Zhou Paper

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Quantitative StrategiesResearch Notes

GoldmanSachs

Strike-Adjusted Spread:

A New Metric For Estimating

The Value Of Equity Options

Joseph Zou

Emanuel Derman

July 1999

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QUANTITATIVE STRATEGIES RESEARCH NOTESSachsGoldman

-2

Copyright 1999 Goldman, Sachs & Co. All rights reserved.

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illegal. Certain transactions, including those involving futures, options and high yield securities, give rise to substantial risk and arenot suitable for all investors. Opinions expressed are our present opinions only. The material is based upon information that we

consider reliable, but we do not represent that it is accurate or complete, and it should not be relied upon as such. We, our affiliates,

or persons involved in the prepar ation or issuan ce of this ma terial, ma y from t ime to time h ave long or sh ort positions and buy or

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SUMMARY

Investors in equity options experience two problems tha t compoundeach other. In contrast to fixed-income and currency markets, there

are thousands of underlyers and tens of thousands of options, and

each underlyer can have a potentially large volatility skew. How can

an options investor gau ge which option provides the best r elative

value?

In t his pap er, we mak e use of a m eth od for estima ting th e fair volat ility

smile of any equity underlyer from information embedded in the time

series of tha t under lyers h istorical r etur ns. We can then comput e th e

relative richness or cheapn ess of an y par ticular strike a nd expiration

by exam ining t he options S trike-Adjusted S pread, or SAS , the differ-

ence between its market implied volatility and its estimated histori-cally-fair volatility.

We obta in fair volat ility smiles by estima ting th e appr opriate r isk-neu-

tral distribution for valuing options on any equity underlyer from that

under lyers historical retu rn s. The distribution includes t he effect of

both past price jumps and past shifts in realized volatility. Using this

distribution, we can estimate the fair volatility skews for illiquid or

thinly-tra ded single-stock an d basket options. We can also forecast

cha nges in t he sk ew from chan ges in a sin gle options price.

___________________

J oseph Zou (212) 902-9794

Eman uel Derman (212) 902-0129

Editorial: Barbar a Dun n

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0

Table of Contents

THE RICHNESS AND CHEAPNESS OF OPTIONS..............................................1

Cur rent vs. Pas t Im plied Volat ilities .............................................. 1

Implied vs. Historical Volatilities ...................................................1

Strike-Adjusted Spread (SAS) ........................................................ 1

ATM Strike-Adjusted Spread ..........................................................3

OPTIONS PRICES AND IMPLIED DISTRIBUTIONS ............................................5

STOCK RETURNS AND H ISTORICAL DISTRIBUTIONS ......................................6

MAXIMAL UNCERTAINTY AND MARKET EQUILIBRIUM ..................................8

Entropy as a Measure of Uncertainty ............................................8

The Risk-Neu tr alized Histor ical Distr ibut ion ............................... 9

APPLICATIONS OF THE RIS K-NEUTRALIZED DISTRIBUTION ....................... 11

Is Th e In dex Im plied Volat ility Skew F air ? .................................11

Str ike-Adjust ed Spr ead As A Measur e Of Options Valu e ...........13

Valuin g Opt ions on Ba sket s of Stocks ..........................................15

Forecast ing The Sha pe of The Skew

Fr om A Cha nge In A Single Opt ion P rice ............................... 17

End-of-Day Mark To Market .........................................................19

Filling Gap s I n Invest ors Mar ket Views ......................................19

CONCLUDING REMARKS .............................................................................. 20

APPENDIX A: INFORMATION AND ENTROPY ................................................21

APPENDIX B: DETERMINING THE RIS K-NEUTRAL

DISTRIBUTION FROM H ISTORICAL RETURNS ..........................24

APPENDIX C: A DERIVATIVE ASSET ALLOCATION MODEL

AND THE EQUILIBRIUM RIS K-NEUTRAL DENSITY ..................25

REFERENCES ................................................................................................29

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THE RICHNESS AND

CHEAPNESS OF

OPTIONS

The equities world is a ma ss of dat a. Sur rounded by fluctua ting shar e

prices, dividend yields, ear nings forecasts, P /E ra tios, an d hosts of

more sophisticat ed mea sures, a nalysts, investors a re in n eed of some

gauge or metric with which to compare the relative attractiveness ofdifferent stocks. Int o the breach, in newslett ers, books an d websites,

step countless economists, technical analysts, fundamental analysts,

cha rtists, wave t heorists, alpha -maximizers and other optimists, hop-

ing to impose order a nd r at iona lity, to tell you wh at to buy an d sell.

Investors in equity options face an equally difficult task, with less

resour ces. For each un derlying st ock, basket or index, ma ny sta ndar d

str ikes and expira tions are ava ilable. For a given under lyer, each str ike

and expirat ion tra des at i ts own implied volati li ty, all of which,

together, comprise an implied volatility surface [Derman, Kani and

Zou, (1996)] that moves continually. Each underlyer has its own idio-

syncrat ic sur face. In a ddition, un derlyers can be grouped to creat e bas-kets, new underlyers with their own (never before observed) volatility

surface.

For a given stock or index, how is an investor to know which strike and

expiration provides the best value? What metric can options investors

use to gauge their estimated excess return ? What is the appropriate

volatility sur face for a n illiquid basket ? Help is spar se.

Current vs. Past

Implied Volatilities

The most comm on gauge of options valu e ha s been th e sprea d between

curr ent an d pa st im plied volat ilities. This is the met ric of options spec-

ulators, who hope to get in at historically low volatilities, hedge for a

while, an d get out h igh. When all options of a given expirat ion t ra de at

the same implied volatility, it is not too hard to compare changes in

implied volatility over time. Since the advent of the volatility smile,

however, it ha s become har der to ha ve a clear opinion of the relative

richn ess of two complex volat ility sur faces.

Implied vs. Historical

Volatilities

A second gauge is the spread between current implied and past real-

ized volatilities. This is the metric of options replicators, who hope to

lock in th e differen ce between futu re r ealized and curr ent implied vola-

tilities by delta-hedging their options to expiration. This comparison,

becomes imprecise in t he pr esence of a volatility skew, when t her e ar e

a range of implied volatilities, varying by strike, that must be com-

par ed with only a single hist orical r ealized volatility.

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2

Strike-Adjusted

Spread

(SAS)

The h istorical time ser ies of a stocks r etur ns conta ins m uch u seful

inform at ion. In th is paper we tr y to come to the pr actical aid of options

investors by estimating the fair value of options from the historical

retu rn s of their u nderlyers. This met hod for options pricing ha s beenextensively developed by St utzer (1996), a nd also employed by Der-

man , Kama l, Kani & Zou (1997), an d St utzer an d Chowdhury (1999).

Here we apply it in th e pra ctical si tuations that occur on an equity

derivat ives tr ading desk, where options on man y different under lyers

mu st be va lued da ily.

This met hod leads us t o th e notion ofS trike-Adjusted S pread, or S AS , a

nat ur al one-dimensiona l metric with which t o rank the relative value

of all standard equity options, irrespective of their particular strike or

expirat ion. We propose t o use S AS in roughly the sa me way tha t st ock

investors use alph a an d mortga ge investors use OA S (option-adjusted

spread). To be specific, the S AS of an option is the spread between thecurr ent ma rk et implied volatility of tha t option an d our m odels esti-

mate of its historically appropriate volatility. Our estimate includes

both th e effect of past pr ice jumps a nd t he influ ences of cha nges in vol-

at ility an d corr elations for ba sket options.

