closed-form transformations from risk-neutral to real-world distributions

8/17/2019 Closed-Form Transformations From Risk-neutral to Real-World Distributions

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Closed-form Transformations from Risk-neutral to

Real-world Distributions

Xiaoquan Liu*, Mark B Shackleton*,

Stephen J Taylor* and Xinzhong Xu**

*Department of Accounting & Finance, Lancaster University

**Guanghua School of Management, Peking University

December 2002

Revised May 2003

Contact information for correspondence :

Stephen J Taylor, Department of Accounting & Finance, Management School,

Lancaster University, England LA1 4YX, telephone + 44 1524 593624, e-mail

[email protected]


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Abstract

Risk-neutral (RN) and real-world (RW) densities are derived from option prices and

risk assumptions, and are compared with densities obtained from historical time

series. Two parametric methods that adjust from RN to RW densities are investigated,

firstly a CRRA risk aversion transformation and secondly a statistical calibration.

Both risk transformations are estimated using likelihood techniques, for two flexible

but tractable density families. Results for the FTSE-100 index show that densities

derived from option prices have more explanatory power than historical time series.

Furthermore, the pricing kernel between RN & RW densities may be more regular

than previously reported and a more reasonable risk aversion function is estimated.


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1. Introduction

Options prices provide a rich source of information for estimating risk-neutral

densities (RNDs) because a complete set of strikes can be used to infer the

distribution of the underlying asset price when the options expire. As a result many

density specifications have been estimated from options prices. Parametric

specifications include a mixture of lognormals [Ritchey (1990), Melick and Thomas

(1997)], polynomials multiplied by a lognormal [Madan and Milne (1994)], a

generalized beta [Anagnou, Bedendo, Hodges and Tompkins (2002)] and the densities

of continuous-time price processes when volatility is stochastic [Bates (2000),

Jondeau and Rockinger (2000)]. Other approaches include maximum entropy

densities [Buchen and Kelly (1996)], non-parametric estimates [Ait-Sahalia and Lo

(1998)], multi-parameter discrete distributions [Jackwerth and Rubinstein (1996)] and

densities implied by smile functions, defined by either polynomials [Shimko (1993),

Malz (1997)] or spline functions [Bliss and Panigirtzoglou (2002a)].


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In recent years, however, attention has shifted to the relationship between risk-

neutral densities and real-world densities 1 . One strand of literature investigates

methods that transform RNDs into real-world densities [Anagnou et al (2002),

Bakshi, Kapadia and Madan (2003), Bliss and Panigirtzoglou (2002b)]. These

methods are important for central bankers and other decision takers who wish to infer

market beliefs about future distributions from market prices. Another strand uses

RNDs and real-world densities obtained from historical time series of asset returns to

obtain insights into empirical asset pricing kernels and the aggregate risk preferences

of market participants [Jackwerth (2000), Ait-Sahalia and Lo (2000), Perignon and

Villa (2002), Rosenberg and Engle (2002), Brown and Jackwerth (2002)]. The results

are also important for assessments of the rationality of option prices.

This paper also explores the relationships between risk-neutral distributions,

real-world distributions, aggregate market risk preferences and empirical pricing

kernels. We provide answers to two questions – (a) how can real-world densities be

calculated rapidly from option prices ?, (b) are these densities more informative than

those provided by historical time series ?. There are also two primary methodological

contributions. First, two parametric closed-form methods are used to obtain real-world

distributions from RNDs, namely a utility transformation and a statistical calibration

transformation. Our RNDs are obtained both from a mixture of two lognormal

densities and a generalized beta density, which results in four series of real-world

densities that incorporate risk factors. Second, maximum likelihood estimation (MLE)

1 The adjectives ‘real-world’, ‘risk-adjusted’ and ‘subjective’ are used interchangeably in the research literature.

They all refer to price distributions in which market risk preferences are embedded.


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is used to efficiently estimate the transformation parameters and to compare the log-

likelihoods of risk-adjusted distributions with those of historical distributions obtained

by simulation of an asymmetric GARCH model.

Our paper is most closely related to the contemporaneous research of Anagnou

et al (2002) and Bliss and Panigirtzoglou (2002b). These papers only consider the

utility transformation and they do not use MLE to estimate their risk aversion

parameters : Bliss and Panigirtzoglou use indirect estimates based upon calibration

concepts, while Anagnou et al evaluate parameter values but do not estimate them.

Historical real-world densities are not compared with option densities by Bliss and

Panigirtzoglou. Comparisons are made by Anagnou et al but they do not compare the

likelihoods of their sets of densities.

Risk-neutral and historical densities provide sufficient information to estimate

representative risk aversion functions. We obtain the first estimates of these functions

for the U.K. equity market, complementing the studies of the U.S. market by

Jackwerth (2000), Ait-Sahalia and Lo (2000) and Rosenberg and Engle (2002) and of

the French market by Perignon and Villa (2002). The most detailed analysis is by

Rosenberg and Engle (2002) who estimate the empirical pricing kernel on a daily

basis for the S&P 500 index, so as to capture time-varying features in RNDs and real-

world densities. They fit a semi-parametric spline function to the implied volatility

smile, to obtain risk-neutral densities, and estimate an asymmetric GARCH model for

index returns allowing Monte Carlo simulations of real-world distributions. They find

that empirical pricing kernels are time-varying, are generally downward sloping and


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that the assumption of constant relative risk aversion (CRRA) is not an accurate

depiction of actual market risk preferences.

The empirical results in this paper are obtained from FTSE-100 futures and

options contracts traded in London that cover the period from 1993 to 2000. The

results show that the real-world densities defined by utility transformed RNDs and

statistically calibrated RNDs have significantly higher log-likelihoods than the RNDs

themselves. Also, the log-likelihoods of the RNDs and their transformed densities

exceed those for historical distributions and hence option prices contain incremental

information about real-world distributions. The average empirical pricing kernel for

the London market is found to be generally downward sloping and does not exhibit

the hump shape observed by Brown and Jackwerth (2002). However, the implied risk

aversion and relative risk aversion are U-shaped, an anomaly also found by Jackwerth

(2000) and Ait-Sahalia and Lo (2000) that confronts economic theory.

