moment generating functions of a binormal distribution of parameter (m, σ1, σ2)⊤. Let

Mτ (u) and Mτ (u; x) be the moment generating function and truncated moment generating

functions of a binormal distribution of parameter (mτ , σ1τ , σ2τ )⊤. It follows that the price

at date t of a European call option with maturity τ and strike price K is given by:

C (t, τ ; K) = exp (−rτ)

[

S (t) Mτ

(

1; ln

(

K

S (t)

))

− KMτ

(

0; ln

(

K

S (t)

))]

. (3.4)

The formula (3.4) is a simple generalization of the Black-Scholes-Merton option pricing

formula and provides the same simplicity and tractability. It simply accounts for asym-

metries in returns of the underlying asset and this asymmetry is particularly important

to explain the option smirk. The Black-Scholes-Merton formula obtains from p = 0. Feu-

nou, Fontaine and Tédongap (2009) also introduce the Homoscedastic Gamma (HG) model

where the distribution of returns is characterized by its mean, variance and an independent

skewness parameter. The HG model also leads to a simple extension of the (discrete-time)

BSM model with closed-form prices but with arbitrary skewness.3

3Feunou, Fontaine and Tédongap (2009) also show empirically that the HG analog to the Practitioner’sBSM model provides substantial improvement of in-sample, out-of-sample and hedging performances, evenwith an equal or lower number of parameters.

7

Other generalizations of the BSM option pricing formula, that assume i.i.d. returns,

have been studied in previous research. Jarrow and Rudd (1982) use a fourth-order Gram-

Charlier expansion to approximate the return distribution. The resulting density is not

a proper density as it can be negative, unless strong restrictions are needed for skewness

and kurtosis as shown by Jondeau and Rockinger (2001). Leon, Mencia and Sentana

(2005) overcome the issue by using the SNP densities of Gallant and Nychka (1987). These

authors do not address temporal aggregation issues and specify a new return density for

each maturity. Madan and Milne (1994) propose a formula based on Hermite polynomials

approximation, and Ane (1999) finds empirically that the model provides good accuracy

for hedging purposes. Bertholon, Monfort and Pegoraro (2005) use a mixture of normal

distributions and show it yields an option price that is also a mixture of BSM prices.

Contrary to the binormal distribution, which we show in the next section leads to closed-

form expressions for Value-at-Risk (VaR) and expected shortfall (ES), the mixture of normal

do not allow a closed-form derivation of VaR and ES. Lim, Martin and Martin (2005) use

the generalized Student-t distribution which is flexible enough to accommodate a wide

range of skewness and kurtosis values, but there is no closed-form solution to European

option prices.

3.2 Value-at-Risk and Expected Shortfall Calculations

We next use the binormal distribution to derive closed-form formulas for Value-at-Risk

(VaR) and expected shortfall (ES). Now we assume that under the historical measure,

log returns R (t) = ln (S (t) /S (t − 1)) are i.i.d. binormal with parameter (m, σ1, σ2)⊤.

Equation (2.8) expresses the quantile function of a binormal distribution of parameter

(m, σ1, σ2)⊤. Let Fτ (α) be the quantile function, and let M ′

τ (u) and M ′τ (u; x) be the

first order derivatives with respect to u of moment generating function and truncated

moment generating functions of a binormal distribution of parameter (mτ , σ1τ , σ2τ )⊤. The

conditional VaR and the conditional expected shortfall at 1−α confidence level for horizon

τ is given by:

V aR (τ ; α) = −F−1τ (α) and ES (τ ; α) =

M ′τ (0) − M ′

τ (0;−V aR (τ ; α))

1 − Mτ (0;−V aR (τ ; α)). (3.5)

This is also a simple and tractable generalization of VaR and ES formulas when returns

are normally distributed, and which obtain from p = 0. While the VaR is a linear function

of volatility when returns are normally distributed,

V aR (τ ; α) = −mτ − στΦ−1 (α) if p = 0,

8

in general, it is not the case when there are asymmetries in returns. However, our formula

shows that the VaR increases for more volatile returns as well as for more negatively skewed

payoffs.

3.3 Estimation of the i.i.d. Model

When observations are i.i.d. binormal, the mode and the Pearson’s first coefficient can be

estimated by method of moments or by maximum likelihood (see Mudholkar and Hutson

(2000) for more details). For the MM estimation, we first estimate the mean, the variance

and the skewness and exploit their relations to the mode and the Pearson’s coefficient to

estimate the latter. Concretely, Let R1, ..., RT be a sample of T i.i.d. binormal returns of

parameter (m, σ, p)⊤. Estimates µ, σ and s of the mean, the variance and the skewness are

given by:

µ =T∑

t=1

Rt, σ =

√

√

√

√

1

T

T∑

t=1

(Rt − µ)2 and s =1

T

T∑

t=1

(

Rt − µ

σ

)3

.

Exploiting the third and the first equations in (2.2), estimates p and m of the Pearson’s

first coefficient and the mode satisfy:

p(

1 − (π − 3) p2)

= s and m = µ − σp.