Theoretically, the historically appropriate implied volatility for a given

option is determined by the cost of replicating that option throughout

its lifetime. Not only is th is replication cost difficult an d t ime-consu m-

ing to simulate, but, in our experience, the hedging errors due to inac-

cura te volatility forecast ing and infrequent h edging ma ke the resu lting

sta tistics inconclusive. Instea d, our met hod for obtainin g the ap propri-

ate implied volatility of a stock option involves the estimation of anappropriate risk-neutral distribution from the past realized return dis-

tribution of the stock. We will explain the method in more detail below,

an d describe its application to SAS . The same technique can be used to

mark and hedge i ll iquid equi ty opt ions whose market prices are

unknown.

The strike-adjusted spread of an option depends on both its strike K

and t ime to expirat ion T, and can be wr it t en more precisely as

. S AS can be th ought of as a n extension of the commonly

quoted implied-to-historical volatility spread, which is unique only in

th e a bsence of skew. In non-skewed worlds, both sprea ds become iden -

tical.

In br ief, the S AS of a st ock option is ca lcula ted a s follows.

1. First, choosing some historically relevant period, we obtain the dis-

tribution of stock retu rns over time T. This empirical ret urn distri-

but ion char acter izes th e past behavior of th e stock.

S A S K T ,( )

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2. Option theory dictates that options are valued as the discounted

expected value of the option payoff over the risk-neutral distribu-

tion. We do not know the appropriate risk-neutra l distribution.

However, we use the empirical return distribution as a statisticalprior to provide us with an estimate of the r isk-neut ra l distribution

by minimizing the entr opy1 associated with the difference between

th e distr ibutions, subject to ensur ing that th e risk-neut ra l distribu-

tion is consistent with the current forward price of the stock. We call

this risk-neutral distribution 2 obtained in th is way the risk-neu-

tral ized historical distribution , or RNHD.

3. We then use the RNHD to calculate the expected values of standard

options of all strikes for expiration T, and convert t hese values to

Black-Scholes implied volatilities. We denote the Black-Scholes

implied volat ility of an option whose price is compu ted from t his dis-tr ibution as . This is our estimated fair option volatility.

4. For an option with strike Kan d expiration T, whose market implied

vola t ilit y is , t h e s tr ike-ad ju st ed s pread in vola t ilit y is

defined as

This spread is a measu re of the curr ent r ichness 3 of th e option

based on historical r etur ns.

ATM Strike-Adjusted

Spread

The volat ility skew, the relat ive gap bet ween a t-th e-money an d out-of-

the-money implied volatilities for a given expiration, is more stable

tha n the absolute level of at -the-money implied volatilities. Often,

therefore, irrespective of historical return distributions, the current

level of at -the-money implied volat ility is t he most believable estima te

of fut ur e volatility. It is likely tha t h istorical distr ibutions tell us m ore

about the higher moments of future distributions than it does about

their stan dar d deviation.

Therefore, we will often use a modified version ofS AS for which the

risk-neutralized historical distribution is further constrained to repro-

1. As we explain later, mark ets in equilibrium a re chara cterized by maxi-mum un certa inty or minima l informa tion, an d minimal entr opy cha nge

is an expression of minimal informa tion.

2. Stu tzer (1996) refers to this as t he canonical distribut ion. an d th is

method of options valuation as canonical valuation.

3. A positive S AS connotes richness only for standard options whose value

is a monotonically increasing function of volatility. Exotic options may

have values th at decrease a s volatility increases.

H

K T,( )

S A S K T ,( ) K T,( ) H K T,( )=

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4

duce the current market value of at-the-money options. We call this

(additionally constrained) distribution the at-the-money adjusted,

r i sk-neutra l ized hi s torica l d i s tribut ion, or RNHDATM. Th e

strike-adjusted spread computed using this distribution, denoted, is a mea sur e of the r elative value of different str ikes,

assum ing t hat , by definition, at-the-money-forward implied volatility

is fair.

We pr opose u sin g t o r an k opt ion s on t he sa me u nder -

lyer, in order t o deter mine wh ich st rikes pr ovide the best value by his-

torical standards. More radically, we can also use the same measure to

compa re options of different un derlyers.

In the rema inder of th is paper, we flesh out these concepts. The n ext

section explains t he r elation between options pr ices an d implied distr i-butions. Thereafter, we compar e implied distributions to h istorical

return distributions. We then explain that markets in equilibrium are

characterized by maximal investor uncertainty, and, introducing the

notion of ent ropy, show th at we can obta in an estima te of th e risk-neu-

tral distribution from the historical distribution by minimizing the

entropy difference between the distributions. The main body of the

paper th en develops several applications of th e risk-neut ra lized histor-

ical distribution, including SAS . After some concluding rema rks, we

provide several mathematical appendices.

S A S ATM K T,( )

S A S ATM K T,( )

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5


OPTIONS PRICES AND

IMPLIED DISTRIBUTIONS

According t o th e t heory of options va luat ion, stock options prices con-

ta in informa tion about t he m ar ket s collective expecta tion of th e stocks

futur e volat ility an d its retu rn dist ribut ion. If no riskless ar bitra ge can

occur, th ere exists a risk-neut ra l retur n probability distr ibution Q suchthat the value V of an option on a stock with price S at t ime t is given

by the discount ed expected valu e of th e opt ions payoff, writt en a s

(EQ 1)

where r is the risk-free interest rate and EQ[ | ] denotes the expected

value of th e fut ur e payoff at t ime T, given that the stock price at time t

is S .

In t he Black-Scholes theory, the r isk-neutr al implied probability distr i-

bution Q is the lognorm al den sity funct ion with a specified volat ility. In

implied tree m odels4, Q is skewed relative to the Black-Scholes density,

and can be estimated at any time from a set of traded European option

prices. Figure 1a illustrates an implied volatility skew for S&P 500

index options, typically about five volatility points for a 10% change in

strike level; Figure 1b shows the correspondingly skewed risk-neutral

implied distribut ion Q. When ever the sh ape of th e skew cha nges, there

is a corresponding change in the distribution. Knowing Q, you can cal-

culate t he fair value of any stan dard Eu ropean option.

4. See for exam ple, Derm an , Kani, and Zou (1996).

V S t,( ) e r T t( ) EQ Option Pa yoff at T | S t,[ ]=

FIGURE 1. (a) The three-month implied volatility skew for S&P 500 indexoptions on 3/10/99. (b) Q, the corresponding risk-neutral impliedprobability distribution of returns. We assume a riskless interest rate of 5%.