The paper is organized as follows. Section 2 describes parametric risk-neutral

densities that can be transformed into closed-form real-world densities. Section 3

documents the transformations and explains how their parameters can be estimated.

Section 4 explains data processing for options written on the FTSE 100 index and

summarizes the derivation of the risk-neutral and real-world densities. Section 5

discusses the estimated densities and measures of risk aversion in the context of

recent research. Finally, Section 6 concludes.


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2. Theoretical risk-neutral densities

Breeden and Litzenberger (1978) show that a unique risk-neutral density g for a

subsequent asset price T S can be inferred from European call prices )( X c when

contracts are priced for all strikes X and there are no arbitrage opportunities. The risk-

neutral density (RND) is then

2

2

)(

X

ce X g rT

∂

∂= (1)

and

∫ −=∞−

X

rT dx x g X xe X c )()()( (2)

with r the risk-free rate and T the time remaining until all options expire. The forward

price F , for time T , is the risk-neutral expectation of T S ; it is also a futures price,

assuming non-stochastic interest rates and dividend payments. These relationships

between the RND and derivative prices are the basis for empirical derivations of

implied RNDs, despite the impossibility of obtaining option data for a continuum of

strikes.

Two parametric families of RNDs are estimated in this paper. Once a month,

and for each family, a parameter vector θ is estimated by minimizing the average

squared difference between observed market prices and theoretical option prices,

namely

∑ −=

N

j j jmarket

X c X c N 1

2))|()((1

θ , (3)


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with ∫ −=∞−

j X j

rT j dx x g X xe X c )|()()|( θθ , N j ≤≤1 . (4)

In these equations, N is the number of prices obtained from option quotes or trades

during a particular day and )( θ x g is a parametric density function that produces the

theoretical option pricing formula )( θ X c given by equation (2). We choose specific

parametric densities for the RNDs because they enable us to obtain closed-form real-

world densities. Other families of RNDs, including non-parametric specifications, will

provide similar empirical results whenever the range of the exercise prices is wide

enough to enclose almost all of the estimated densities.

2.1 Mixture of lognormal densities

Following Ritchey (1990) and Melick and Thomas (1997), the density of the

asset price when options expire can be defined as a mixture of lognormal densities

(hereafter MLN). The MLN densities are flexible and easy to estimate, with the

possibility of giving an economic interpretation to the parameters when the

component densities are determined by specific states of the world when the options

expire.

The density function in this study is a linear combination of two lognormal

densities, LN g , with weights w and w−1 ,

),,|()1(),,|()( 2211 T F x g wT F xwg x g LN LN MLN σσθ −+= (5)

with


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−−−=

22 ]

2

1)[log()log(

2

1exp

2

1),,|(

T

T F x

T xT F x g LN

σ

σ

πσσ . (6)

The parameter vector is ),,,,( 2121 w F F σσθ = . The parameters 1 F , 1σ and w denote

the mean, volatility and weight of the first lognormal density and likewise 2 F , 2σ

and w−1 are the mean, volatility and weight of the second lognormal density. The

theoretical European option pricing formula is simply a weighted average of two

prices given by Black’s formula, with weights w and w

−1 ,

),,,,()1(),,,,(),,|( 2211 σσθ r X T F cwr X T F wcT r X c B B −+= . (7)

There are three constraints on the parameter values : first, 10 ≤≤ w , to ensure the

mixture density is always positive; second, F F wwF =−+ 21 )1( , the constraint that

the risk-neutral expectation equals the current futures price F ; and third,

0,,, 2121 >σσ F F .

The mixture density avoids the rigid shape of a single density because it has

three additional free parameters; the risk-neutral constraint reduces the free

parameters of the mi xture to four, compared with only one for a single risk-neutral

lognormal. The standard deviation, skewness and kurtosis of the mixture can be

derived from

))(2

1exp()1())(

2

1exp(][ 22

22

21

21

T nn F wT nnwF S E nnnT σσ −−+−= . (8)

2.2 Generalized beta density

The generalized beta distribution of the second kind (hereafter GB2) was first

proposed by Bookstaber and McDonald (1987) and is utilized by Anagnou et al


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(2002). The GB2 density incorporates four positive parameters, ),,,( q pba=θ , that

permit general combinations of the mean, variance, skewness and kurtosis of a

positive random variable, thus enabling the shape of the density to be flexible. The

GB2 density function is defined as

q pa

ap

apGB

b

x

x

q p Bb

aq pba x g

+

−

+

=

)(1),(

),,,|(1

2 , 0> x , (9)

with )(/)()(),( q pq pq p B +Γ Γ Γ = .

The density is risk-neutral when

),(

)1

,1

(

q p B

aq

a pbB

F

−+= (10)

and its moments are

),(

),(

][q p B

a

nq

a

n p Bb

S E

n

nT −+= for aqn < . (11)

The parameter b is seen to be a scale parameter, while the product of a and q

determines the maximum number of moments and hence the asymptotic shape of the

tails; moments do not exist when aqn ≥ .

The theoretical option pricing formula depends on the cumulative distribution

function (c.d.f.) of the GB2 density, denoted 2GBG , which is a function of the c.d.f. of

the beta distribution, denoted βG and also called the incomplete beta function :

),),,((),,1,1)/((),,,|( 22 q pba x z Gq pb xGq pba xGa

GBGB β== (12)

with ))/(1()/(),,( aa b xb xba x z += . If the density is risk-neutral, so that the

constraint in equation (10) applies, then European call option prices are given by


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∫ −=∞−

X GB

rT dxq pba x g X xe X c ),,,|()()( 2θ (13)

[ ]),,,|(1)1

,1

,,|(122

q pba X G Xea

qa

pba X G FeGB

rT

GB

rT

−−

−+−=

−−

( )[ ]q pba X z G Xea

qa

pba X z G Fe rT rT ,|),,(11

,1

|),,(1 ββ −−

−+−= −− .