The drawback of the MM procedure is that it does not guarantee that |s| ≤ 0.9953, and

the value of the estimator s can be inconsistent with restrictions required by the binormal

distribution. To the contrary, the ML procedure imposes the required constraint when

maximizing the likelihood. Mudholkar and Hutson (2000), compute the exact form of the

ML estimator in the i.i.d. case and they derive its limiting distribution. In our empirical

study, we explore both MM and ML estimators for the i.i.d. case.

4 Conditional Mode and Pearson’s Coefficient: the Bin-GARCH Model

Following Hansen (1994), many researchers provided both theoretical and empirical evi-

dence that returns asymmetry vary overtime and that modeling the asymmetry is impor-

tant in asset pricing to capture salient features of financial data (see among others, Harvey

and Siddique (1999, 2000), Jondeau and Rockinger (2003), Brooks et al. (2005), Feunou

and Tédongap (2009)). The common point in all this literature is that central tendency

and asymmetry in returns are modeled through conditional mean and conditional skewness

respectively. However, it is also known that these measures as well as excess kurtosis are

very sensitive to outliers, such as for example the crash of October 1987 in S&P500 returns

as well as in many other asset returns and exchange rates.

9

Kim and white (2004) conduct an empirical survey of alternative measures of asym-

metry and fat-tailedness, including measures based on quantiles and the Pearson’s first

coefficient. In particular, they show that this coefficient is more robust to ouliers than the

classical measure of asymmetry (skewness). As shown earlier, the binormal distribution can

alternately be parameterized by its Pearson’s first coefficient and its quantile function exists

in closed-form. This is a substantial advantage since we can directly model the Pearson’s

first coefficient, and also compute all other measures enumerated by Kim and white (2004),

conditionally and in closed-form. Several studies also conclude that conditional skewness

and excess kurtosis only take moderate values. In that sense, bounds on skewness implied

by bounds on the Pearson’s coefficient of the binormal distribution are not restrictive. In

consequence, the binormal distribution appears to be a good parametric alternative for

modeling conditional distribution of returns.4

4.1 Bin-GARCH Model Specification

We model returns as conditionally binormally distributed and we allow for time variation in

the return distribution. To be more specific, we allow for heteroscedasticity as in GARCH

models, and depart from previous literature by not simply allowing time-variation in, but

directly modeling, the mode and the Pearson’s coefficient of the conditional return distribu-

tion. As argued earlier, we rely on the mode and the Pearson’s coefficient to model central

tendency and asymmetry because they are less sensitive to outliers than the mean and the

skewness.

We assume that, conditionally to information up to time t − 1, returns Rt follow a

binormal distribution with mode mt−1, variance σ2t−1 and Pearson’s first coefficient pt−1.

We borrow our specification for heteroscedasticity from the NGARCH model of Engle and

Ng (1993):

σ2t = ω + βσ2

t−1 + ασ2t−1 (zt − θ)2 (4.1)

where zt = (Rt − Et−1 [Rt]) /σt−1 are standardized residuals. This specification accounts

for leverage effect and Christoffersen and Jacobs (2004) show that its has a better out-of-

sample performance in option pricing compared to several alternative GARCH models.

Given that the Pearson’s coefficient is bounded (|pt−1| ≤ 1/

√

π /2 − 1), and using the

hyperbolic tangent transformation to guarantee the bounds, we assume that the Pearson’s

coefficient evolves as follows:

pt =

√

2

π − 2tanh

(

δ0 + δ1z∗t 1{z∗

t>0} + δ2z

∗t 1{z∗

t<0} + δ3pt−1

)

(4.2)

4Other nonnormal distributions have been explored as well. We can cite the Student-t (Bollerslev et al(1992)), the generalized error (Duan (1999)), the normal inverse gaussian (Bollerslev and Forsberg (2002))and the skewed variance gamma (Christoffersen et al. (2006)).

10

where z∗t = (Rt − mt−1) /σt−1 . Asymmetries in the Pearson’s coefficient are generated

by deviations of realized returns with respect to the conditional mode. We recall that

dynamics of volatility and Pearson’s first coefficient of asymmetry lead to indirect downside

and upside volatility modeling through the following relation:

σ1t = σt

(

−√

π /8pt +√

1 − (3π /8 − 1) p2t

)

σ2t = σt

(

√

π /8pt +√

1 − (3π /8 − 1) p2t

)

.(4.3)

We finally specify the conditional mode as:

mt = λ0 + λ1σ1t + λ2σ2t (4.4)

Our specification of the conditional mode is motivated by the ARCH-in-Mean model

of Engle, Lilien and Robins (1987) which relates expected returns to volatility. We recall

from Section 2 that by definition:

σ1,t−1 =

√

π

π − 2V art−1 [Rt | Rt < mt−1], σ2,t−1 =

√

π

π − 2V art−1 [Rt | Rt ≥ mt−1]. (4.5)

Since the mode, as the mean, also characterizes the central tendency, we assume in equa-

tion (4.4) that the future conditional mode is linear in volatilities of returns conditional

to realizations above and below the current conditional mode. It then follows that the

conditional mean is given by:

Et−1[Rt] = mt−1 + pt−1σt−1 = λ0 +

(

λ1 −

√

2

π

)

σ1,t−1 +

(

λ2 +

√

2

π

)

σ2,t−1. (4.6)

4.2 Bin-GARCH and Risk-Return Tradeoff

The first equality in (4.6) holds by definition of the Pearson’s first coefficient of asymmetry

whereas the second results from our Bin-GARCH model specification. As we discuss below,

these two equalities are particularly important for understanding the risk-return tradeoff.