Strike Level

ImpliedVolatility(%)

1000 1100 1200 1300 1400 1500

20

25

30

35

Index Return (%)

Probability(%)

-40 -20 0 20

0

2

4

6

(a) (b)

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6

STOCK RETURNS AND

HISTORICAL

DISTRIBUTIONS

Stock options prices determine the implied distribution of stock

retu rn s. Independently, we can also observe the actual distribution of

stock. Consider th e hist orical ser ies S i of daily closing pr ices of a st ock

or stock index. We can construct the rolling series of continuously com-pounded st ock ret ur ns R i from da y i for a subsequent period of N trad-

ing days by calculatin g

(EQ 2)

Figure 2 shows th e distr ibution of actu al thr ee-month S&P 500 retu rn s

for periods both before a nd since th e 1987 stock mar ket cra sh, where

the latter period includ es the crash itself.

The pre-crash return distribution is approximately symmetric and nor-mally distributed. In contrast, the post-crash distribution (1987-crash

data included) has a higher mean return and a lower standard devia-

tion, as well as a n a symmetric secondar y peak a t its lower end.

There is a rough similar ity in sha pe between th e implied distribution

of Figur e 1b, whose mean reflects th e risk-free r at e at which its options

were priced, and the historical distribution ofFigure 2b, whose (differ-

ent) mean is the a verage historical ret urn over th e post-cra sh per iod.

R i

S i N+

S i---------------log=

Index Return (%)

Probability(%)

-20 0 20

0

1

2

3

4

Index Return (%)

Probability(%)

-20 0 20

0

1

2

3

4

5

6

mean =1.8%std. div. = 7.3%

mean = 3.3%std. div. = 7.8%

FIGURE 2. Three-month, S&P 500 index, observed return distributions.(a) Pre-1987 crash (Jan. 1970 to Jan. 1987); (b) Post-1987 crash (June 1987to June 1999)

(a) (b)

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Options theory does not enforce an unambiguous link between histori-

cal and implied distributions. Nevertheless, historical distributions,

suitably interpreted, can provide plausible informat ion about fair

options prices. Our aim in th is paper is to develop a h eur istic but logi-cal link between the two distributions, utilizing the notions of market

equilibrium an d u ncertaint y.

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8

MAXIMAL

UNCERTAINTY AND

MARKET EQUILIBRIUM

Markets are supposed to settle into equilibrium when supply equals

demand, when there are equal numbers of buyers and sellers at some

price. In a n efficient ma rk et, the potent ial buyers of a st ock mu st th ink

the stock is cheap, and potential sellers must think it rich. This differ-ence of opinion m ean s th at , in equilibrium , th e distr ibution of expected

retu rn s displays great un certa inty.

How do we quantify this simple intuition that equilibrium involves

uncerta inty in the expected retu rn distribution?

Entropy as a Measure

of Uncertainty

The probability of a single event is a m easur e of the un certa inty of its

occurrence. Entropy is a mathematical funct ion that measures th e

uncer ta in ty of a probabil it y d i st ribu t ion5. The entropy of a random

variable R , whose ith occurrence in the distribution has probability p i,

is defined t o be

(EQ 3)

Since any pr obabilityp i is less th an or equa l to 1, the ent ropy is always

positive. If the distribution R collapses to one certain single event j,

whose pr oba bility with a ll oth er , th en H(R) = 0. There-

fore, certainty corresponds to the lowest possible entropy. You can also

show that the en tropy takes it s maximum value, log(n), when

for all i, that is, when all outcomes have an equal chanceand uncertainty reigns. This is consistent with the notion that maxi-

mum entropy corresponds to maximum uncertainty and minimum

information.

H(R ) is t he entr opy of a single distr ibution R . We can also define t he

relative entropy S (P,Q) between an initial distr ibution P and a subse-

quent distribution Q. S measures the decrease in entropy (or the

increase in inform at ion) between the initia l distr ibution P and the final

distribution Q, an d is given by

(EQ 4)

In Appendix A we show th at the relative entr opy is always non-nega-

tive, and is zero if an d only if the t wo distr ibutions P and Q ar e identi-

5. In Appendix A we explain th e link between en tr opy an d informat ion.

H R( ) p i p ilogi 1=

n

=

pj 1= p i 0=

p i 1 n=

S P Q,( ) EQ Q Ploglog[ ] Q x( )Q x( )P x( )------------- log

x= =

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cal. This agrees with our intuit ion tha t any chan ge in a probability

distribution conveys some new informat ion. The relative entropy

between two distr ibutions measu res th e inform at ion gain (or redu ction

in uncertainty) after a distribution change. Thus, minimum relativeent ropy corresponds to the least increase in inform at ion.

The Risk-Neutralized

Historical Distribution

Consider a stock option with time to expiration T on a st ock whose spot

price is . To value th e option, we need to average the option payoff

over t h e r is k-n eu t ra l p roba bilit y d en sit y . I n t h eor y, Q( )

is foun d by solving th e differen tial equa tion tha t const ra ins th e inst an -

tan eously hedged option t o ear n t he insta nta neously riskless retu rn. In

the Black-Scholes world, a stocks futu re probability distribution is

assu med to be lognorma l, an d consequ ent ly, th ough not obviously, Q( )

itself is a lognormally distributed probability density, and its options

prices have no volatility skew.

This theoretical lack of skew conflicts with the data from markets,

where stocks and indexes tha t have sufficiently l iquid out-of-the-

money strikes display clear, and often large, skews. How can we esti-

mate a suitable risk-neutral probability density that is more consistent

with market skews than the Black-Scholes lognormal distribution?

It is n atur al to tu rn for insight t o the distribution of actual return s,

. The two distributions Q( ) and P( ) can not be str ictly

identical, because the expected value of the stock price under the risk-

neutral distribution Q( ) at a ny time m ust be the stocks curren t for-

ward pr ice, as determined by t he curr ent risk-free r ate, whereas theexpected value of the stock price under P( ) is the aver age h istorical for-

ward price, which bears no relation to current risk-free rates.

The r igorous wa y to obta in Q( ) from the past evolution of stock prices

is to obta in fair historical options pr ices for a var iety of str ikes by sim-

ulating the instantaneously riskless hedging strategy over the life of

these options, and to then infer the risk-neutral density that matches

these prices. This r equires a detailed knowledge of every past instan t

of the st ock pr ice evolution, at a ll times an d ma rk et levels, and is time-

consu ming, difficult, er ror-prone a nd u ltima tely impra ctical.

Instead, we will estimate the current risk-neutral return distributionQ( ) for a s tock from it s hist orical distr ibut ion P( ) by assuming th at the

latter is a plausible estimate for the former, and then requiring that

the r elative entr opy S (P,Q) between th e distr ibutions is minimized. We

impose this criterion in order to avoid any spurious increase in appar-

ent informa tion in creat ing the r isk-neut ra l distribution from th e his-

torical distribution. We perform the minimizat ion subject to th e r isk-

S 0

Q S 0 0 S T T,;,( )

P S 0 0 S T T,;,( )

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neut ra lity constr aint, t ha t is, the condition t ha t t he expected value of

the stock price under the risk-neutral distribution Q( ) is consistent

with t he st ocks curr ent forwa rd pr ice6. We call Q( ) found in this way

the risk-neu tral ized historical distribution 7, or the RNHD. It is

our plausible guess for the distribution to use in options valuation,

given our knowledge of the past . Our kn owledge of a stocks h istorical

volatility, the second moment of its distribution, is often used to esti-

mate options values using the Black-Scholes formula. Here we go one

step further by using t he ent ire historical return dis t r ibut ion. A

description of the general approach outlined here can also be found in

Stu tzer (1996).