3. Theoretical real-world densities

All expected returns equal the risk-free rate in a risk-neutral economy. Risk-neutral

investors do not require a premium to induce them to bear risks and therefore all

derivatives can be priced regardless of the market risk preference. When we change

our attention from a risk-neutral economy to the real-world economy, the discount

rate for asset payoffs changes. It then involves a risk factor that is associated with the

sentiment of investors over future price uncertainties and their relationship with a

market portfolio. Correspondingly, risk preferences need to be taken into account in

estimating real-world price distributions. This can be achieved by transforming the

risk-neutral density (often called a Q density) into the real-world density (the P

density) by making a risk adjustment. Two parametric methods are evaluated in this

paper. One transformation is motivated by a representative utility function and the

other involves statistical calibration. The two transformations are applied to RNDs

that are MLN or GB2 densities. Both transformations provide closed-form real-world

densities whose transformation parameters can be estimated by maximizing the


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likelihood of real-world observations. Furthermore, for both our selected RND

methods the densities remain in the same family after the utility transformation.

3.1 A utility transformation

Let )( T S M be the stochastic discount factor, or pricing kernel, for payoffs at

a future time T . With sufficient assumptions, reviewed by Ait-Sahalia and Lo (2000),

Jackwerth (2000) and Rosenberg and Engle (2002), the stochastic discount factor is

proportional to the marginal utility of a representative agent,

dx

d x M

υλ=)( (14)

with λ an irrelevant constant and υ the representative utility function. European

option prices are real-world discounted option payoffs, given by

∫ −=−=

∞

X T T

P dx x g X x x M X S S M E X c )(~))(()]0,max()([)( (15)

for the real-world density )(~ x g and also by the risk-neutral formula

.)()()]0,[max()( ∫ −=−=∞−−

X

rT T

QrT dx x g X xe X S E e X c (16)

Consequently,

)(~)(

)( x g

x g e x M

rT

−= . (17)

From (14) and (17), it can be seen that the risk-neutral density g , the real-world

density g ~ , and the utility function υ are linked by

∫ ′

′= ∞

0

)()(

)()()(~

ydy y g

x x g x g

υ

υ. (18)

We assume the representative agent has a power utility function,


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γ υ

γ

−=

−

1)(

1 x x , 0≥γ but 1≠γ , (19)

)log( x= , when 1=γ ,

with γ the risk aversion parameter. The marginal utility is then γ υ −=′ x x)( and the

relative risk aversion function is a constant : γ υυ =′′′−= )()()( x x x x RRA . The real-

world density then becomes

∫

=∞

0

)(

)()(~

dy y g y

x g x x g

γ

γ

. (20)

Note that if g is lognormal then so is g ~ . The volatility parameters of g and g ~

are then equal but their expected values are respectively F and )exp( 2T F γσ for the

parameterization defined by equation (6). From this result for one lognormal density,

it is apparent that a transformed mixture of two lognormals is also a mixture of two

lognormals. For a risk-neutral mixture density )( θ x g MLN given by equation (5),

with ),,,,( 2121 w F F σσθ = , it can be shown that the real-world density is

)~

(),(~ θγ θ x g x g MLN = (21)

with

)~,,,~

,~

(~

2121 w F F σσθ = ,

)exp(~2T F F iii γσ= , for 2,1=i ,

and

)))((2

1exp()(

11~

1 21

22

2

1

2 T F

F

w

w

wσσγ γ γ −−−+= .


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Likewise, it is easy to show that the utility transformation changes a GB2 density into

another GB2 density. For the density )(2 θ x g GB given by equation (9), with

),,,( q pba=θ , the real-world density is

)~

(),(~ 2 θγ θ x g x g GB= (22)

with

),,,(~

aq

a pba

γ γ θ −+= ,

assuming γ >aq .

3.2 Statistical calibration

Let G and 1−G initially denote any cumulative distribution function (c.d.f.)

and its inverse function. Also let actual G be the actual, but unknown, c.d.f. of T S .

Observe that the c.d.f. of the random variable )( T S GU = is

))(())((Prob)(Prob11

uGGuGS uU actual T −− =≤=≤ for 10 ≤≤ u . (23)

The two c.d.f.s G and actual G are identical when the density of T S is correctly

specified, and then uuU P =≤ )( . Thus U is uniformly distributed between 0 and 1 if

and only if the density of T S is correctly specified.

Furthermore, suppose densityi

g is produced at timei

t for the asset price at

time *it with 1*

+≤ ii t t . Then the stochastic process }{ iU is i.i.d., with the above

uniform distribution, when all the densities are correctly specified. The two

assumptions of uniformity and independence can be checked either separately

[Diebold et al (1998)] or jointly by using tests described in Berkowitz (2001). The


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data for these tests are given by the cumulative probabilities )( ,iT ii S Gu = , evaluated

at asset prices iT S , observed at the times*it .

Fackler and King (1990) describe a recalibration method that improves a set of

densities when they are judged against the assumption that the random variables

defined by their c.d.f.s are uniformly distributed. Their method can be used to directly

transform risk-neutral densities into real-world densities; they do this for lognormal

RNDs obtained from a variety of commodity options. The key assumption when

recalibrating densities is that the iu are observations from a common probability

distribution.

Now let g and G respectively denote the RND and the cumulative distribution

function of a particular T S . For the random variable )( T S GU = , let its real-world

c.d.f. be the calibration function )(Prob)( uU uC ≤= . The random variable )(U C has

a uniform real-world distribution because

uuC C uC U uU C ==≤=≤ −− ))(())((Prob))((Prob 11 .

Define the calibrated real-world c.d.f. G~

and another random variable U ~

by

))(()(~

xGC xG = and )())(()(~~

U C S GC S GU T T === . (24)

Then U

~

is uniformly distributed and hence G

~

is correctly specified. However, the

function C is unknown.

Fackler and King (1990) recommend the c.d.f. of the Beta distribution as a

candidate calibration function,

),(/)1()( 1

0

1 k j BdvvvuC k u

j −− −∫ = . (25)


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This parametric distribution is defined on the interval [0, 1] and has a flexible shape.