First, if both the conditional mode and the condition Pearson’s coefficient are constant,

the first equality in (4.6) says that they are respectively the drift and the slope of the

linear regression of returns onto conditional volatility. In this case, a negative Pearson’s

coefficient implies that expected returns fall due to an increase in volatility. In consequence,

the positive linear relation between expected returns and volatility, as suggested by the

intertemporal capital asset pricing model (ICAPM) of Merton (1973), would be inconsistent

with the fact that both the conditional mode and the conditional Pearson’s coefficient are

constant and the latter is negative. There are conflicting findings in the literature about

the true nature of the linear relation between expected returns and volatility. Ghysels,

11

Santa-Clara and Valkanov find a significantly positive relation between risk and return

in the stock market. French et al. (1987) and Campbell and Hentschel (1992) find that

the relation is positive and insignificant. In contrast, Campbell (1987) and Nelson (1991)

find a significantly negative relation. Based on our previous discussion, we argue that

modeling the mode and the Pearson’s first coefficient of asymmetry appears important for

characterizing this relation.

Second, it is worth noting from equation (4.5) that σ1,t−1 and σ2,t−1 are respectively

measures of market downside and upside volatilities using the conditional mode of returns

as the cutoff point between up-markets and down-markets (see Ang, Chen and Xing (2006)

for further discussion about downside and upside volatilities). Measuring risk through

volatility, if equity tends to be more volatile in a bear market than it is in a bull market

(that is σ1,t−1 > σ2,t−1), then, investors require a compensation for holding it, since equity

tends to have very low payoffs precisely when they feel poor and pessimist, compared to

when they feel wealthy and confident. Then, the relative downside volatility, σ1,t−1−σ2,t−1,

should be remunerated by appropriate returns. From equation (4.4) and from the second

equality in (4.6), if λ2 = −λ1, then conditional mode and expected returns are determined

by relative downside volatility:

mt−1 = λ0 +λ1 (σ1,t−1 − σ2,t−1) and Et−1[Rt] = λ0 +

(

λ1 −

√

2

π

)

(σ1,t−1 − σ2,t−1) . (4.7)

Later in the empirical section, we discuss results of the Bin-GARCH model that imposes

this restriction.

The binormal distribution used in our return modeling appropriately takes part in un-

derstanding how market volatility and asymmetry affect downside and upside risks. Panel

A and B of Figure 3 show that downside risks are important in more volatile and negatively

asymmetric markets, and so more than upside risks. In contrast, Upside risks dominate

downside risks in more volatile and positively asymmetric markets. An alternate inter-

pretation is provided in Panels C and D of Figure 3. Panel C shows that an increase in

either downside or upside volatility also increases market Volatility. In contrast, as shown

in Panel D, an increase in downside volatility lowers market skewness whereas an increase

in upside volatility also increases market skewness.

5 Empirical Results

5.1 Data and Preliminary Analysis

We use S&P500 index daily returns from January 2, 1980 to December 30, 2005, for a total

of 6564 observations. Table 2 summarizes the MM estimation described in Section 3.3. The

12

MM estimate of the skewness is huge, and equal to -1.736 as shown in the column “All”,

due to the crash of October 1987. Because this value is out of the bounds imposed by the

binormal distribution, there is no solution for the Pearson’s first coefficient and then for

the mode. When this observation is removed from the sample (column “Crash Out”) as well

as when the three observations greater or equal to 7% in absolute value are removed from

the sample (column “≥7% Out”), then the skewness estimate falls sharply to more than

10 times lower in absolute value. This corroborates the findings of Kim and White (2004)

that skewness is very sensitive to outliers. This lower value of skewness (about -0.160)

is far less (in absolute value) than the maximum skewness of -0.9953 generated by the

binormal distribution. The binormal distribution is then a good alternative to the normal

distribution for modeling asymmetries in S&P500 index returns. Since individual stock are

far less negatively skewed than the market index, the binormal distribution would also be

accurate for individual stocks.

In Table 2 we also provide estimates of the Pearson’s coefficient and the mode for feasi-

ble values of the skewness. The value of the Pearson’s first coefficient of asymmetry is very

closed to the one of the classic skewness as argued by Garvin and McClean (1997) for mod-

erately skewed distributions. This is definitely an advantage of the binormal distribution

as, without outliers in the data, the Pearson’s first coefficient can be viewed as the classic

skewness. Moreover, it is common to exclude outliers in the data while dealing with the

estimation (even for ML) of the classic skewness, there is no need to do that as far as the

binormal distribution is used because the Pearson’s first coefficient is robust.

We estimate the i.i.d. binormal model by maximum likelihood, and results as shown in

Table 3 indicate that all parameters are significant at the 1% level, and for the Pearson’s

first coefficient, that the ML estimate is smaller (in absolute value) than the MM estimate.

5.2 Implied Black-Scholes Volatility and Hedging

We evaluate the ability of the binormal option pricing model introduced in Section 3.1 to

reproduce an observed implied volatility curve. We compare binormal, Black-Scholes and

“model free” deltas and gammas which are very useful in practice to replicate a portfolio

of options.