It is possible to impose further constraints on Q( ). If you believe that

the current at-the-money volatility for some particular stock is fair, you

can constrain the distribution Q( ) to ma tch n ot only th e st ock forward

price, but also to mat ch th e curr ent at -the-money implied volatility. We

denote this additiona lly constr ained distribution by Qat m ( ), and refer

to it as the at-the-money-consistent , r isk-neu tral ized historical

distribution , or RNHDATM. It can be used to compare the relative

values of options with different strikes on one underlyer, assuming

th at at -the-money volatility is fair.

Appendix B sta tes t he m inimization condition on Q( ) in mat hema tical

ter ms. In Appendix C we presen t a model of an Arr ow-Debreu economy

and show that it is possible to obtain the risk-neutralized historical

distr ibution by optimally allocating investors wealth u nder a n equilib-

rium condition with a n exponentia l utility fun ction.

Having obtained our estimate of the r isk-neut ra l distribution, we can

est imate the fai r price for any s tandard opt ion as the discounted

expected value of its payoff at expiration. We then extract the fair

implied volatility as the volatility which equates the Black-Scholes

option price to th e estima ted fair pr ice. This pr ocedur e can be repea ted

for all strikes and mat urities t o yield an entire fair implied volatility

surface.

6. Several aut hors have stu died the relevance of entr opy in finan cial eco-

nomics and derivatives pricing. See Stu tzer (1996), Derma n et al .

(1997), Buchen an d Kelly (1996), an d Gu lko (1996).7. For a norma l historical distribut ion of simply compoun ded retu rn s, one

can show that the risk-neutralized historical distribution obtained by

entr opy minimization is equivalent to a t ran slation of the historical dis-

tribution to re-center i t at the appropriate risk-neutral rate, without

altering its shape. This t ranslation invariance of the sha pe in moving

from th e historical to th e risk-neutr al distribut ion does not hold in gen-

eral.

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APPLICATIONS OF

THE RISK-NEUTRALIZED

HISTORICAL

DISTRIBUTION

The RNHD contains informat ion which can be used to estimate the

value of illiquid options whose prices a re unobtainable, as well a s to

compare the relative value of options with known market prices. We

present several representative examples below.

Is The Index Implied

Volatility Skew Fair?

Since the 1987 crash, equity index markets have displayed a pro-

nounced, persistent implied volatility skew. Is this skew fair? Are the

options prices determ ined by th e skew justified by historical r etur ns?

Figure 3a sh ows the r isk-neut ra lized three-month S&P 500 return dis-

tr ibution for the pre-cra sh period corr esponding t o Figure 2a, con-

structed using our method of relative entropy minimization. Figure 3b

shows the sa me distr ibution corresponding to the post-cra sh er a ofFig-

ure 2b. The post-crash distribution has a substantially longer tail at

low retur ns th an t he pre-cra sh distribution.

Skew slopes seem m ore sta ble th an volat ility levels. Therefore, we willfocus here on the relation between the implied volatilities of different

strikes that follows from these distributions, and pay little attention to

th e pr evailing a bsolute level of implied volatility. We est imat e t he fair

volati l i ty skew by using the distributions of Figure 3 to calculate

options prices, and by th en converting these options prices t o Black-

Scholes implied volatilities.

The results are shown in Figure 4. The pre-crash skew is approxi-

ma tely flat, but th e post-cra sh volat ilities increase for low str ikes, with

a slope similar to actual index skews in stable ma rkets. The observed

degree of skew, a bout five to six volatility points per 10% chan ge in

Index Return (%)

Probability(%)

-20 0 20

0

1

2

3

4

Index Return (%)

Probability(%)

-40 -20 0 20

0

2

4

6

FIGURE 3. The three- month risk-neutral distribution of S&P 500 returnsconstructed from the empirica l distributionsof Figure 2 using a 6% risklessrate. (a) Pre-crash (b) Post-crash.

(a ) (b)

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str ike level, seems a pproximately fair in t he light of post-crash ma rk et

behavior. Our fair post-crash skew is bilinear and more convex than

the recent skew ofFigure 1a, but index mark ets do sometimes display

skews like th at ofFigure 4.

We find that estimated one-month skews tend to resemble a smile

more than a skew: our fair implied volatilities of both out-of-the-money

calls a nd p ut s for one-month expirat ions exceed a t-th e-money volatili-

ties. Short -dated index options often d isplay th is type of behavior. We

have applied our method to several other major stock indexes and

found that their fair volati li ty skews a re roughly consistent withobserved market skews during normal market periods, as shown in

Table 1.

FIGURE4. The fair implied volatility skew for three-month S&P 500 options ascalculated from the risk-neutralized historical distributions of Figure 3.

Strike Level

Volatility(%)

90 95 100 105 110

10

12

14

16

18

20

22

24

post-crashpre-crash

TABLE 1. Comparison of actual skews with estimated fair volatility skews forthree major indexes. The spread shown is the difference in volatility pointsbetween a 25-delta put and a 25-delta ca ll.

IndexNormala

Spread

a. Average during normal market conditions (excluding the periods of

extreme volatility in late October 1997 and August-September 1998).

Extreme b

Spread

b. Average during periods of extreme market volatility.

Fairc

Spread

c. Based on historical returns over the period June 1987 to June 1999.

SPX 4-7% 14% 6.0%

DAX 3-6% 10% 3.5%

FTSE 2-6%. 10% 4.0%

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Strike-Adjusted

Sprea d As A Measure

Of Options Value

The Strike-Adjusted Spread for an option with strike Kan d expiration

T is defined as

where is the Black-Scholes market implied vola t ility of the

op tion , a n d is t h e im plied vola t ilit y com pu t ed fr om t h e

RNHD over some chosen r elevan t per iod. S ASATMis constrained to be

consist ent with t he m ar kets at -the-money-forward im plied volat ility

for t ha t pa rt icu la r u nder lyer a nd expir at ion , so t ha t

, wh er e is t he for wa rd va lu e of t he

underlyer at time T. This spread is a measur e of the curr ent r ichness,

relative to history, of an option, assum ing t hat at-the-money-forward

options, usua lly th e most liquid, are fairly valued.

Figure 5a shows a plot of fair and market skews for Sept. 1999 S&P

500 options, on May 18, 1999, using the twelve years of historical

r et u rn s from May 1987 t o May 1999 t o ca lcu la t e . Figure 5b

shows the S ASATMfor the same options. For out-of-the-money puts, the

entr opy-adjusted volatilities slightly exceed the mar ket volatilities,

which suggests that out-of-the-money puts are slightly cheap. Con-

verse ly, out -of-th e-money calls seem rich.

Figure 6 shows the sam e plots based on a h istorical ret ur n distr ibution

tak en from May 1988 thr ough May 1999, th ereby excluding the 1987

global stock market crash. In this case, out-of-the-money puts seemmuch too rich, while out-of-the-money calls are slightly cheap.

In our view, SAS is a qua nt itat ive tool for r an king th e relat ive value of

options, but t his does not a bsolve the u ser from choosing th e hist orical

period relevant to the computation of the risk-neutralized distribution.

