Accounting for Biases in BlackScholes

y z x

David Backus Silverio Foresi Kai Li and Liuren Wu

First draft August

This version Novemb er

Abstract

Prices of currency options commonly dier from the BlackScholes formula along two

dimensions implied volatilities vary bystrike price smiles and maturity

of atthemoney options increases on average with maturity

We account for b oth using GramCharlier expansions to approximate the conditional

distribution of the logarithm of the price of the underlying security In this setting

volatility is approximately a quadratic function of a result we use to infer

skewness and kurtosis from volatility smiles Evidence suggests that b oth kurtosis

in currency prices and biases in BlackScholes prices decline with maturity

JEL Classication Co des G G F C

Keywords currency options skewness and kurtosis GramCharlier expansions

implied volatility

Wewelcome comments includin g references to related pap ers we inadvertently overlo oked

We thank Jose Campa Steve Figlewski Ken Garbade Cli Hurvich Josh Rosenb erg and Raghu

Sundaram for helpful comments and suggestions The most recentversion of this pap er is available

at httpwwwsternnyuedudbackus

y

Stern Scho ol of Business New York University and NBER dbackussternnyuedu

z

Salomon Brothers Inc sforesisbicom

x

Stern Scho ol of Business New York University klisternnyuedu

Stern Scho ol of Business New York University lwusternnyuedu

Intro duction

Since Black and Scholes revolutionized the theory of option pricing increas

ingly rened applied work has do cumented a numb er of anomalies or biases exam

ples in which observed prices of options dier systematically from the BlackScholes

formula These observations havemotivated in turn development of more complex

mo dels in which among other things jumps and sto chastic volatility in the price

of the underlying generate theoretical option prices dierent from BlackScholes

For currency options biases are apparent across dierentstrike prices and ma

turities or moneyness biases include familiar smiles smirks and

skews in which implied volatilityvaries across strike prices of otherwise similar op

tions For options on dollar prices of ma jor currencies the most common pattern

is a smile with implied volatilitylower at the money than in or out of the money

Equally interesting in our view are biases across maturities Although these are

obscured byvariation over time in the term structure of volatilitywe do cumentan

average pattern of lower atthemoney volatility for shorter options This maturity

bias suggests a time element to departures from BlackScholes that has largely b een

overlo oked

We show that b oth biases are natural consequences of stationary mo dels in

whichchanges in the logarithm of the price of the underlying securityherea

currency have greater kurtosis than normal random variables In continuous

ypically mo deled with jumpdiusions We time departures from normalityaret

cho ose instead discretetime GramCharlier expansions an approach pioneered in

nance byJarrow and Rudd Whatever its source excess kurtosis is well

understo o d to pro duce volatilitysmiles We derive a particularly simple version

of this result in which implied volatility is approximately a quadratic function of

moneyness with co ecients related to the skewness and kurtosis of the price Less

well understo o d we think is how the eect of kurtosis varies with maturityWe

show in two related theoretical environments that kurtosis biases downward the

implied volatility of atthemoney options but the bias eventually declines with

maturityFor long options the BlackScholes formula is a go o d approximation in

these mo dels

We argue that currency and option prices conform broadly with this theory The

excess kurtosis of logchanges in dollar prices of foreign currencies declines rapidly

as the time interval is extended Further an example of volatility smiles for several

maturities at the same date shows clearly that they get less pronounced as the ma

turity increases The rate of convergence in this case is slower than wewould exp ect

from mo dels with iid increments but is p otentially consistent with the secondorder

timedep endence of sto chasticvolatility mo dels with fattailed innovations In this

resp ect the b ehavior of currency options accords with related work on currency

prices Baillie and Bollerslev and Drost Nijman and Werker among

others

Biases in BlackScholes

We b egin with a selective list of systematic dierences in observed prices of currency

options from the BlackScholes formula Much of this dates back to Bo durtha and

Courtadons pp comprehensive assessment of currency options and

even b efore

Lists of BlackScholes biases invariably b egin with dierences in implied volatil

ities across strike prices or moneyness Although there have b een striking examples

of volatilityskews in currency options with volatility increasing or decreasing mono