Following Bates (2005), we use call option prices written on S&P500 index with 23 days

to maturity (DTM) quoted on August 29, 2002, a total of 18 contracts. The index price

was 917.80, the dividend yield 0.004360%, the interest rate 0.004521% and the volatility

1.34%. Call option data and its characteristics (the implied volatility, σ, and Black-Scholes

sensitivities (the hedging ratio, ∆ and the gamma, Γ) are summarized in Table 1.

The “model free” Black-Scholes volatility is obtained by regressing the Black-Scholes

13

implicity volatility onto the strike price and its square (see Shimko (1993)):

σ (K) = A0 + A1K + A2K2. (5.1)

The regression-based estimate of the volatility function using the implicit volatility from

Table 1 is σ (K) = 0.18257 − 0.00033312K + 1.6294 × 10−7K2.

We estimate parameters of the binormal option pricing model (mode m, volatility σ

and Pearson’s first coefficient p) by minimizing the mean squared error (MSE, the distance

between the observed price vector and the model price vector). We find a daily volatility

of 1.4247% and a Pearson’s first coefficient is −1.3236 (which is the highest possible value

of the Pearson’s first coefficient generated by the binormal model). The RMSE of 1.0932,

indicates a great improvement of the binormal model compared to Black-Scholes-Merton

(RMSE of 1.4975).

To test the ability of the binormal option pricing model to generate actual “smile” and

“skew”, we represent in Panel A of Figure 1 the “model free”, the observed and the binor-

mal’s BSM implied volatilities. As expected, a negative Pearson’s first coefficient generates

an upward implied volatility curve (as function of moneyness), although the observed im-

plied volatilities are still deeply skewed compared to the binormal implied volatility. One

possible explanation is that the binormal distribution cannot generate enough skewness to

restore the pattern observed in the data. The risk neutral skewness necessary to fit the

actual pattern is more higher, in absolute value, than the maximum skewness of −0.9953

generated by the binormal distribution. We advocate the use of the binormal for conditional

distribution instead of unconditional.

We now turn to the hedging performance of the binormal model. Despite the well

known BSM model bias, the BSM Greeks are still widely used to hedge option positions.

Bates (2006) shows that BSM deltas and gammas are also subject to important biases, and

proposes a bias correction based on a nonparametric model. The “true” Greeks are related

to the BSM Greeks as follows:

∆ = ∆BS − CBSσ

K

S

∂σ

∂Kand Γ = ΓBS +

(

K

S

)2[

2CBSKσ

∂σ

∂K+ CBS

σσ

(

∂σ

∂K

)2

+ CBSσ

∂2σ

∂K2

]

,

where ∆BS and ΓBS are Black-Scholes delta and gamma computed at that option’s implicit

volatility, CBS is the call price and CBSσ , CBS

σσ and CBSKσ are its first and second order partial

derivatives relative to σ and K.

Since we have the binormal option price in closed-form, it is straightforward to compute

binormal Greeks and compare them to BSM and the “true” Greeks. We plot deltas in Panel

B of Figure 1 and the gammas in Panel C of Figure 1. The implied volatility curve in

14

Panel A of Figure 1 reveals a substantial slope, also implying that Black-Scholes delta and

gamma are biased estimates of the true delta and gamma (Model’s free Greeks) as shown

in Panels B and C of Figure 1 respectively. The binormal Greeks improve significantly

on the Black-Scholes delta and gamma, although there are still biased estimates of the

“Model free” Greeks. Nevertheless, Bates (2005) also advocates to interpret the “model

free” with caution, since in fact, it relies on hypotheses such as homogeneity which may

not be verified.

5.3 Bin-GARCH Model Estimation and Discussion

We now estimate the Bin-GARCH model by maximum likelihood and discuss results shown

in Table 4. Specification (I) in column 2 corresponds to the canonical NGARCH model (the

conditional Pearson’s first coefficient is zero and the mode, which in this case is equal to

the mean, is constant) and constitutes our benchmark for models comparison. In column

3, we depart from the NGARCH model by allowing a constant but nonzero Pearson’s coef-

ficient of asymmetry in specification (II). The estimated value of the constant conditional

Pearson’s first coefficient is -0.0957. Results for this specification confirms that S&P500

index returns are conditionally negatively skewed. The gain in likelihood resulting from

the inclusion of a single parameter from (II) to (I), the associated LR test and the infor-

mation criterion all indicate that the NGARCH with i.i.d. gaussian standardized residuals

is strongly rejected in favor of the GARCH with i.i.d. binormal standardized residuals.

Estimates of constant conditional mode and Pearson’s coefficient are respectively positive

and negative and strongly significant. As discuss in Section 4, this leads to a negative

relationship between expected returns and volatility. A positive risk-return relation would

simply mean that either the mode or the Pearson’s coefficient is misspecified, or both.