There is no escaping the judgement necessary to decide which past

period is most relevant t o the curr ent m ar ket from both a fundam enta l

and psychological point of view.

In Figure 7, we plot th e skews an d SASATM for t he sa me set of options

used in Figure 5 and 6, but evaluated one month later. Although at-

the-money volatility has now fallen from 25.5% to 21%, the size ofskews has remained relatively stable. Roughly irrespective of which

historical distribution was used, th e strike-adjusted spreads have

changed so that out-of-the-money puts have become about two SAS

point s cheaper, wher eas th e SAS of out -of-th e-money calls ha s chan ged

less. If you had thought the relevant historical distribution was the

crash-inclusive one ofFigure 5, and had bought cheap puts, you would

S A S ATM K T,( ) K T,( ) H K T,( )=

K T,( )H K T,( )

S A S ATM S F T[ ] T,( ) 0= S F T[ ]

H K T,( )

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Strike Level

Volatility(%)

1200 1250 1300 1350 1400 1450

22

24

26

28

30

32

"fair skew"market skew

Strike Level

SAS(%)

1200 1250 1300 1350 1400 1450

-0.5

0.0

0.5

1.0

FIGURE 5. (a) Fair and market skews for S&P 500 index options on May 18,1999. (b) SASATM for the same options.The options considered expire on September 17, 1999. Both fair andmarket implied volatilities are constrained to match at the money,forward. The RNHD is constructed using returns from May 1987 to May

1999, including the 1987 crash.

(a ) (b )

Strike Level

Volatility(%)

1200 1250 1300 1350 1400 1450

24

26

28

30


Strike Level

SAS(%)

1200 1250 1300 1350 1400 1450

0

2

4

6

FIGURE 6. (a) Fair and market skews for S&P 500 index options on May 18,1999. (b) SASATM for the same options.The optionsconsidered expire on September 17, 1999. Both fair and marketimplied volatilities are constrained to match at the money, forward. The

RNHD is constructed using returns from May 1988 to May 1999, therebyexcluding the 1987 crash.

(a ) (b)

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have lost SAS . If, on th e other ha nd, you ha d thought th at t he relevan t

distr ibution was t he cra sh-exclusive one ofFigure 6, an d sold rich pu ts,

you would h ave gained several points ofS AS

.

Valuing Options on

Baskets of Stocks

The valu e of an OTC option on a custom bask et of stocks is difficult to

estima te, since there is n o liquid options m ar ket from which to extr act

pricing inform at ion. Consider an investor inter ested in bu ying a collar

on a basket of bank stocks he owns. Suppose he wants to buy a 10%

out-of-th e-money pu t an d fin an ce it by selling a 10% out -of-th e-money

call on the basket. What volatility spread or skew should one use to

price th e collar?

One of the problems in valuing basket options is that correlations

am ong component stocks var y widely with t he st ock levels. When t her e

is a sharp downward market move, the correlations tend to increase.

Consequ ent ly, a ba sket ma y exhibit la rge volat ility skews even if each

component stock shows little skewness in its distribution. Thus OTM

puts on a basket should, in general, trade a t a premium because th e

Strike Level

Volatility(%)

1200 1250 1300 1350 1400 1450

18

20

22

24

26

28

30


Strike Level

SAS(%)

1200 1250 1300 1350 1400 1450

-2

-1

0

1

Strike Level

Volatility(%)

1200 1250 1300 1350 1400 1450

20

22

24

26

28


Strike Level

SAS(%)

1200 1250 1300 1350 1400 1450

0

1

2

3

4

5

FIGURE 7. Re-evaluated SASATM on June 21, 1999 for September 17, 1999S&P 500 options. The top two figures correspond to the crash-inclusivedistributions of Figure 5; the bottom two correspond to the crash-exclusivedistributions of Figure 6.

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increasing corr elations in a down ma rket mak es the basket retu rn dis-

tr ibut ion sk ewed to the lower en d. How do we model this effect? We ut i-

l ize th e informa tion embedded in the historical t ime series of th e

basket.

To be specific, we consider an example in which th e bas ket consists of

an equal number of shares of five bank stocks: J.P. Morgan, Wells

Fargo, Bank One, Bank America, a nd Ch ase. We first retr ieve th e his-

torical data for all five stocks and aggregate th em to form the t ime

series of bask et ret ur ns an d their hist orical distr ibution. We use histor-

ical data from June 1987 to June 1999 in this example. By minimizing

th e relat ive ent ropy, we convert th e historical distr ibution into an est i-

mate of the risk neutral distribution. Figure 8 displays the estimated

three-month implied volatility skew for the bank basket calculated

from the risk-neutral distribution. The volatility spread between the

10% OTM call an d th e 10% OTM put is appr oximat ely seven volat ilitypoint s. In th e absen ce of any ma rk et inform at ion on t he pr ice of options

on this bask et with a variety of strikes, this seems a useful method of

obtaining some sense of the appropriate skew. We note that in using

this approach, we managed to bypass the problem of predicting future

correlations between the component stocks in the basket, a major hur-

dle in valuing basket options. In this particular example, t he three-

month correlations between the stocks in the basket almost doubled

during the Fall of 1998 following the Russian currency devaluation.

Our a pproach ta kes into account t he chan ges in corr elations embedded

in the basket time series.

We ha ve also applied our model to options on t he BKX index (a bask etof 24 large U.S. banks with opt ions l is ted on the Phi ladelphia

exchange). We constructed a basket with the same weighting as the

BKX index and calculat ed both its emp irical retu rn dist ribut ion an d its

estimated risk-neut ra l distribution. The r esulting thr ee-month volatil-

ity skew is close to the skew observed in the listed options market,

even when the at-the-money volatility levels differ. This further dem-

onstr at es the reasonableness of our approach.

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Forecasting The

Shape of The Skew

From A Change In A

Single Option Price

Market makers in index options keep a steady eye on the skew. Sup-

pose th at for a given expira tion ther e are n options, with st rikes Ki and

known implied volatil it ies , that chara cterize the skew. Suppose

th at imp lied volatilities and th e skew have been rela tively sta ble; th en,

one of the options implied volat ilities su ddenly chan ges in r esponse to

new mar ket sentiments or pressures. How should th e mar ket m akeradjust t he quotes for a ll oth er options given t he su dden change in th e

price of one? This question is par ticularly relevan t for a ut oma ted elec-

tronic market-making systems. The maximum entropy method pro-

vides a possible answer.

We start with the implied distribution 8 computed from the known

implied volatil it ies . Now suppose the implied volati li ty of one

op tion wit h st r ik e level h a s ch a nged t o a n ew va lu e . We

would like to regard this one move in implied volatility as the visible

tip of th e iceberg, the observable segment of a new s kew th at will soon

manifest. To identify this new skew, we seek to find the new risk-neu-

t ra l dist ribu tion t ha t is con sist en t wit h t he sin gle n ew

and known implied volatility, while minimizing the entropy change

FIGURE 8. The estimated fair three- month implied volatility skew for thebasket of five bank stocks listed in the text, estimated from the risk-neutralized historical distribution using returns from June 1987 to June1999.