The uniform distribution is a special case ( 1== k j ) that is appropriate when the

original density g does not require recalibration. From (24) and (25), the calibrated

real-world density is

)(),(

))(1()())((

)(~

)(~11

x g k j B

xG xG

dx

dG

dG

dC xGC

dx

d

dx

xGd x g

k j −− −==== . (26)

This real-world density has a closed-form when g is either a GB2 density or a mixture

of lognormal densities, because G has a closed-form in both cases. The two

parameters j and k permit transformations from g to g ~ that change the location,

volatility, skewness and kurtosis between densities. The possibility 1>> k j is of

particular interest and corresponds to a transformation that reduces volatility and

negative skewness for our data.

3.3 Estimation of the transformation parameters

Estimates of the risk aversion parameter γ and the calibration parameters j

and k can be obtained by maximizing the likelihood of the observed asset levels

when options expire. For expiry time *it , densities are evaluated at asset prices

denoted by iT S , . Providing the densities do not overlap, so they are formed at times

it with 1*

+≤< iii t t t , the likelihood of a set of observed asset prices is simply the

product of density values. The required log-likelihood function for a set of n densities

is

∑==

n

i iT inT T T

S g S S S L1

*

,

*

,2,1,

))(~log()),...,,(log( θθ (27)


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with *θ the transformation parameter(s), either γ or ),( k j . This function can be

maximized to provide the ML estimate of *θ .

The log-likelihood of a set of RNDs is obtained when there is no

transformation, so 0=γ or 1== k j . Likewise, the log-likelihood of non-

overlapping real-world densities obtained from histories of asset returns is also given

by summing the logarithms of density values.

4. Empirical methods

4.1 Data

The futures and options data used in this study were obtained from the London

International Financial Futures and Options Exchange (LIFFE). The contracts are

written on the FTSE 100 index. Daily tick data for bid quotes, ask quotes and actual

trades are used. The averages of bid and ask prices are employed to avoid bid-ask

bounce effects. The data covers the 83 months from June 1993 to April 2000 inclusive

and includes prices from both American options (until November 1995) and European

options (from December 1995). The data used are switched from American to

European options when the trading volume of European options overtook that of their

American counterparts.

Risk-free interest rates were collected from DataStream. We prefer the London

Eurocurrency rate to the UK Treasury bill rate, because the Eurocurrency rate is a

market rate accessible to triple-A corporate borrowers.


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The options are written on the spot index. Futures and options having the same

expiry month share a common expiry time, at 10:30 a.m. on the third Friday.

European options can then be valued by assuming that they are written on the futures

contract, and hence spot levels of the index are not needed.

4.2 Empirical risk-neutral densities

On a set of 83 prediction days it , one per month, risk-neutral densities i g are

constructed for expiry days *it . The days*it refer to the third Friday of every month,

when the index options expire, while each it is selected to be exactly four weeks2

before *it . Fixing the option lives at four weeks gives us 83 non-overlapping data sets.

The non-overlapping structure is essential for our likelihood calculations.

Futures contracts, unlike the options, are traded for only one expiration date

each quarter, in March, June, September and December. Synthetic futures prices must

therefore be calculated for the remaining months. Fair futures prices are the future

value of the spot price minus the present value of dividends expected during the life

of the futures contract, i.e.

))(( dividends PV S e F

rT

−= . (28)

We have obtained actual dividend payments for the 100 component companies of the

index from DataStream, and computed the present value of dividends by assuming

that future expected dividends can be approximated by actual dividend payments.

Either the actual spot level or the spot level implied by the nearest futures contract

2 We go back an additional day to acquire the data on the rare occasions that a holiday makes this necessary.


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(and the dividends until it ceases trading) can be used in (28). These two approaches

are identical in theory but they are not empirically equivalent. The spot levels implied

by futures are preferred, particularly for two reasons. First, the futures implied level

has been claimed to lead the spot index level for various reasons [see, for example,

Stoll and Whaley (1990) and Booth et al (1999)]. Second, options are hedged using

futures and not the spot index.

Non-synchronous trading also requires attention. Since options with different

strikes trade at different times of the day, they contain information at each

corresponding point of time that makes it impossible to directly extract an RND at a

common time. It is therefore necessary to convert option prices to equivalent prices at

a chosen point of time; 10:30 a.m. is selected as the standard for synchronization as

options and futures expire at 10:30 a.m. on their final trading days.

On any prediction day there are several options prices for the same expiry day.

The implied volatility from each option is obtained from the observed option

premium *market c , exercise price j X , contemporaneous futures price j F , time to

maturity T and interest rate r , according to the formula of Black (1976) for European

call options on futures,

))(,,,,()(* jimplied j j B jmarket X r X T F c X c σ= . (29)

The futures price at 10:30 a.m. then replaces j F in the formula Bc to provide the

synchronized market price of the option, )( jmarket X c . For American options, the

approximate pricing formula of Barone-Adesi and Whaley (1987) is used to extract

the market implied volatility and hence to obtain a synchronized European option


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price. All European put prices are converted to call prices using the put-call parity

equation.

Certain exclusion rules were applied to the raw bid, ask and trade prices, to

remove uninformative data and outliers. First, options violating the boundary

conditions for European and American options were deleted. Second, options which

are 100 index points or more in-the-money were deleted as these options have less

liquidity and depth. Third, repeated quotes, which are defined as quotes that are

exactly the same but appear within 30 minutes, were excluded for the reason that

keeping the redundant quotes would give undue weight to such quotes.

Finally, visual screening of the implied volatility smiles shows that a very

small number of extreme outliers ought to be removed because of their excessive

influence on the density estimates. In total 69 option prices are deleted, 63 for

midquotes and 6 for trades. These leave 30,341 options prices in all, with 18,832

midquotes and 11,509 trade prices. The summary statistics in Table 1 provide further

information about the moneyness of the options prices.

All our methods have been implemented for the averages of option bid and ask

quotes and for option trade prices. The two types of price data provide very similar

results and hence we concentrate upon the results obtained from the midquotes in the

following sections.