tonically with moneyness options on dollar exchange rates more commonly exhibit

smiles with minimum volatility roughly at the money See for example Campa

Chang and Reider Malz and Rosenb erg Figure provides

an example from the overthecounter market onemonth deutschemark options at

vestrike prices on April This observation was supplied by Jose Campa

and comes from data describ ed in Campa Chang and Reider The gure

contains a classic example of a

Smiles and other patterns in implied volatilities are often attributed to depar

tures from normality in the logarithm of the price of the underlying In Table

we rep ort sample moments for three currencies of daily logchanges or depreciation

rates of the sp ot exchange rate

S x logS log

t t t

where S is the dollar price of one unit of foreign currency at date t and time is

t

measured in business days The mean and are expressed in

annual units With business days p er year in our sample on average we

multiplied the estimated mean by and the standard deviation by

Of particular interest to us are indicators of skewness and kurtosis whichwe

dene in terms of the cumulants of a random variable x The rst cumulantis

j

the mean E x The second is the variance E x The third is the

third central moment E x The fourth is E x

The standard indicators of skewness and kurtosis are scaled versions of the third

and fourth cumulants

skewness

kurtosis

Both are invarianttochanges in the scale of x and equal zero if x is normal

The estimates in Table replace exp ectations with sample moments Standard

deviations are just ab ove annually for the mark and the yen less than half that

for the Canadian dollar Skewness is small but estimated values of suggest

that logchanges in currency prices are not normal all three currency prices exhibit

greater kurtosis than normal random variables More formal statements to this

eect are made by Akgiray and Bo oth Baillie and Bollerslev Drost

Nijman and Werker Hsieh and Jorion

Less commonly noted are dierences in option prices across maturities Table

summarizes twoyears of daily volatility quotes for atthemoney options oered bya

large international bank Zhu The quotes come from atthemoney

a combination of a put and a call with the same strike price see for example Campa

and Chang p Most interesting to us is the term structure of atthe

money implied volatility Although the term structure at dierent p oints in time

has b een increasing decreasing or even at mean and median volatility exhibit a

steady rise with maturity The latter is pictured in Figure where we see a similar

pattern in all three currencies Average dierences in atthemoney volatility across

maturities are small relative to the time series variation in volatility but they are

nevertheless a recurring feature of option data including longer time series from

currency options traded on the Philadelphia Exchange and a nonoverlapping over

thecounter dataset summarized by Campa and Chang Tables I and I I This

feature of the data has not b een emphasized in earlier work but w as implicit in

remarks made by Black p Options with less than three months to

maturity tend to b e overpriced by the BlackScholes formula and Bo durtha and

Courtadon p atthemoney currency options with less than days to

maturity are overpriced by their mo dels

A RiskNeutral Framework

Like many of our predecessors we study option prices in a riskneutral world This

assumption is not inno cuous but it allows us to fo cus our attention on the distri

bution of the underlying sp ot price

The rst element of our framework is the conditional distribution of future sp ot

exchange rates The conditional distribution of the onep erio d change in the log

price of the sp ot rate is determined by the prop erties of x in equation Over n

p erio ds the change is

n

X

log S log S x

tn t tj

j

n

log S x

t

t

n n

with the obvious denition of x Then S S exp x and the conditional

tn t

t t

n

distribution of S dep ends on that of x

tn

t

The second elementisthevaluation of options The price of a Europ ean call

option on a currency with strike price K is

C E M S K

nt t ttn tn

where M is a multip erio d sto chastic discount factor and x maxx Put

ttn

call parity

n r n r

nt

nt

P C Ke S e

nt nt t

gives us the price P of a put with the same maturity n and strikeprice K

nt

The pricing relation is quite general but leaves us with the dicultyof

disentangling the eects of M and S We follow tradition and sidestep this issue by