Strike Level

Volatility(%)

80 90 100 110 120

20

2

2

24

26

28

30

32

8. See Derma n, Kan i, an d Zou (1996).

i Ki( )

i Ki( )

Kj Kj T,( )

Q S T T S 0 0,,( )

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between the old and new distributions. Once th e new r isk-neut ra l dis-

tribution is obtained, we can update the quotes for the rest of the

options by valuing th em off the n ew distribut ion.

Here are some examples. Consider a hypothetical index whose current

value is 100. Suppose th e th ree-month , at -the-money volatility is 24%,

and the thr ee-month skew is linear in str ike with a slope corr espond-

ing to a two-volatility-point increase per ten-strike-point decline, as

displayed in Figure 10a. The heavy X in th e figure shows the one newlyobserved implied volatility, assumed to rise of four implied volatility

points, from 26% to 30%, for the 90-strike put. Figure 10b shows the

cha nge in the risk-neut ra l implied distr ibution obta ined by minimizing

the cha nge in distributiona l entr opy consistent with one n ew implied

volatility. The increase in 90-strike implied volatility has led to a sig-

nificant hump in the risk-neutra l distribution below the 90 level.

Finally, Figure 10c shows both the old and new skews, the lat ter com-puted from the n ew risk-neut ra l distribution. The new estimat ed skew

differs from t he old in a non-obvious way: it h as not sh ifted pa ra llel to

accommodate the one new item of information, but instead suggests

th at th e skew slope will increa se as a r esponse to th is shock.

Figure 9 shows several more examples where the volatility at one par-

ticular strike is shocked. Note that shifts in at-the-money volatility

seem t o lead t o par allel shifts in t he skew, wherea s shifts in out-of-th e-

money volatility lead t o cha nges in slope a s well as level.

End-of-Day Mark To

Market

At the end of a trading day, volatility traders need to re-mark all of

th eir options positions. Often, only th e liquid str ikes ha ve tr aded close

to the end of th e day, and, if th e last t ra ded option has u nder gone a sig-

nificant change in implied volatility, one needs to estimate the appro-

priate skew for the remaining, less liquid options, based on the new

information. Our model of minimizing the relative entropy provides a

ready solution to the problem. In the examples shown in Figure 9, one

could plau sibly mar k to ma rk et th e rest of options using th e solid lines

as our best guesses for closing volatilities.

Filling Gaps In

Investors Market

Views

Risk ar bitr ageurs, speculators, an d other situat ion-driven investors

often have very specific, but not necessarily complete, views on the

mar ket. For inst ance, risk ar bitr ageurs t aking a position in t he stocks

of two companies involved in a merger or acquisition may estimate a

90% probability that the deal will be completed and the stock will move

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19


above a certain target level within three months. Suppose tha t theywould like to take positions in stock and options to implement this

belief.

Although the arbitrageurs have a firm opinion about only one segment

of the probability distribution, a more complete distribution can be

helpful in constructing strategies and determining reasonable prices.

How can we fill the gap and extract the arbitrageursmarket distribu-

ImpliedVolatility(%)

15

20

25

30

Strike Level

80 90 100 110 1201

5

20

25

30

80 90 100 110 120

FIGURE 9. The shifts in skew that result from minimizing the entropy changeof the new risk-neutral distribution to acc ommodate a shift in one impliedvolatility, as shown by the arrows below.

FIGURE 10. (a) The initial skew (dashed line) and the one newly observed implied volatility (X).(b) The initial (dashed line) risk-neutral distribution and the new (solid line) risk-neutraldistribution. (c) The new skew (solid line), with grea ter negative slope.

Strike Level

ImpliedVolatility

(%)

80 90 100 110 120

20

25

30

X

X

(b)

Index Level

Probability(%)

50 100 150

0.0

0.5

1.0

1.5

2.0

2.5

Strike Level

ImpliedVolatility

(%)

80 90 100 110 120

20

25

30

X

X

(a) (b)

X

(a) (b) (c)

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t ion based on their specific prediction that the market has a 90%

chance of moving above the target level? One good starting point is to

use t he option implied distribut ion, P, as a pr ior distribution, and th en

find a new distribution, Q, tha t sa tisfies the a rbitra geurs 90% proba-bility estimate

where S T is th e a rbitra geurs t ar get stock price. Again t he model of

minimizing the relative entropy provides a natural solution to this

problem.

Q S T S 0 0,;,( ) SdS T

0.9=

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CONCLUDING

REMARKS

Investors in equity options have two problems that compound each

other : the ma ny th ousan ds of equity underlyers an d th e presence of a

un ique volatility skew for ea ch of them. For ma ny th inly-tr aded sin gle

stock and basket options, it is difficult or impossible to get adequat einformation on the market skew.

In this paper, we have employed a systematic, semi-empirical method

for estimating the risk-neut ra l distribution of any under lyer, stock or

basket, whose historical returns are available. This method, originally

used by St ut zer (1996), involves t he det erm ina tion of a n ew, risk-neu-

tr alized historical distr ibution (RNHD) for an u nder lyer by minimizing

the relative entropy between the historical distribution and the risk-

neut ra l distribution.

Using the RNHD, we can comput e t he estimated fair implied volatili-

ties of options of any strike and expiration. We can apply this methodto i ll iquid or thinly-t raded derivat ives where market prices are

unavailable.

We have defined a new metric, the strike-adjusted spread, or SAS, for

gauging t he valu e of options whose prices ar e kn own. SAS is th e differ-

ence between an options implied volatility an d its fair volatility as

estimated using the RNHD. This spread represents the richness in vol-

atility points of an option, compared to the history of its underlyer.

Most often, in liquid markets, we calibrate the SAS to be consistent

with cur rent at-the-money volatility, so tha t it becomes a measu re of

skew richness as compared with history. The SAS ranking cannot be

used blindly; it depends on th e u sers selection of the historical periodmost r elevant to the curr ent ma rket.

There a re m an y other applicat ions of the m ethod of minimal relative

ent ropy we have illust ra ted in th is paper. One ma y choose as a pr ior a n

existing optionsimplied distribution, or any other distribution reflect-

ing subjective mar ket views. We h ope th at this pr actical m ethod an d

its extensions will help investors make more rational decisions about

value in volatility markets.

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APPENDIX A:

INFORMATION AND

ENTROPY

Pr obability measur es th e un certa inty about t he occur rence of a single

random event. For a given random variable X, what can we deduce

from a s ingle obs erva t ion t ha t ?

Information Information 9 changes our view of the world. It seems obvious that the

amount of informat ion conveyed by the observat ion that should

depend on how likely this event was previously assumed to be. If a

stock was expected to go up the next day by everyone, and it actually

went down, the surprising outcome is certainly more informative than

th e expected out come. We wan t t o quan tify the n otion of events p rovid-

ing inform at ion.

We seek a funct ion that can represent the informat ion provided

by the occurrence of the event whose probabil ity was assumed

to be p. We require that be a non-negat ive and decreasing func-tion ofp; our intuition says th at I( ) is non-negative becau se th e occur -

ren ce of the event mu st provide some inform at ion; similarly, I( ) must

decrease w it h increasing p because the more l ikely we thought the

event wa s, the less inform at ion it s occurr ence provides.