4.3 Summary statistics for densities obtained from option prices

Parameters are estimated for the two parametric density families, month by


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month, by minimizing the average of the squared differences specified in equation (3).

Table 2 provides a summary of these averages for the two families and the 83

available months. These averages are very similar for the two density families. The

standard deviations of the option pricing errors implied by these summary statistics

are less than the typical bid-ask spread. Summary statistics for the moments of the

risk-neutral densities are presented in Table 5 and discussed later in Section 5.1.

Figure 1 shows typical estimates of the risk-neutral densities. They exhibit a marked

negative skewness that has often been reported for equity indices.

Most RND estimation methods give similar results within the range of

exercise prices that provide the market data, because the implied volatility smile is

typically a smooth function that can be approximated by a low-order polynomial. All

methods must extrapolate outside the available range of the data and their tail shapes

will depend on the type of densities chosen. Our empirical conclusions are probably

insensitive to the RND method because there are almost no outcomes in the regions of

extrapolation. The probability within the range of exercise prices for our RNDs is

always more than 90%. Table 3 gives typical probabilities for four expiry dates, most

of which exceed 98%.

Table 4 includes estimates of the parameters used to transform the risk-neutral

densities into real-world densities. It also includes the maximized values of the log-

likelihood function. The results are shown under the heading 1=α . They are robust

with respect to data and density choices, being similar for midquote and trade prices

and also for MLN and GB2 and densities. The MLEs of the CRRA parameter γ are


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all near four. For the calibration transformation, all the MLEs are near 1.4 for j and

1.1 for k . The statistical significance of the log-likelihood increases, given by

transforming the densities, is discussed later in Section 5.3. Figure 2 illustrates a

typical GB2 risk-neutral density and the two derived real-world densities. The same

comparisons are shown for densities obtained from lognormal mixtures on Figure 3.

4.4 Historical real-world densities

It is important to compare the log-likelihood of densities obtained from

historical returns with those obtained from options prices. The real-world option

densities must outperform historical densities if they are to be recommended as real-

world predictive densities. ARCH models for daily index returns are here estimated

and simulated to provide historical real-world densities. The simulated ARCH models

must accommodate the stylized facts documented in the literature, including a time-

varying conditional mean, a persistent conditional volatility and an asymmetric

response of volatility to positive and negative returns. We choose the GJR-

GARCH(1,1)-MA(1)-M specification, following Glosten, Jagannathan and Runkle

(1993) and Engle and Ng (1993). The conditional meant

µ and the conditional

variance t h of the daily index return t r are as follows,

21111 ))(( −−−

−− −+++= t t t t t r Dhh µααβω , (30)

)()( 112/1

1 −−− −Θ++′= t t t t r h µλµµ

otherwise. 0

,0 if 1

=≤= t t r D


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22

Ten years of daily index returns prior to each estimation date it are used to estimate

the seven ARCH parameters, by maximizing the quasi-log-likelihood function which

assumes the conditional distributions are normal. These estimates are consistent when

the normality assumption is false (Bollerslev and Wooldridge (1992)).

The parameters obtained from information up to time it are used to simulate

the ARCH equations until time *it . Ten thousand simulations of the final asset level

iT S , are obtained for each month i. The historical real-world density i g ~ is then the

smooth function obtained by using the Gaussian kernel with bandwidth

5 10000

9.0 σ= H , (31)

and with σ the standard deviation of the simulated final levels. Figure 1 shows a

typical historical real-world density. The bandwidth H should be carefully chosen.

Bands that are too wide create oversmooth densities, while narrow bands can create

spurious multimodalities. Our value follows Silverman (1986, page 48) and it is half

the bandwidth used in Jackwerth (2000), which we consider may lead to over-

smoothing.

The log-likelihood of the historical densities is evaluated at the same stock

index levels as the option densities, using the same function defined by equation (29).

Table 4 shows that the historical log-likelihood is slightly less than that obtained from

each of the sets of RNDs. Summary statistics for the moments of the historical real-

world densities are included in Table 5. Most of the skewness values are near zero,

with the average skewness slightly negative. This occurs because the asymmetric

volatility parameter −α in equation (30) is always positive. Most of the kurtosis


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23

values are in the vicinity of four, which reflects the uncertain average level of

volatility during the four-week periods covered by the simulations.

5. Comparisons and discussion

Comparisons of the risk-neutral densities, the risk-adjusted densities and the historical

densities are performed using three sets of statistics. First we compare the moments of

the density functions, then we compare the cumulative probabilities of the realized

index levels iT S , and finally we compare the log-likelihoods of these levels.

5.1 Comparison of the moments of sets of densities

Table 5 summarizes the mean, standard deviation, skewness and kurtosis of

densities for random variables iiT F S /, , produced on the prediction days it when the

futures prices are i F . Summary statistics are provided for seven sets of 83 densities.

The statistics are given for the GB2 and lognormal mixture RNDs, the two

transformations applied to each set of RNDs and the historical real-world densities.

The RNDs are obtained from midquotes.

The historical and the transformed densities have similar average standard

deviations that are slightly lower than the averages for the RNDs. However, the

dispersion of the standard deviations is much less for the historical densities and

reflects low estimates of the persistence of the ARCH conditional variances.


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24

Almost all of the RNDs are negatively skewed, with the average skewness

about –0.7 for both sets of risk-neutral densities. The transformed real-world densities

are less skewed, with average levels between –0.6 and –0.5, that are far from the

average level of –0.1 for the historical real-world densities. The negative skewness in

the option-based densities is consistent with beliefs that there is more chance of a

substantial negative one-month return than of a corresponding substantial positive

return. Past experiences of crashes are reflected in option prices, although crash

anxieties may be unduly pessimistic when related to the ten-year histories of index

changes that define the historical densities.

The RNDs and their transformations are slightly more leptokurtic than the

historical densities. Overall, the risk adjustments alter the RNDs to make them more

similar to the historical real-world densities, in terms of all four moments. The utility

transformed and calibrated distributions are less volatile, less negatively skewed and

less leptokurtic.