We now keep the mode constant and allow the Pearson’s coefficient to vary over time

and follow the nonlinear autoregressive dynamics specified in (4.2). This correspond to the

specification (III) in the fourth column of Table 4. All parameters are strongly significant

and the inclusion of three more parameters relative to specification (II) induces a substantial

gain in likelihood. The corresponding LR test and information criterion also strongly reject

the constant Pearson’s coefficient against the actual GARCH-in-asymmetry. Results also

suggest that, realizations of returns relative to the conditional mode have different impacts

of conditional asymmetry measured through the Pearson’s first coefficient. Returns above

the conditional mode increase the Pearson’s coefficient a lot more than returns below,

and of same absolute difference relative to the conditional mode, reduce this asymmetry

(estimates of δ1 and δ2 are both positive and δ1 is 3 times higher than δ2).

We now turn to specification (IV) in column 5 which only has the meaningful restriction

15

λ2 = −λ1, and to the full Bin-GARCH specification (V) in column 6 of Table 4. Again, all

parameters are strongly significant but λ0, the drift in the mode, only in specification (V).

Two major facts are worth noting relative to specification (III). First, an increase in the

downside volatility increases the future conditional mode (the estimate of λ1 is positive),

whereas an increase in the upside volatility decreases the conditional mode (the estimate of

λ2 is negative). Second, responses of the asymmetry to return realizations above and below

the conditional mode increase when the conditional mode becomes time-varying (estimates

of δ1 and δ2 are more than twice their respective values when the mode is constant over

time). The LR test at 1% level rejects specifications (I) to (III) in the table against

specification (IV) and against the full Bin-GARCH specification. The same test rejects

specification (IV) at the 5% level against specification (V), but not at the 1% level, and

the information criterion tends to favor specification (IV) over the full Bin-GARCH. This

latter observation deserves more attention. It simply means that data evidence the fact

that the relative downside volatility σ1,t−1−σ2,t−1 predicts returns Rt as shown in equation

(4.7). The positive coefficient λ1, significantly estimated at the 1% level in column 5 for

specification (IV) in Table 4, is such that λ1 −√

2 /π < 0, meaning from equation (4.7)

that expected returns are negatively related to the downside volatility, but instead, the

conditional mode is positively related to the downside volatility.

As mentioned earlier, several papers throughout the literature find conflicting results

about the relationship between risk and return. Previous research has focused almost

all attention on measuring the risk-return tradeoff simply through a relation between the

expected return and return volatility, as predicted by major asset pricing models. However,

despite the theoretical motivation and the simplicity of its relation to volatility, the expected

return, or the conditional mean, might not be a critical or even reliable measure for risk

compensation. If the return distribution is negatively skewed (with a negative Pearson’s

first coefficient of asymmetry), we argue that the mode of the return distribution may

provide a better measure for risk compensation than the mean.

To make our point, and for simplicity, assume that return can only take two values

and that the probability of the highest value is close to one. First, the mode, here the

highest value, is greater than the mean since returns are negatively skewed with a negative

Pearson’s first coefficient. Second, the mode has the highest probability to realize. Then,

investors would like to see the mode increasing with volatility as they know it is probably

going to realize, and their risk is compensated anyway, even though the mean decreases with

volatility. Third, the mode is less sensitive to outliers that may affect return prediction.

The Bin-GARCH model introduced in this paper provides a good alternative for un-

derstanding the risk-return tradeoff, although the lack of an explicit theory suggesting risk

16

compensations through its channels. Figure 4 shows the plots of annualized daily expected

returns in Panel A and conditional mode in Panel B. On average, annualized daily ex-

pected returns and conditional mode over the sample are respectively 10.23% and 35.50%.

The two series are negatively correlated precisely because expected returns fall following

an increase in relative downside volatility, whereas the conditional mode is driven up and

is more sensitive to fluctuations in downside volatility, as predicted by the Bin-GARCH

estimates. As discussed earlier, this is an evidence of relative downside volatility being

rewarded through an increase in the conditional mode instead of expected returns.

The annualized daily volatility and the daily Pearson’s coefficient are plotted in Panels

C and D of Figure 4. On average, the annualized daily volatility over the sample is 15.19%

and the daily Pearson’s coefficient of −0.0926 matches the value estimated for the i.i.d.

specification (II) in column 3 of Table 4. Fluctuations in the Pearson’s coefficient show

that, although the conditional asymmetry is centered to a negative value, stock returns can

be positively skewed. This contrasts with the IG-GARCH model of Christoffersen, Heston

and Jacobs (2006) which imposes a negative conditional skewness over time. Instead,

the direction of asymmetry in the Bin-GARCH model is commanded by relative downside

volatility. Returns are negatively skewed only if equity is more volatile in a declining market

than it is in a rising market, and are positively skewed otherwise. We finally represent the

extracted series of downside and upside volatilities in Panels E and F of Figure 4. On

average, annualized daily downside and upside volatilities over the sample are respectively

16.13% and 14.13% (a relative downside volatility of 2%). The two volatilities are also

highly correlated(a correlation of 0.82), and this suggests that the two volatilities usually

move in the same direction.