assuming that the two are indep endent With this riskneutral structure the call

n

price dep ends in a clear way on the conditional distribution of x

t

r n

nt

S K C e E

t tn nt

Z

r n x

nt

e S e K f xdx

t

logKS

t

n

where f is the conditional densityofx and r is the continuouslycomp ounded

nt

t

np erio d yield measured in the same units as n

n

If the np erio d depreciation rate x is conditionally normal with say mean

t

and standard deviation then S is conditionally lognormal and the solution

n n tn

to is the familiar BlackScholes formula as adapted to currency options

by Garman and Kohlhagen

r n r n

nt

nt

C S e d Ke d

nt t n

where

logS K r r n

t nt

nt n

d

n

is the standard normal cum ulative distribution function and r is the foreign

nt

currency yield This follows from the arbitrage condition interest parity

log E S log S r r n

t tn t nt

nt

implying

r r n

n nt

nt n

and evaluation of the integral dened by Details are provided in App endix A

Largely for mathematical convenience we use the variable d as a measure of

moneyness in later sections with d indicating a at the money d

in the money and d out of the money This choice diers from standard practice

but the dierences are small The atthemoney straddles used to construct the

implied volatilities used in Table for example dene atthemoney by the forward

rate

n r r

nt

nt

K S e

t

implying d For an annual option on the deutschemark d is approximately

n

whichistiny relative to the range of d pictured in Figure For shorter options

the choice of atthemoney d is even smaller

Accounting for Moneyness Bias

We turn to prices of options of xed maturity when logchanges in the sp ot exchange

rate are not normal In continuous time nonnormality is mo deled with jumps

or p oint pro cesses In discrete time the conditional distribution of logchanges

can b e sp ecied directly and wehave some exibilityover its form A convenient

choice for our purp oses is a GramCharlier expansion in which a normal density

is augmented with additional terms capturing the eects of skewness and kurtosis

Johnson Kotz and Balakrishnan pp and Kolassa ch describ e

the underlying statistical theory This approachwas intro duced to nance byJarrow

and Rudd and has since b een applied byAbken Madan and Ramamurtie

Brenner and Eom Knight and Satchell Longsta

and Madan and Milne Our application diers from Jarrow and Rudds

in approximating the conditional distribution of the logarithm of the price rather

than the price itself This relatively subtle dierence leads to a particularly simple

expression for volatility smiles in terms of higher moments of the logprice of the

underlying

A GramCharlier expansion generates an approximate density function for a

standardized random variable that diers from the standard normal in having p o

tentially nonzero skewness and excess kurtosis In our application let the np erio d

n

equation have conditional mean and stan logchange in the sp ot rate x

n

t

dard deviation and dene the standardized variable

n

n

x

n

t

w

n

A GramCharlier expansion denes an approximate densityforw by

f w w D w D w

n n

j

where w expw is the standard normal density and D denotes

the j th of what follows Equation is typically viewed as an approx

imation to an arbitrary density with nonzero higher moments but for mo derate

values of and it is a legitimate density in its own right In such cases it

n n

provides a more parsimonious representation of a distribution with skewness and

kurtosis than the normal mixtures implied by jumpdiusions Most imp ortantto

us is that departures from normality are closely related to measures of higher mo

ments

Some of the prop erties of the GramCharlier expansion f are apparentfromits

cumulant generating function the log of its moment generating function

sw

s w log E e

s s s

n n

In w is normal this p ower series is zero after the quadratic term The GramCharlier

expansion extends the series for two additional terms Since the derivatives of

dene the cumulants the co ecients are

j

s w

jn

j

s

s

For a standardized random variable the standard deviation is one so

n n

n

and the standard indicators of skewness and kurtosis And since x is a

n n

linear translation of w these indicators apply to it to o Weleave the maturity n

explicit for later use

We use the relation b etween the density and its higher moments to derive ex

pressions relating a call options price implied volatility and delta to the same

moments Call prices are given by

Prop osition Consider currency options in an environment characterized by the

riskneutral pricing relation with the evolution of the spot rate governedby

equations and the GramCharlier expansion Cal l prices in this

setting are approximately

r n r n

nt

nt

S e d Ke d C

t n nt

n n

n r

nt

d d d d S e

n n n t

n

where d is dened by

A pro of is given in App endix A As with our approach to the BlackScholes

formula it relies on a combination of an arbitrage condition and the patience to

evaluate the integral dened by The approximation refers to our elimination of

terms involving p owers three and higher of which are negligible in practice

n

Equation expresses the call price as BlackScholes plus terms involving the

skewness and kurtosis of the logarithm of the underlying sp ot exchange

n n

rate Kurtosis lowers prices of atthemoney options d relativetoBlack

Scholes but raises prices of far in and outofthemoney options jdj The

combination stems from the eect of kurtosis on the density it increases the prob

abilities of b oth small and large realizations For atthemoney options the former