Consider Xand Y to be two independen t ra ndom variables, an d assum e

that

(A 1)

Since Xand Y are independent, we have the joint probability

(A 2)

Wh en bot h in dep en den t even t s, a n d , occu r, t h e a ss oci-

ated information I( ) of each mu st a dd to the t ota l, so th at

(A 3)

Differentiate Equation 3 first with respect to p and similarly with

respect t o q to obta in

(A 4)

Dividing the fir st equ at ion by th e second, we obtain

x=

9. See Kullback, S. (1967), an d Cover et al. (1991).

X x=

I p( )X x=

I p( )

P X x=( ) p and P Y y=( ) q= =

P X x Y y=;=( ) pq=

X x= Y y=

I p q( ) I p( ) I q( )+=

qpq( )

I pq( )p

I p( )=

pp q( ) I p q( )

q I q( )=

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or

(A 5)

Since q and p are independent variables, each term in Equa tion 5 must

be a constant , denoted , so that

Since an event with probability q = 1 p rovides no in form a t ion , .Sin ce , a nd sin ce we r equ ir ed t ha t I( ) be a non-negative and

decreasing fun ction ofq, the consta nt c mu st be positive. From now on

we set , which defines the conventional size of a unit of informa-

tion. The information provided by the occurrence of an event whose

probability wa s p is th erefore given by

(A 6)

Entropy The probability a ssigned to a single event is a measure of the u ncer-

ta inty of its occurr ence. The entropy of a random variable R , whose ith

occur rence in th e distribution h as probability p i, is defined t o be the

expected value of the information from the occurrence of an event in

the distribution, na mely

(A 7)

Since any pr obabilityp i is less th an or equal to 1, th e entr opy is always

positive.

A large expected value of information means that the distribution was

broad, with a wide spread of probabilities. A small expected informa-

tion means the distribution was relatively narrow, so that not much

informa tion can be gained from t he occurr ence of an expected event.

Qua litat ively, th erefore, you can see th at H represents t he uncertainty

in th e distribu tion: lar ge (sma ll) avera ge informa tion H corr esponds to

q

p----

p

I p( )

q

I q

( )

-------------------=

qq

I q( ) p

p

I p( )=

c

I q( ) c q( )ln A+=

A 0=0 q 1

c 1=

I p( ) p( )ln=

H R( ) p i p ilogi 1=

n

=

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high (low) uncertainty.Entropy is th e math emat ical function th at m ea-

sures the uncertainty of a distribut ion . This is consistent with our int u-

ition t hat maximum entr opy corr esponds t o maximum uncerta inty.

If t he distribution collapses to one certain single event j whose

, a ll other , then H = 0, a m inimum . You can a lso show

tha t t he entr opy takes its ma ximum value, log(n), when for

a ll i, i.e. when a ll out comes ha ve an equa l cha nce and th ere is maxima l

uncertainty.

Having established the notion tha t the entr opy of a pr obability distri-

bution reflects the expected amount of information, we can now quan-

tify the informa tion gained u pon chan ging th e distribution a s a result

of new information. Let us assume we have a prior distribution of a

ra ndom var iableXwhich we denote by P. Upon ar rival of new inform a-tion, a posterior distribution Q is established. Wha t is th e consequent

reduction in uncertainty (decrease in entropy) in this process? One

obvious choice is the relative entropy:

(A 8)

The -log( ) fun ction is convex, so th at , by Jen sen s inequ alit y, th e aver -

a ge of t he is gr ea ter t ha n t he -log( ) of t he a ver age of

. Therefore,

Therefore, is s t r ict ly non-negat ive, and is zero if and on ly if

ident ical ly. can be thought of as a dis tance between

two probability dist ribut ions.

To maintain maximum uncertainty given some new information, our

goal is to minimize this relat ive ent ropy or dista nce between t he pr ior

and posterior distributions.

pj 1= p i 0=

p i 1 n=

S P Q,( ) EQ Q Ploglog[ ] q iq i

p i-----log

i= q i

p i

q i-----log

i= =

p i q i( )log

p i q i( )

S P Q,( ) q ip i

q i-----

ilog> p i( )

ilog 1log 0= = =

S P Q,( )P Q S P Q,( )

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APPENDIX B:

DETERMINING THE

RISK-NEUTRAL

DISTRIBUTION FROMHISTORICAL RETURNS

Consider a stock whose spot price is . Assume tha t investors believe

the fu ture re turn d is t r ibu tion of the asse t is given by a p r ior

. Our goal is to infer the r isk-neutral dis t r ibut ion

from subject to the const ra in t tha t the

mean of the risk-neutral stock distribution must equal the stock for-

ward price. We determ ine th e risk-neutra l distribution by minimizing

the relative entr opy between P( ) an d Q( ) subject t o Q( ) sat isfying the

forward condition.

We seek so tha t

(B 1)

such th at

(B 2)

and

(B 3)

where is the current r iskless interest rate.

In s ome cases we will also have a good idea of the cur ren t va lue of th e

stocks at -the-money implied volatility. In tha t case we will add to

Equation B2 and Equation B3 the further constraint that the at-the-

money implied volatility produced by the distribution Q( ) is equa l to

th e a t-the-money implied volatility of th e st ock.

Solving the above equa tions, we obta in th e risk-neut ra l distr ibution

(B 4)

where the consta nt can be found numer ically by ensur ing tha t it sat-isfies th e forwar d condition

(B 5)

Because in Equa tion B4 is a lways non -nega t ive, s o is .

S 0

P S0

0 ST

T,;,( )

Q S 0 0 S T T,;,( ) P S 0 0 S T T,;,( )

Q S 0 0 S T T,;,( )

M in S P Q,( ) EQQ S( )P S( )-------------

log=

Q S T( )S T S Td S 0erfT

=

Q S T( ) S Td 1=

rf

Q S 0 0 S T T,;,( )P S 0 0 S;, T T,( )

P S( ) S( )exp Sd--------------------------------------------------- S T( )exp=

Q S 0 0 S T T,;,( )S T S Td S 0erfT

=

P S T( ) Q S T( )

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APPENDIX C:

A DERIVATIVE ASSET

ALLOCATION MODEL

AND THE EQUILIBRIUMRISK-NEUTRAL DENSITY

In t his appen dix, we present a simple derivat ive ass et allocation model

and give a possible financial economic interpretation of the results

obtained in this paper based on the maximal entropy principle. Con-

sider an economy in equilibrium with total investor wealth W0. Thereis a market for an equity index, a riskless bond, and a complete range

of derivative instr umen ts on th e index. We assum e th at ther e is a r ep-

resentative investor10 with a subject ive market view expressed

through a conditional probability density , where is

the spot index level. We also assume the return on the riskless bond

over the period T is rf. The representative investor allocates a portion

of the total init ial wealth to the riskless bonds and the remaining

wealth to the risky assets. Let be the portion allocated to risklessbonds, and be the por t ion a lloca ted to r isky asse t s. S ince the

risky a ssets can be r eplicat ed with a portfolio of Arr ow-Debreu securi-

ties, th e investor can sim ply find a n optimal port folio of Arrow-Debreu

securities. An Arrow-Debreu security with parameter E has a price

given by

(C 1)

where is the discount factor and is the

risk-neu tr al pr obability density. The payoff of the Arr ow-Debreu secu-

rity at th e expira tion dat e is, by definition,

(C 2)