5.2 Comparison of ranked cumulative probabilities

As previously stated in Section 3.2, evaluating the cumulative distribution

function for month i at iT S , produces a probability iu . The iu should be uniformly

distributed on the interval [0, 1] if the process that generates iu is well calibrated.

Thus a plot of the sample c.d.f., )(ˆ uC , which is the proportion of the iu equal to or

less than u, should be near the 45-degree line when densities are well calibrated.

Figure 4 shows )(ˆ uC for the GB2 midquote RNDs and its two sets of risk-adjusted


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25

densities. Figure 5 shows )(ˆ uC for the historical real-world densities, that appear to

be well calibrated.

For the RNDs, it is clearly seen that )(ˆ uC is much less than u for small values

of u : there are relatively few realized index levels below the first quartiles of the

RNDs and the minimum of the 83 probabilities iu on Figure 4 is 0.093. The minima

for three further sets of RNDs are 0.091, 0.079 and 0.092, respectively for the

GB2/trade, MLN/midquote and MLN/trade combinations. The eight sets of risk-

adjusted densities are more satisfactorily calibrated in the left tail region, with the

minimum values of iu between 0.039 and 0.056. The minimum is only 0.003 for the

historical real-world densities.

A standard goodness-of-fit statistic for a set of densities is the Kolmogorov-

Smirnov (KS) statistic defined as the maximum value of uuC −)(ˆ . The KS statistic

for the historical densities equals 0.062. The KS statistics for the RNDs range from

0.097 to 0.118. They are all reduced by the transformations to real-world densities,

varying from 0.071 to 0.083 for the utility transformation and from 0.086 to 0.102 for

the calibration method. None of the maximum values occur in the left tail region. All

the KS statistics accept the null hypothesis of i.i.d. observations from a uniform

distribution at the 15% level.

5.3 Comparison of log-likelihoods

Several comparisons are made between the densities by referring to the log-

likelihoods of various sets of densities, shown in Table 4. As the log-likelihoods are


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26

central to our methodology, they are provided for option-based densities derived

separately from midquote and trade prices. Note immediately that the historical real-

world densities have the least log-likelihood, which suggests that option prices

contain more information about real-world densities than the history of daily returns

themselves.

The log-likelihoods are always slightly higher for the lognormal mixtures than

for the GB2 densities. This may simply reflect the fact that each mixture density has

four free parameters, compared with three for each GB2 density. The calibrated real-

world densities have higher log-likelihoods than the utility-adjusted densities, which

again may reflect the additional transformation parameter employed in the

transformation from RNDs to calibrated densities. The log-likelihoods are very

similar for the midquote and trade data, with a slight advantage for trades when using

the GB2 method and for midquotes when applying the lognormal mixture method.

Likelihood-ratio tests show that the transformed densities provide a significant

improvement upon the RNDs, when the significance level is 10%. The test statistics

are defined by twice the increase in the log-likelihood achieved by optimizing over

the transformation parameter(s). For comparisons of the RNDs with the utility

transformed densities, the four test values (for different types of data and different

density families) are 3.30, 3.32, 3.64 and 4.00. These all exceed the 10% asymptotic

critical value from 21χ , but only one exceeds the 5% point. Likewise, the test values

are 5.64, 5.64, 5.80 and 6.16 when comparing RNDs with recalibrated densities. All

of these exceed the 10% point from 22χ but again only one is beyond the 5% point.


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The relatively high p-values for these tests can be attributed to the limited sample size

of 83 densities.

The set of historical real-world densities can be compared with a set of option-

densities by nesting the two sets within an encompassing family. We maximize the

likelihood of the densities

)()1()()( / x g x g x g ARCH RWD RNDcombined ααα −+= (32)

as a function of α . The results are shown in the final five columns of Table 4.

For the risk-neutral densities the maximum likelihood estimates of α are 0.51

and 0.52 for the two GB2 sets, and 0.61 and 0.64 for the two MLN sets. The null

hypothesis 0=α is rejected by each of the log-likelihood-ratios at the 2% level;

twice the increase in the log-likelihood given by optimizing α equals 6.06, 6.34, 6.89

and 7.67 for the four sets of RNDs. These tests show there is incremental predictive

information in the RNDs.

Comparisons of the historical and real-world option densities involve

maximizing the likelihood over α and the transformation parameters (either γ or j

and k ). The rejections of 0=α are then more decisive as the test values necessarily

increase. The other null hypothesis of interest, 1=α , is accepted at the 5% level for

all eight combinations of transformation (utility or calibration), density family (GB2

or MLN) and price type (midquote or trade). The test values are compared with 21χ

and are all less than one for the MLN densities; they vary from 0.88 to 3.38 for the

GB2 densities. These tests show the evidence for incremental predictive information

in the historical densities is not statistically significant. The MLEs of α are higher


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when the real-world densities replace the RNDs, with average estimates of 0.63 and

0.79 respectively for the GB2 and MLN densities.

5.4 Empirical pricing kernels and risk aversion

Constant relative risk aversion is assumed in the transformation from RNDs to

real-world densities via a power utility function. The four maximum likelihood

estimates of γ shown in Table 4 are 3.83, 3.84, 3.86 and 4.04 when 1=α . These are

very similar to the 3.94 estimated by Bliss and Panigirtzoglou (2002b), for options

with four weeks to expiry. Their FTSE 100 index options data commences in June

1992 and ends in March 2001. They obtain RNDs from a smooth volatility smile that

is a function of the option delta and then select γ to minimize the calibration test

criterion of Berkowitz (2001). All of the estimates of γ are rather high compared with

expectations from lognormal RNDs and real-world densities; then, for a typical equity

index premium of 6% per annum and a volatility of 15% per annum, the expectations

after equation (20) give 67.215.006.0 2 ==γ . One way to reconcile the difference

is to assert that some options are consistently mispriced, especially out-of-the-money

put options, thereby creating excessive negative skewness that can only partially be

corrected by a high power in the utility function. This explanation implies the relative

risk aversion function is not constant across wealth levels.