dominates leading to lower prices than BlackScholes For outofthemoney op

tions the latter dominates and the price exceeds its BlackScholes value Figure

illustrates the dierences b etween BlackScholes call prices the solid line and those

of Prop osition with p ositiveskewness dashed line and kurtosis dashdot line

The picture is more familiar if w e translate the bias into volatility units Dene

implied volatility v as the value of that equates the BlackScholes formula

n n

to the observed price given values of the other parameters We refer to the relation

between implied volatility and moneyness as the implied volatility smile If the call

price is in fact generated by then the implied volatility smile is given by

Prop osition Let the priceofa cal l option betheGramCharlier formula

Then the implied volatility smile is

n n

d d v d

n n

with d dened by

A pro of is given in App endix A

Although the form of is new its content is reminiscent of Mertons

Section discussion of the qualitative features of volatility smiles in jumpdiusion

mo dels The most attractive feature of our version is its simplicity it expresses

option price biases in terms of the widelyused implied volatility and relates them

to higher moments in the distribution of the underlying The equation is an approx

imation for several reasons the GramCharlier expansion a linear approximation of

call prices in terms of volatility and the elimination of terms involving p owers of two

and higher in Of these only the rst and last seem to havemuch eect Figure

n

compares the volatility smiles implied by excess kurtosis under our approximation

dashed line and under an exact inversion of the call price formula For

jdj the dierences are small but for larger values the true implied volatility

is smaller than that implied by Prop osition Figure illustrates the dierences

implied by equation b etween p ositive kurtosis solid line and a combination

of p ositive kurtosis and negativeskewness dashed line that mimics the volatility

smirks and skews do cumented in other markets

As an example of how equation might b e used in practice consider the

volatility smile for onemonth deutschemark options pictured in Figure A least

squares t of a quadratic function to the smile implies moments of

n

and In this way the volatility smile gives us estimates

n n

of the true volatility and higher moments of the sp ot exchange rate pro cess Just

as the BlackScholes formula made volatility visible from option prices Prop osition

makes skewness and kurtosis visible from the shap e of volatilitysmiles

Before turning to maturity bias wetake a short digression to consider the eects

of skewness and kurtosis on the delta of an option

Corollary The delta of the GramCharlier cal l priceis

C

nt

r n

nt

e d

r n

nt

S e

t

n

d d d

n

n

n

dd d d

n n

n n

with d dened by

The pro of consists of dierentiating

The delta like the call price and implied volatility is the conventional Black

r n

nt

Scholes result here e d plus additional terms involving skewness and kur

tosis Examples are pictured in Figure For atthemoney options the delta

simplies considerably Setting d and eliminating terms involving p owersoftwo

and higher of

n

r n

nt

e

n n n

Since is generally small kurtosis has little eect on the delta at the money

n

despite having a substantial eect on the price Skewness on the other hand has

little eect on the atthemoney price but can have a sizable eect on the delta

Equation is a reminder that the rstorder approximations often used in

hedging and risk management of options can dier substantially from the Black

Scholes b enchmark when the logprice of the underlying exhibits skewness or kurto

sis We will see shortly that skewness and kurtosis are most evident in shortdated

options where the diculties of hedging are widely thought to b e most severe

The Quality of the Approximation

Prop ositions and base approximations of call prices and implied volatility smiles

resp ectively on a GramCharlier approximation to the conditional density of the

logprice of the underlying In this section we examine the accuracy of the Gram

Charlier approximation to the smile equation of Prop osition when the dis

tribution is generated by a jumpdiusion Merton This distribution has

some supp ort in the empirical literature Akgiray and Bo oth for example

and is capable of generating a wide range of nonnormal b ehavior Weshow that

with what we regard as reasonable parameter values higher moments inferred from

option prices using are similar to the true moments of the pro cess for the

underlying This do es not constitute a general endorsement of the GramCharlier

approximation but gives us some reason to b elieve that estimates of higher moments

from option prices are useful indicators of the same prop erties of the underlying