We now analyze the news impact curves resulting from the Bin-GARCH model. Panel A

of Figure 7 shows reactions of market volatility, as well as downside and upside volatilities,

to return shocks. The asymmetric pattern that emerges for market volatility is interest-

ing and corroborates existing findings. Positive return shocks and slightly negative return

shocks lower market volatility. In contrast, extreme positive return shocks increase market

volatility just as (or less than) do negative return shocks. This asymmetric pattern also

transmits to downside and upside volatilities. Positive return shocks lower downside volatil-

ity while negative return shocks of same magnitude increase downside volatility and even

more. In contrast, negative return shocks lower upside volatility as well as slightly positive

return shocks. Extreme negative return shocks tends to not impact upside volatility while

positive return shocks increase upside volatility sharply. Panel B of Figure 7 finally con-

firms the asymmetric pattern in market asymmetry. It shows that negative return shocks

today increase the likelihood of negative return shocks tomorrow, whereas positive return

17

shocks today increase the likelihood of similar shocks tomorrow and even more.

5.4 Bin-GARCH Diagnostics

We perform several experiments to assess the ability of the Bin-GARCH model to fit the

data. First, we evaluate the ability of the Bin-GARCH to reproduce the observed implied

volatility curve discussed in Section 5.2 for the case of the i.i.d. binormal model. We

recall that the i.i.d. binormal model, although it produces an upward IV curve as the

actual, it does not generate enough skewness unconditionally to match the observed IV

curve. We then advocated to use of the binormal for conditional (instead of unconditional)

distribution of returns. Figure 2 compares the IV curve when returns are Bin-GARCH

to the actual IV curve. The Bin-GARCH model clearly restores the actual pattern of the

IV curve. Although the constraint imposed by the Bin-GARCH model on the conditional

Pearson’s coefficient (and by then the skewness) never binds as shown in Panel D of Figure

4, the model is still able to generate the amount of unconditional skewness necessary to

restore the observed pattern of the IV curve.

Next, test if the model would produce accurate measures of the VaR. We have shown

in Section 3.2 that the i.i.d. binormal model provides simple closed-form VaR and ES

formulas. These formulas hold conditionally for one-day ahead VaR and ES. To test for

the ability of the Bin-GARCH model to generate a correct conditional coverage, we follow

Christoffersen (1998) who derives a test for correct conditional coverage as a joint test of

unconditional coverage and independence, and which we recall in the following. Under the

null hypothesis of unconditional coverage one has:

E

[

1

T

T∑

t=1

1{Rt>−V aRt−1(1,α)}

]

= 1 − α (5.2)

where 1{Rt>−V aRt−1(1,α)} follows a Bernoulli distribution of parameter 1 − α. A likelihood

ratio test of the null hypothesis uses the following statistic:

LRuc = 2[

ln(

πT1

1 (1 − π1)T−T1

)

− ln(

(1 − α)T1 αT−T1

)]

(5.3)

where T1 is the number of observations such that Rt > −V aRt−1 (1, α), T is the total

number of observations and π1 = T1 /T . Under the null of unconditional coverage, LRuc is

a chi-square with one degree of freedom.

Since we are examining conditional VaR, Christoffersen(1998) also suggests testing for

independence. The null hypothesis of independence is:

Prob[

1{Rt>−V aRt−1(1,α)} = j | 1{Rt−1>−V aRt−2(1,α)} = i]

= Prob[

1{Rt>−V aRt−1(1,α)} = j]

.

(5.4)

18

Similar to the test for unconditional coverage, a likelihood ratio test for independence uses

the following statistic:

LRind = 2[

ln(

πT01

01 (1 − π01)T0−T01 πT11

11 (1 − π11)T1−T11

)

− ln(

(1 − α)T1 αT−T1

)]

, (5.5)

where Tij is the number of observations 1{Rt−1>−V aRt−2(1,α)} = i followed by observations

1{Rt>−V aRt−1(1,α)} = j, and πij = Tij /Ti . Under the null of independence, LRind is a

chi-square with one degree of freedom.

The test for conditional coverage has the joint hypothesis (5.2)-(5.4) and uses the statis-

tic LRuc + LRind, which under the null, is a chi-square with two degrees of freedom. We

perform the three tests on the Bin-GARCH, both for α = 1% and α = 5%, and we present

results in Table 5.

The percentage π1 of realized returns above the Value-at-Risk is high as expected. The

unconditional coverage for the conditional VaR at 99% confidence level is not rejected

at conventional levels of significance (the p-value is 0.3698). The same conclusion holds

for independence and conditional coverage (p-values of 0.2558 and 0.3506 respectively).

Instead, unconditional coverage, independence and conditional coverage are all rejected at

conventional levels of significance for the conditional VaR at 95% confidence level (p-values

are less than 0.015). Finally, the Bin-GARCH model is accurate for measuring conditional

VaR at 99% confidence level for the S&P500 index.

Finally, we turn to evaluate the ability of the Bin-GARCH to fit, not only “worst”

events as the VaR, but to forecast the conditional return distribution (the complete pattern

of conditional quantiles). Our test follows the work of Diebold et al. (1998) and Bai

(2003) who show that the transformed variables Ut−1 = F (Rt | It−1) , t = 1, .., T , are i.i.d.

uniform over (0, 1) if and only if the forecasts F (Rt | It−1) are correct. It−1 represents the

information up to time t − 1, and F (· | It−1) is the cumulative distribution function of

returns Rt. This cumulative distribution function is binormal in the Bin-GARCH model

and exists in closed-form. More specifically, it is given by:

F (Rt | It−1) = 1 − Mt−1 (0; Rt) (5.6)

where Mt−1 (u) and Mt−1 (u; x) which refer to equations (2.5), (2.6), and (2.7), are respec-

tively the moment generating function and the truncated moment generating function of

a binormal distribution of parameter (mt−1, σ1,t−1, σ2,t−1)⊤. The processes mt, σ1t and σ2t

are specified in Section 4.