Let be th e port ion of th e fr act ion a lloca ted to r isky

assets t ha t is invested in Arr ow-Debreu security with pa ra meter E. At

th e end of period T, the t ota l investor wea lth will be

(C 3)

where

(C 4)

Note th at

(C 5)

10. See Consta nt inides (1982).

P S 0 0 S T T,;,( ) 0

1

S t t E T,;,( ) DQ S t E T ,;,( )=

D 1 1 rf+( )= Q S t t E T,;,( )

S T T E T,;,( ) S T E( )=

E( )d E 1

WT S T( ) W0 1 rf 1 ( ) E( )rE S T( ) Ed+ +[ ]=

rE S T( ) S T T E T,;,( ) S t t E T,;,( )

S t t E T,;,( )-----------------------------------------------------------------------------

S T E( )

DQ S t E T ,;,( )-------------------------------------- 1=

E r E ( )Q S t t E T,;,( )d rf=

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If the representat ive investor changes the al locat ion , the sup-

ply and demand for the Arrow-Debreu security will also change and

t hus s o will t he s hape of t he r is k-neu t r a l densit y funct ion

. Therefore, to achieve a market equilibrium, the repre-

sent at ive investor mu st solve th e asset allocation problem by maximiz-

in g t he expect ed u tilit y :

(C 6)

subject to th e constr aints:

(C 7)

where the expectation operator Ep is under m easur e P. The first order

conditions for t he const ra ined optimization problem ar e

(C 8)

(C 9)

(C 10)

(C 11)

(C 12)

and

(C 13)

wh er e a re t hr ee La gr an ge m ult iplier s cor respon din g t o

the three constra ints in Equation C7, respectively. From Equation C8

and C9 we obtain

E( )d E

Q S t t E T,;,( )

U WT( )

M a x E p U WT( )[ ]{ }

E( ) Q E( ),,{ }

budget constraint: E( ) E= 1dnormalization: Q E( ) E= 1d

forwar d constr aint: Q E( )E E=S 0 1 rf+( )d

EP U' WT( )W0 rf E( )rE S T( ) Ed( )[ ] 0=

EP U' WT( )W0 1 ( )rE S T( )[ ] 1 for E=

E( ) Ed 1=

EP U' WT( )'W0 1 ( ) E( ) S T E( )

DQ2

E( )---------------------------------------- 2 3E for E+=

Q E( ) Ed 1=

Q E( )E Ed S 0 1 rf+( )=1 2 and 3, ,

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(C 14)

and

(C 15)

From Equat ions C8, C14 and C15 we have

(C 16)

where

(C 17)

Using Equat ions C11, C15 and C16, we get

(C 18)

Using th e const ra ints (C12) an d(C13), Equ at ion (C16) leads t o

(C 19)

These results so far ar e independent of the pa rt icular functional form

of the utility function as long as the utility function is monotonically

increasing and concave. We now specialize in an exponential utility

given by

(C 20)

Using Equat ions C16, C17, C18 and C20, one can sh ow th at

(C 21)

where consta nt c0 and c1 satisfy

(C 22)

EP U' WT( )rE S T( )[ ]1

W0 1 ( )---------------------------=

EP U' WT( )[ ]1

W0 1 ( )rf---------------------------------=

Q S 0 t S;, T T,( )U' WT S T( )( )

EP U' WT( )[ ]---------------------------------- P S 0 t S T T,;,( )=

WT E( ) W0 1 rf+( ) 1 ( ) E( )

DQ E ( )-------------------+=

E( )Q E( )--------------

rf

1 rf+---------------

21------

31------E+=

21------ 31

------S 0 1 rf+( )+ 1 rf+rf---------------=

U WT( ) bWT( ) with b 0>exp=

Q S t t E T,;,( ) P S t t E T,;,( ) c0 c1E( )exp=

c0 b W0 1 rf+( ) EP U' b[ ]( )2

EP U' b[ ]---------------------------log=

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(C 23)

and t ha t we have, using Equa tion C17,

(C 24)

We can now car ry out th e optimization program as follows.

Step 1. From Equations C21, C12 and C13, we can n um er ica lly solve

for const an t c0 and c1, and thus obtain the risk-neutral distribution

Q. We point out th at the consta nt c0 and c1 (and thus the r isk-neu-

tr al distribution Q) is independent of the par amet er, b, of the u tility

function. It only depends on th e prior distr ibution P, and t he forwar dprice constraint Equa tion C13! The pa rameter b is characteristic of

th e represent at ive investors risk a version. It is essential tha t t he

risk-neu tr al distribut ion be indepen dent of th e investors risk aver-

sion! In our derivation of distribution Q, we relied only on the fact

th at th e investors u tility fun ction is of exponentia l form .

Step 2. From Equa tion C24, we ca n solve for , wh ich does

depend on para meter b.

Step 3. Fr om Equa tion C23, we can solve for .

Step 4. From Equa tions C15, C19 and C22 we solve for , , a nd

.

Fina lly, th e r epresen ta tive investors a llocation of the Arr ow-Debreu

security with parameter E, (E), can be found from Equation C18,which of cour se depends on th e risk avers ion pa ra met er b.

The most important feature of this solution to the asset allocation

problem is tha t the r isk-neut ra l p robability dens ity funct ion

is of th e form given by Equa tion C21 subject t o th e nor-

malization constr aint and Equa tion C13. This is th e sam e solution a s

that given by the minimal relative entropy approach in Equation B4,

provided tha t the represent at ive investor chooses th e r ealized histori-cal distr ibution as t he su bjective prior dist ribut ion P.

c1

3EP U' b[ ]---------------------------=

EP U' b[ ] bW0 1 rf+( ) Q E( )Q E( )P E( )--------------log Ed

exp=

EP U' b[ ]

3

1 2

Q S t t S T T,;,( )

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REFERENCES Buchen, P.W. and M. Kelly, 1996, Journal of Financial and Quanti ta -

tive Analysis, 31 , pp. 143 - 159.

Const an tinides, G.M., 1982, The Journal Of Business, 55 , No. 2.

Cover, T.M. and J.A. Thomas, 1991, Elements of Information Theory,

New York, J ohn Wiley & Sons.

Derman, E., I. Kani and J. Zou, 1996. The Local Volatility Surface,

Financial Analysts Journal, July/August, pp. 25-36.

Derm an , E., M. Kam al, I. Kani, an d J. Zou, 1997, Is the Volatility Skew

Fair?, Goldm an, S achs Qu antitative S trategies R esearch N otes

Gulko, L., 1996, Yale U niver sity Work ing Pa per.

Kullback, S., 1967, Informat ion Theory and Statist ics, New York,Dover Publications, Inc.

Stutzer, M., 1996, A Simple Nonparametric Approach to Derivative

Security Valua tion,J. of Finance, 51 . pp. 1633 - 1652.

Stutzer, M., and M. Chowdhury, 1999, A Simple Non-Param etric

Approach to Bond Fut ures Options Pr icing, Journal of Fixed Income,

March 1999, pp. 67-76.

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