The jumpdiusion pro cess denes the densityofthenp erio d logprice change

asacountable mixture of normals

X

f x p x j j

j

j

j

n

where p e j is the p oisson probabilityofj jumps is the jump intensity

j n

n

the exp ected numb er of jumps in n p erio ds are the mean and standard

deviation of the diusion are the mean and standard deviation of the jump

The jumpdiusion density and x expx

exhibits greater kurtosis than the normal and if is nonzero nonzero skewness as

well The rst four cumulants are

n n

n

n

n

If skewness is zero and kurtosis dep ends on the intensity and variance

n

its sign carries over to the third cumulant of jumps When

This departure from normality results in option prices that dier from Black

Scholes The price of a Europ ean call with strikeprice K is

X

r n

t

C e p W

nt j j

j

where

 

j j





W S e d K d j

j t j j

log S K j j

t

d

j

j

The result follows from rep eated application of equation in App endix A

Similar formulas are rep orted by Bates b equation Jarrow and Rudd

equation and Merton equation

Table summarizes the accuracy of GramCharlierbased approximations to

skewness and kurtosis in the jumpdiusion mo del For a varietyofchoices of pa

rameter values we compute call prices for the jumpdiusion mo del equation

implied volatilities inverting and GramCharlier estimates of higher moments

a least squares t of equation In each case the moneyness vector d cor

resp onds to d The jumpintensity parameter is chosen to

corresp ond to estimates in the literature Other parameters are chosen to repro

duce plausible values of domestic and foreign interest rates the standard deviation

skewness and kurtosis Given interest rates the nonjump mean

is adjusted to ob ey







r r n e

t n

t

an application of the arbitrage condition Our b enchmark values are r r

t

t

annualized and Estimates of the jump

intensity parameter are rep orted by Akgiray and Bo oth Table Bates

a Table and Jorion Table Annualized values range b etween

and with a median of ab out just under one jump p er month on average

whichwe use as our starting p oint

With b enchmark parameter values Panel A the GramCharlier approximation

of the volatility smile implies skewness and kurtosis estimates of and

Both values are close to the moments of the jump distribution and resp ectively

In that sense they suggest that Prop osition is a passable approximation In

fact some of the error comes not from the GramCharlier approximation but from

the additional approximations made in moving from Prop osition to Prop osition

A least squares t of the more accurate equation generates estimates of

In the remaining panels of Table we examine the sensitivity of these estimates

to the choice of parameters In Panel B we consider dierentvalues of jump in

tensity With larger values the approximation gets b etter with rises

to but with smaller values it gets worse when Values

ab ove are incompatible with Values b elow imply larger less frequent

jumps and the GramCharlier estimates of kurtosis are well b elow those of the

jumpdiusion mo del

In Panel C we examine volatility Neither smaller nor larger values has an

appreciable eect on estimates of higher moments In Panel D we examine op

tions with longer maturities Threemonth options allow more time for jumps and

therefore require greater jump variance to generate the same amount of kurtosis

The result is a slightoverestimate of kurtosis For the sixmonth option we set

the mo del is incapable of repro ducing with the b enchmark value

of The GramCharlier estimate is In all of these examples the errors in

GramCharlier estimates of skewness and kurtosis are small

Panel E suggests that the accuracy of the approximation changes little when

wevary the amount of kurtosis Skewness however p oses some diculties Panel

F The GramCharlier approximation systemically underestimates the absolute

amountofskewness and for larger values underestimates kurtosis as well This

diculty highlights a dierence b etween the GramCharlier and jump mo dels In

the GramCharlier approximation skewness has approximately a linear eect on

implied volatility In the jumpdiusion mo del skewness alone pro duces concave

volatility smiles The approximation therefore reduces its estimate of to comp en

sate

In short the GramCharlier approximation works well in some cases less well in

others Others here refers to examples with low jump intensity or substantial skew

ness In our opinion the approximation is reasonably go o d in the range of parameter

values indicated by currency data but logic strongly suggests that there can b e no

general defense of the approximation The GramCharlier expansion arises from

aTaylor series approximation to the cumulant generating function and examples

are commonplace of functions for whichashortTaylor series is an extremely p o or

approximation For similar reasons there must certainly exist examples in which

GramCharlier approximations to call prices are p o or Whether these examples are

realistic is imp ossible to say without knowing what they are Wetake some com

fort however in the ability of the mo del to approximate estimates of kurtosis from

jumpdiusions when plausible parameter values are used

Accounting for Maturity Bias

The second bias concerns maturity average atthemoney implied volatilityistyp

ically smaller at short maturities than long ones Table and Figure To study

this issue wemust sp ecify a sto chastic pro cess for logprice changes a description

of the conditional distribution of price changes over p erio ds of arbitrary length n