Based on the previous result, to evaluate density forecasts, the uniformity and serial

independence of the Ut’s, we rely on graphical tests. We draw the uniform QQ plot of

Ut in Figure 5 (the quantiles of the series Ut are plotted against those of the uniform

19

distribution over (0, 1)). This diagnoses if Ut follows a uniform distribution over (0, 1).

The plot reveals that the Ut’s are very closed for being uniformly distributed, despite small

deviations observed around the quantile 75%.

Also remark that if Ut is uniform over (0, 1), then Φ−1 (Ut) is standard normal, where

we recall that Φ is the cumulative distribution function of the standard normal. We then

draw the autocorrelation function of Φ−1 (Ut) in Figure 6. This diagnoses if the Ut’s are

serially independent. The graph suggests that autocorrelations are insignificant. Thus the

binormal distribution is a candidate alternative for conditional returns. The dependence of

higher-order moments are well captured by the Bin-GARCH model through a time-varying

Pearson’s first coefficient.

6 Conclusion and Future Work

This paper presents a discrete-time dynamic model of asset prices with binormal return

innovations, and the new (Bin-GARCH) model nests the canonical NGARCH model. Com-

pared to existing GARCH models, the Bin-GARCH explicitly relates downside and upside

risks to both conditional heteroscedasticity and conditional asymmetry. The model also

relates the mode of conditional return distribution to downside and upside volatilities.

Although previous studies assess risk compensations through expected return channels,

empirical tests strongly suggest that relative downside risks are remunerated by an in-

crease in the conditional mode. This finding is particularly appealing as it opens the room

to other channels through which risks are compensated in equity markets. In particular

investors utility may increase with other return characteristics than expected returns.

Several experiments assess the performance of the Bin-GARCH model in different di-

mensions. In particular, the Bin-GARCH model generates enough unconditional skewness

to restore the observed implied volatility curve. Also, back testing and density forecast

evaluation of the model show its accuracy in measuring conditional value-at-risk and in

fitting conditional return distribution.

In a future research it would be interesting to exploit the analytical formula of option

prices provided by the i.i.d. binormal model, to fit daily index option data and measure

downside and upside volatilities implicit in option prices. The resulting relative downside

volatility may be used as return predictor, or to fit an empirical measure of the mode of

conditional return distribution.

20

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24

Table 1: August 29, 2002 Call Options on S&P 500 Index, Maturity = 23

Strike Price Call Price Implicit Parameters

Volatility (σ) Delta (∆) Gamma (Γ)

870 57.60 0.0160 0.76833 0.0043167875 53.70 0.0158 0.74738 0.0045828880 49.90 0.0157 0.72392 0.0048266885 46.20 0.0155 0.70028 0.0050836890 42.60 0.0153 0.67526 0.0053327900 35.30 0.0146 0.62309 0.0059011905 32.20 0.0145 0.59318 0.0060711910 29.00 0.0143 0.56271 0.0062520915 26.70 0.0144 0.53107 0.0062681920 23.40 0.0139 0.49869 0.0065140925 20.80 0.0138 0.46589 0.0065379935 16.50 0.0136 0.40032 0.0064504940 14.30 0.0134 0.36686 0.0063793945 12.40 0.0132 0.33379 0.0062505950 10.90 0.0132 0.30392 0.0060141960 8.10 0.0130 0.24508 0.0054923965 6.80 0.0128 0.21582 0.0051955970 5.70 0.0127 0.19001 0.0048527

25

Table 2: MM Estimation of the i.i.d. Model

Parameter All Crash Out ≥ 7% Out

µ 0.00037 0.00041 0.00041

σ 0.01047 0.01008 0.00997

s -1.73579 -0.15995 -0.16499

m 0.00203 0.00206

p -0.16054 -0.16563

26

Table 3: ML Estimation of the i.i.d. Model

Parameter All Crash Out ≥ 7% Out

µ 0.00019 0.00037 0.000370.00013 0.00013 0.00012

σ2 1.091E-4 1.016E-04 9.932E-051.908E-6 1.774E-06 1.734E-06

s -0.13246 -0.03672 -0.038820.01880 0.02043 0.02067

m 0.00158 0.00074 0.000760.00022 0.00022 0.00022

p -0.13279 -0.03673 -0.038830.01894 0.02044 0.02069

Log-Lik. 20637.5360 20860.4622 20928.1333BIC -3.1400 -3.1745 -3.1858

27

Table 4: ML Estimation of Various Bin-GARCH Models

(I) (II) (III) (IV) (V)