Empirical pricing kernels have two components, RNDs obtained from the

options market and historical real-world densities obtained from index returns. These

densities come from different information sets. As options data are more informative


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29

than the index history for U.S. markets (Christensen and Prabhala (1998), Fleming,

(1998), Blair et al (2001)), the issue of different information sets may be important

when interpreting empirical pricing kernels.

Two pricing kernels )(~)()( x g x g e x M rT −= are constructed for each option

expiry date, with )( x g either the mixture RND or the GB2 RND and with )(~ x g the

historical density from GARCH simulations. The geometric mean of each set of

kernels is computed to reduce noise created by the data and the different information

sets. We plot the geometric means of the two sets of ratios )(~)( yF g yF g against the

moneyness variable F x y = on Figure 6. Both empirical kernels are generally

decreasing functions of F x , although they are almost flat at some points between

0.95 and 1. These results are very different to those of Brown and Jackwerth (2002).

They observe a very clear hump-shaped marginal utility function, using S&P 500 data

from April 1986 to December 1995, which challenges economic theory and indicates

that the representative agent is risk seeking in some wealth region.

The risk aversion function is defined by the first and second derivatives of the

utility function,

))(

)(~

log()(

)(

)(~)(~

)(

)()( x g

x g

dx

d

x g

x g

x g

x g

x

x x RA =

′−

′=′

′′−= υυ

. (33)

The relative risk aversion is )()( x xRA x RRA = , evaluated by Jackwerth (2000) and

Ait-Sahalia and Lo (2000). Figure 7 shows the risk aversion and the relative risk

aversion functions plotted against moneyness. These functions are calculated from the

geometric means of RNDs divided by historical real-world densities. We observe a

downward sloping risk aversion function as far as 96.0/ = F x for the GB2 densities


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and further to 1/ = F x for the lognormal mixture densities, then both risk aversion

functions steadily increase. These estimates are incompatible with power and other

rational utility functions, which predict monotonically decreasing risk aversion. Thus

the previous assumption of constant relative risk aversion is not a very accurate

depiction of the combined dataset of option prices and index levels.

Jackwerth (2000), Ait-Sahalia and Lo (2000) and Rosenberg and Engle (2002)

all estimate risk aversion functions once a month from one-month options data on the

S&P 500 index. Jackwerth (2000) finds that the risk aversion function is credible

before the 1987 crash but becomes U-shaped post-crash. He attributes this to

persistent mispricing of options. However, his smoothed estimates of historical real-

world densities ignore stochastic volatility while his GARCH densities do not permit

negative skewness in monthly index returns. Ait-Sahalia and Lo (2000) use

overlapping data for the single year 1993. They rely on kernel estimates for both risk-

neutral and real-world densities and make the very strong assumption that the implied

volatility function is constant through time. Their relative risk aversion function is U-

shaped and incompatible with a power utility function. Rosenberg and Engle (2002)

also estimate irrational risk aversion functions, obtaining negative estimates over the

range 02.1/96.0 ≤≤ F x during the period from 1991 to 1995. They show that risk

aversion is a time-varying function, that is counter-cyclical with a risk premium that is

low (high) near business cycle peaks (troughs).


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6. Summary and conclusions

Option prices provide a rich source of information with regard to the future

distribution of the underlying asset price, particularly with respect to future volatility.

This information is here translated initially into a risk-neutral probability density,

assuming a parametric form that is either a mixture of two lognormal densities or a

generalized beta density. Investors do not require a risk premium in this risk-neutral

world.

In order to move from an artificial risk-neutral world to the real world, where

premia are earned for bearing risks, a risk aversion function is needed. We therefore

choose a parametric utility function and estimate the relative risk aversion, which is

assumed to be constant. We also statistically calibrate risk-neutral densities to obtain a

second set of real-world densities from option prices. Using the FTSE 100 index as a

proxy for the aggregate wealth process, the risk aversion estimates from a power

utility function are found to be reasonable and robust to different risk-neutral

estimation methods. Both utility-transformed densities and calibrated densities exhibit

less skewness and kurtosis than risk-neutral densities. In addition, likelihood-ratio

tests show that the real-world densities have significantly higher log-likelihood values

than the risk-neutral densities. Both sets of risk-neutral densities have higher

likelihoods than historical densities estimated from ARCH models and hence the real-

world densities obtained from options prices are more informative than those obtained

from the time series history of the index.


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Table 1. Summary statistics for the dataset of FTSE-100 options prices.

Midquotes Trades

Total Number 18832 11509

Calls 9285 (49.3%) 5728 (49.8%)

Puts 9547 (50.7) 5781 (50.2%)

Average number of prices per day 227 139

10% or less ITM options 3331 (17.7%) 2061 (17.9%)

ATM options 1174 (6.2%) 859 (7.5%)

10% or less OTM options 12785 (67.9%) 7880 (68.5%)Deep OTM options 1542 (8.2%) 709 (6.2%)

Range of moneyness

Calls [0.81, 1.04] [0.82, 1.34]

Puts [0.97, 1.72] [0.97, 1.52]

The four moneyness categories are (1) options in-the-money by more than 1%, but by

less than 100 index points, (2) within 1% of being at-the-money, (3) 10% or less out-

of-the-money and (4) more than 10% out-of-the-money.


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Table 2. Risk-neutral probabilities within the range of exercise prices.

Expiry Month Lower Bound Upper Bound Probability Captured

GB2 MLN

May, 1994 2800 3800 98.11% 98.64%

May, 1996 3475 4075 98.22% 98.08%

May, 1998 4625 6825 98.21% 98.85%

May, 2000 5325 6925 94.43% 93.57%

The tabulated probabilities are the total risk-neutral probability within the range of

exercise prices for which midquotes are available. The densities are evaluated four

weeks before the options expire.


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Table 3. Summary statistics for the average squared errors from RNDs.

Quartile GB2 MLN

Minimum 0.01 0.01

1st Quartile 0.55 2.51

2nd Quartile 5.42 5.63

3rd Quartile 14.00 14.42

Maximum 116.94 112.20

Mean 11.85 12.29

Standard Deviation 16.86 17.04

The summary statistics in each column are for the 83 values of the average of the

squared errors given by parametric density functions. The averages are defined by

minimizing the quantity in equation (3). The option prices are midquotes.