We consider two such pro cesses In this section we examine iid innovations The iid

structure is not realistic but allows an esp ecially simple characterization of the ef

fects of maturity In the next section we explore the p ossibility of timedep endence

induced bystochastic volatility In b oth cases departures from normality and Black

Scholes go to zero with maturity Since kurtosis lowers atthemoney volatilitywe

have a p otential explanation for the maturity bias

Supp ose then that np erio d logchanges in the exchange rate are comp osed of

iid onep erio d comp onents with nite moments of all order Sp ecically let daily

depreciation rates x be

log S log S x

t t t

t t

where f g are iid with mean zero and variance one Over n p erio ds the logchange

t

in the sp ot rate is

n

log S log S x

tn t

t

n

X

n tj

j

n

with the obvious denitions of x and

n

t

Now consider the b ehavior of cumulants and moments over dierent time inter

has cumulant generating function vals n An arbitrary with nite cumulants

j

j

X

s

j

s

j

j

Recall that the cumulant generating function of the sum of indep endent random

variables is the sum of the generating functions of the individual random variables

P

n

n

has cumulant generating function Then

tj

j t

j

X

n s

j

n

s

t

j

j

and hence has cumulants n As a result the standard deviation increases in the

j

familiar way with the square ro ot of maturity n In addition the skewness

n

and kurtosis indicators and dened by decline with n

n n

n

n

n n

n

n

n n

Similar expressions are rep orted by Das and Sundaram for jumpdiusions

Thus departures from normality as indicated by higher moments decline with ma

turity

The GramCharlier expansion is a sp ecial case of this setup with cumulants equal

to zero after the fourth In this setting

Prop osition Suppose nperiodlogchanges in the exchange rate arecom

posed of standardized iid oneperiod innovations whose density is given by a

t

GramCharlier expansion with third and fourth cumulants and As the

maturity n of a cal l option approaches innity skewness and kurtosis approach zero

and cal l prices approach the BlackScholes formula

The prop osition follows from the formulas for skewness and kurtosis equations

and and the relations b etween higher moments and option prices and

Prop osition suggests an explanation of the maturity bias the tendency for

average or median atthemoney volatility to rise with maturity The eects of

kurtosis at dierent maturities are pictured in Figure For onemonth options

kurtosis generates the smile wesaw in Figure For threemonth options and the

iid structure of this section kurtosis declines by a factor of This results in a less

sharply curved smile and a general reduction in dierences from the BlackScholes

formula For atthemoney options implied volatility is higher for the longer option

We mightthus exp ect to see that atthemoney volatility rises with maturityaswe

saw on average in the data

Prop osition and the moment prop erties go b eyond these qualitative

prop erties and provide explicit predictions of how prices of currencies and options

b ehave across maturities Each of these prop erties can b e compared with the evi

dence Consider currencies In equation kurtosis is prop ortional to n

n

Figure is an attempt to compare this prediction with the data We computed

n

for dierentvalues of n for the ten years of data used in Table To put them on

the same scale the estimates of for each currency are divided by We see

n

in the gure that kurtosis declines rapidly with n Moreover the pattern of decline

is not much dierent from the solid line n which is what the iid mo del predicts

The yen is very close to this line The Canadian dollar and the mark are similar

but decline less rapidly for small n than the b enchmark line On the whole the

data are roughly in line with equation but exhibit less rapid convergence

A more limited lo ok at option prices conveys a similar message Equations

and tell us that Prop osition should showupasvolatility smiles with

progressively less slop e and curvature as we increase maturity Exchangetraded

options generally do not have enough depth across the moneyness sp ectrum to allow

us to estimate smiles preciselymuch less compare them across maturities but the

Campa Chang and Reider data includes maturities b etween one day and

eighteen months Strike prices in this data set are chosen to set the BlackScholes

r n

nt

e d equal to and with an additional atthemoney strike

corresp onding to d see the discussion at the end of Section Smiles on

n

April our only observation and the rst one in Campa Chang and Reiders

data are graphed in Figure for maturities b etween one day and one year As the

theory suggests the smiles get increasingly atter as the maturity rises Its essential

here that the smiles are expressed in units of moneyness that are comparable across