m 2.87E-04 1.00E-03 8.05E-041.01E-04 1.86E-04 1.94E-04

λ0 4.90E-04 -3.53E-041.13E-04 3.77E-04

λ1 0.6004 0.62260.0532 0.0529

λ2 -0.6004 -0.51230.0654

ω 1.60E-06 1.30E-06 1.32E-06 1.35E-06 1.70E-062.70E-07 2.42E-07 2.44E-07 2.32E-07 3.09E-07

β 0.8872 0.8943 0.8866 0.8915 0.88460.0098 0.0092 0.0103 0.0094 0.0106

α 0.0652 0.0631 0.0600 0.0609 0.06130.0058 0.0057 0.0055 0.0054 0.0054

θ 0.7333 0.7401 0.8608 0.7824 0.78990.0703 0.0702 0.0860 0.0785 0.0813

p 0.0000 -0.09570.0203

δ0 -0.0554 -0.0844 -0.08590.0150 0.0197 0.0193

δ1 0.0916 0.1962 0.19750.0175 0.0327 0.0320

δ2 0.0295 0.0719 0.07200.0105 0.0151 0.0148

δ3 0.2167 0.2116 0.21510.0831 0.0721 0.0703

Lik. Gain 21569.0410 21579.9164 21605.2466 21628.7235 21631.72680.0000 +10.8754 +36.2056 +59.6825 +62.6858

BIC -3.2793 -3.2796 -3.2794 -3.2817 -3.2808

28

Table 5: Test of Independence and Unconditional Coverage

α = 1% α = 5%

π01 0.9726 0.9510π10 0.0109 0.0432π1 0.9889 0.9564

LRuc 0.8044 5.9598p − value 0.3698 0.0146

LRind 1.2916 6.4819p − value 0.2558 0.0109

LRuc + LRind 2.0959 12.4417p − value 0.3506 0.0020

29

Figure 1: IV Curve, Delta and Gamma from the i.i.d. Model

A. Implied Volatility Curve B. Delta

0.94 0.96 0.98 1 1.02 1.04 1.060.0125

0.013

0.0135

0.014

0.0145

0.015

0.0155

0.016

0.0165

Moneyness (S/X)

Impl

ied

Vol

atili

ties

Observed

"Model Free"

"Binormal"

0.94 0.96 0.98 1 1.02 1.04 1.060.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Moneyness (S/X)

Del

ta

Black−ScholesBinormalModel Free

C. Gamma

0.94 0.96 0.98 1 1.02 1.04 1.063

3.5

4

4.5

5

5.5

6

6.5

7

7.5x 10

−3

Moneyness (S/X)

Gam

ma

Black−ScholesBinormalModel Free

30

Figure 2: IV Curve from the Binormal GARCH Model

0.94 0.96 0.98 1 1.02 1.04 1.060.24

0.25

0.26

0.27

0.28

0.29

0.3

0.31

Moneyness (S/X)

Ann

ual I

mpl

ied

Vol

atili

ties

ObservedBin−GARCH

31

Figure 3: Down and Upside Volatilities vs. Market Volatility and Asymmetry

A. B.

020

4060

80100

−1.5−1

−0.50

0.51

1.50

50

100

150

200

Market VolatilityMarket Asymmetry

Dow

nsid

e R

isk

020

4060

80100

−1.5−1

−0.50

0.51

1.50

50

100

150

200

Market VolatilityMarket Asymmetry

Ups

ide

Ris

k

C. D.

020

4060

80100

020

4060

80100

0

20

40

60

80

100

Downside VolatilityUpside Volatility

Mar

ket V

olat

ility

020

4060

80100

020

4060

80100

−1.5

−1

−0.5

0

0.5

1

1.5

Downside VolatilityUpside Volatility

Mar

ket A

sym

met

ry

32

Figure 4: Annualized Daily Series

A. Annualized Expected Returns B. Annualized Conditional Mode

Dec 1983 Nov 1987 Nov 1991 Oct 1995 Oct 1999 Oct 2003−200

−150

−100

−50

0

50

100

150

200

Dec 1983 Nov 1987 Nov 1991 Oct 1995 Oct 1999 Oct 2003−200

−100

0

100

200

300

400

C. Annualized Market Volatility D. Pearson’s Coefficient

Dec 1983 Nov 1987 Nov 1991 Oct 1995 Oct 1999 Oct 2003

10

20

30

40

50

60

70

80

90

100

Dec 1983 Nov 1987 Nov 1991 Oct 1995 Oct 1999 Oct 2003

−1

−0.5

0

0.5

1

E. Annualized Downside Volatility F. Annualized Upside Volatility

Dec 1983 Nov 1987 Nov 1991 Oct 1995 Oct 1999 Oct 2003

10

20

30

40

50

60

70

80

90

100

Dec 1983 Nov 1987 Nov 1991 Oct 1995 Oct 1999 Oct 2003

10

20

30

40

50

60

70

80

90

100

33

Figure 5: Density Forecast of Standardized Residuals

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 6: Autocorrelation Function of Standardized Residuals

The horizontal dashed lines denote 95% Bartlett confidence intervals around zero.

10 20 30 40 50 60 70 80 90 100−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

Lag Order

Aut

ocor

rela

tion

34

Figure 7: Volatility and Asymmetry News Impact Curves

A.

−5 −4 −3 −2 −1 0 1 2 3 4 55

10

15

20

25

30

35

40

Return Shocks

Vol

atili

ty

Market Volatility

Downside Volatility

Upside Volatility

B.

−5 −4 −3 −2 −1 0 1 2 3 4 5

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Return Shocks

Mar

ket A

sym

met

ry

35