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Table 4. Log-likelihoods and transformation parameters for risk-neutral, risk-adjusted,

historical and encompassing densities.

Panel A: α = 0

Historical densities L = -547.51

α = 1 Optimal α

Panel B: Midquotes AL γ j k AL α γ j k

RND

GB2 0.72 3.17 0.52

MLN 2.79 3.84 0.64

Utility transformed

GB2 2.37 3.84 3.53 0.63 2.81

MLN 4.61 4.04 4.73 0.85 3.79

Recalibrated

GB2 3.54 1.38 1.08 5.23 0.63 1.60 1.45

MLN 5.87 1.42 1.11 6.17 0.78 1.52 1.26

Panel C: Trades

RNDGB2 0.97 3.04 0.51

MLN 2.27 3.44 0.61

Utility transformed

GB2 2.97 3.83 3.41 0.64 2.93

MLN 3.93 3.86 4.14 0.81 3.48

Recalibrated

GB2 3.87 1.40 1.11 5.01 0.61 1.60 1.47

MLN 5.09 1.40 1.11 5.53 0.73 1.52 1.31

The adjusted log-likelihood, AL, is the log-likelihood, L, minus the log-likelihood

obtained from historical densities. The estimated parameters are γ for the utility

transformation and j and k for the recalibration transformation. The parameter α is

the weight assigned to RNDs and risk-adjusted RWDs when they are combined with

historical ARCH densities:

ARCH RWD RNDcombined g x g x g )1()()( / ααα −+= .


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Table 5. Moments for RNDs, risk-adjusted densities and historical real-world

densities, obtained from midquotes.

GB2 MLN Historical

RND Utility- Recalibrated RND Utility- Recalibrated

transformed transformed

Mean/F

Minimum 1 1.006 1.006 1 0.964 0.965 0.987

1st Quartile 1 1.005 1.008 1 0.993 0.989 0.998

2nd Quartile 1 1.010 1.010 1 1.002 1.003 1.005

3rd Quartile 1 1.010 1.014 1 1.011 1.010 1.028

Maximum 1 1.040 1.026 1 1.022 1.022 1.041Mean 1 1.011 1.011 1 1.000 1.001 1.010

Stand. Dev. 0 0.008 0.004 0 0.013 0.014 0.016

Stdev/F

Minimum 0.027 0.025 0.022 0.027 0.026 0.021 0.036

1st Quartile 0.036 0.035 0.030 0.036 0.035 0.030 0.040

2nd Quartile 0.048 0.046 0.041 0.048 0.046 0.040 0.042

3rd Quartile 0.063 0.056 0.051 0.061 0.054 0.048 0.043

Maximum 0.116 0.093 0.094 0.112 0.091 0.091 0.063

Mean 0.051 0.047 0.042 0.050 0.046 0.041 0.042

Stand. Dev. 0.019 0.015 0.015 0.018 0.014 0.014 0.004

Skewness

Minimum -1.43 -1.35 -1.24 -1.30 -1.14 -1.13 -0.27

1st Quartile -0.94 -0.78 -0.75 -0.87 -0.70 -0.71 -0.21

2nd Quartile -0.75 -0.60 -0.56 -0.70 -0.59 -0.55 -0.15

3rd Quartile -0.50 -0.35 -0.30 -0.53 -0.42 -0.32 0.08

Maximum -0.16 0.08 0.19 0.40 0.66 0.59 0.51

Mean -0.74 -0.58 -0.53 -0.69 -0.57 -0.52 -0.07

Stand. Dev. 0.28 0.27 0.28 0.28 0.29 0.29 0.17

KurtosisMinimum 3.22 3.20 3.15 3.06 3.20 3.22 3.44

1st Quartile 4.13 3.98 3.79 3.64 3.78 3.74 3.73

2nd Quartile 4.69 4.52 4.39 3.97 4.24 4.19 3.88

3rd Quartile 5.07 4.92 4.74 4.51 4.78 4.75 4.08

Maximum 6.48 6.33 6.03 9.36 9.64 7.52 6.90

Mean 4.62 4.49 4.33 4.27 4.40 4.33 3.98

Stand. Dev. 0.66 0.64 0.63 1.05 0.93 0.78 0.45

The summary statistics in each column are for 83 densities. The first two panels are

respectively for the mean and variance divided by the futures prices.


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Figure 1. Risk-neutral densities from midquote option prices and historical densities

on March 21, 1997.

0

0.001

0.002

0.003

0.004

0.005

3500 4000 4500 5000

Index Levels at Expiry

D e n s i t y

RND (GB2) RND (MixLn) historical


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Figure 2. The risk-neutral GB2 density and its two risk-adjusted densities on March

21, 1997 for midquotes.

0

0.001

0.002

0.003

0.004

0.005

3500 4000 4500 5000

Index level at expiry

D e n s i t y

RND Utility transformed Recalibrated


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Figure 3. The risk-neutral lognormal mixture density and its two risk-adjusted

densities on March 21, 1997 for midquotes.

0

0.001

0.002

0.003

0.004

0.005

3500 4000 4500 5000

Index level at expiry

D e n s i t y

RND Utility transformed Recalibrated


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Figure 4. Cumulative distributions for cumulative probabilities obtained from GB2

RNDs and two sets of risk-adjusted densities.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

u

C ( u )

RND Utility transformed Recalibrated Uniform


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Figure 5. Cumulative distributions for cumulative probabilities obtained from

historical densities.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

u

C ( u )


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Figure 6. Empirical pricing kernel, averaged across all expiry months.

0

0.5

1

1.5

2

2.5

3

0.9 0.95 1 1.05 1.1

x/F

G e o M e a n ( Q / P )

GB2/Historical MixLn/Historical


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Figure 7. Risk aversion and relative risk aversion from the geometric means of the

empirical pricing kernels.

-5

0

5

10

15

20

25

0.9 0.95 1 1.05 1.1

x/F

RA (GB2) RRA (GB2) RA (MixLn) RRA (MixLn)

closed-form transformations from risk-neutral to real-world distributions

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