maturities We use d as suggested by Prop osition but the imp ortant thing is

that the natural spread in the distribution over time the increase in with nbe

n

counteracted Once this is done the smiles atten out with maturity as Prop osition

suggests

n

More concretely the smiles in Figure can b e used to infer moments of x as

we describ ed in Section The results of this are rep orted in Table The

dramatic rise in the level of the smiles and the dierence b etween Figures and

is attributed to a rising term structure of volatility More relevant to Prop osition

both skewness and kurtosis decline with maturity The primary discrepancy with

the theory of this section is the rate of convergence estimates of kurtosis inferred

from option prices Table decline substantially less quickly than the n rate of

equation

Accounting for Maturity Bias

Despite its lack of realism the iid mo del in the last section provides a relatively

go o d qualitative explanation for b oth moneyness and maturity biases in Black

ely it suggests that higher moments decline rapidly with ma Scholes Quantitativ

turity volatility smiles suggest that the rate of decline is less rapid In this section

we consider timedep endence in onep erio d depreciation rates x Since depreciation

rates exhibit little evidence of auto correlation we are led to consider dep endence

through second moments or sto chastic volatilityWe show that suchmodelscan

generate slower convergence of higher moments to zero over the range of maturities

observed in option markets

Sto chastic volatility mo dels are motivated by clear evidence of predictable varia

tion in conditional variance This is apparent in the dynamics of implied volatilities

Table for example and in estimates of GARCH and related mo dels Notable

applications to currency options include Bates a Melino and Turnbull

and Taylor and Xu

The generic sto chastic volatility mo del starts with logprice changes of the form

x z

t t t

t

where is conditionally indep endentofz and f g is a sequence of iid draws

t t t

with zero mean and unit variance The new element z is the conditional variance

t

Popular pro cesses for z include the squarero ot mo del used by Bates a

z z z

t t t

t

the logarithmic mo del of Kim and Shephard

log z log z

t t t

and the GARCH mo del estimated by Baillie and Bollerslev

z z z

t t t

t

z z

t t t

In all of these mo dels f g is iid with zero mean and unit with

t t

t

variance Although these mo dels are not equivalent their prop erties are similar in

many applications

n

Whatever the pro cess variation in z generates kurtosis in x and x With no

particular structure on z but existence of unconditional mean and variance the

unconditional kurtosis in x is

x cz cz

where cz Var z E z is the squared co ecientofvariation of z the ratio of the

standard deviation to the mean See App endix A Equation quanties two

ways of generating excess kurtosis in the unconditional distribution of x kurtosis

in the innovations represented by and sto chastic volatility represented by

cz Note to o that the two eects complementeach other the total is greater

than the sum of the parts

Similar relations hold conditionally and over multiple p erio ds if f g and f g are

t t

indep endent This assumption is violated in the GARCH mo del since

t

t

but we think the gain in simplicitymakes the extra structure worthwhile Then

conditional kurtosis is

P

P

n

n

z E

z Var

t

tj t

j tj j

n

x

P P

n n

E E z z

t t tj tj

j j

which is derived in App endix A As with unconditional kurtosis the interaction

of fattailed innovations and sto chastic volatility results in greater kurtosis than the

sum of the two individuall y

Further progress requires us to b e more sp ecic ab out the pro cess for z In the

interest of transparencywestudyalinearvolatilitymodel

z z

t t t

with f g indep endentoff g When the comp onents of conditional

t t

kurtosis are

n

X

nj n

z z

tj tn t

j

n n

nj n

X X

z z n

t tj tj

j j

n

n

X

E z n z O n

t tj t

j

n n n

j

h i

X X X

j

O n z E z

t t

tj

j j j

n n

j

X X

Var z O n

t tj

j j

k k

The notation f nO n means that f nn has a nite nonzero limit Wesay

k

that f n is of order n When the expressions b ecome

n

X

E z nz O n

t tj t

j

n

X

z E nz nn O n

t

tj t

j

n

X

z Var nn n O n

t t j

j

This limiting case provides an upp er b ound to the eects of sto chastic volatilityon

conditional kurtosis

We can now assess the qualitative features of conditional kurtosis When

conditional kurtosis is

O n O n

n

x

O n O n

The second term has a nite limit but the rst grows without b ound with nIn

sharp contrast to the previous section there is no tendency for higher moments to

decline with maturityWhen however the qualitative features of the

last section return Conditional kurtosis is

O n O n

n

x

O n O n

so b oth terms eventually converge to zero

Consider now the rate at which higher moments converge in this mo del In

the stationary case the mo del combines twomechanisms studied extensively by

Das and Sundaram nonnormal innovations and sto chastic volatility With

out sto chastic volatility that is with whichwe think of as analogous to a

pure jump mo del kurtosis follows the pattern we do cumented in the last section

n

x n Without nonnormal innovations that is with kur

tosis is humpshap ed approaching zero at very short and very long maturities As

AitSahalia and Lo p note neither corresp onds to observed option prices

the former declines to o rapidly to zero the latter is to o small at short maturities

Anumerical example suggests that a combination of nonnormal innovations

and sto chastic volatilitygives a much b etter account of the b ehavior of conditional

kurtosis across maturities than either on its own This complements work by Baillie

and Bollerslev Drost Nijman and Werker Hsieh and Jorion

who prop osed similar mo dels and showed that they accounted for manyof

alues are the observed prop erties of currency prices The parameter v

annualized annualized and and the

state variable is set equal to its mean z The relation b etween kurtosis and

t

maturity is pictured in Figure For maturities b etween one dayandeighteen

months kurtosis declines from to as in the option data summarized in Table

Note that the kurtosis in this example is substantially greater than we get from

jumps nonzero or sto chastic volatility nonzero alone

The tendency for kurtosis to decline with maturity in this mo del is a conse

quence of a stronger result the central limit theorem As Dieb old showed

in similar environments average changes in logprices are normal over long enough

time intervals As a result departures from the BlackScholes formula decline with

the maturity of the option This statement do esnt apply to all theoretical environ

ments the unit ro ot volatility mo del is a counterexample but it app ears to b e a

reasonable approximation for prices of currencies and currency options

Final Thoughts

Wehavecontinued a line of research initiated by Jarrow and Rudd of using

GramCharlier expansions to explore the impact of departures from lognormality

in the underlying on the prices of options We add two things i an extremely

simple relation b etween the shap e of implied volatility smiles and the skewness and

kurtosis of the underlying logprice pro cess and ii an examination of how smiles

and higher moments vary with maturity The latter includes the suggestion that

biases in the p opular BlackScholes formula should b e greatest for short options and

disapp ear entirely at long enough maturities We argue that a mo del with b oth

fattailed innovations and sto chastic volatility can account for the relatively slow

decline in kurtosis with maturity that we infer from option prices

These conclusions p ointusinanumb er of directions One is to use option

prices to study the dynamics of conditional higher moments in asset prices Studies

based on the price alone have b een hindered by the diculty of estimating such

moments from small numb ers of realizations Hansen remains the only study

weknow to attempt this Option prices give us another source of information

and raise the hop e that we will b e able to do cument the time series prop erties of

skewness and kurtosis as earlier work has done with conditional variances Another

direction is the b ehavior of exotic options App endix A extends the theory to

digital options but the more dicult barrier options remain A third direction

concerns the nature of option markets and data Wehave mo deled option prices in

a comp etitive frictionless world but p erhaps bidask spreads and noncomp etitive

pricing account for some of the dierences b etween observed option prices and the

holes formula BlackSc

A Derivations and Pro ofs

A BlackScholes Formula

We adapt Rubinsteins approach to the BlackScholes formula Let the n

n

p erio d logprice change x b e normal with mean and variance so the density

n

t n

function is f x exp x The integral in has two

n n

n

terms The rst is

Z



x

n

n

S e f xdx S e d

t t

logKS

t

with

log S K

n t

n

d

n

The second is

Z

Kf xdx K d

n

logKS

t

One version of the BlackScholes formula is

h i



r n

nt n

n

BS e S e d K d

t n

whichwe note for later use The conventional BlackScholes formula equation

results from using the arbitrage condition to eliminate

n

A Prop ositions and

Prop osition The pro of has the same steps as the derivation of BlackScholes

but requires more work in evaluating integrals The GramCharlier densityis

n n

D D w f w

n

where w x and w expw Application of thus

n n

involves the integral

Z Z

w w

n n n n

S e K w dw S e K f w dw

t t

w w

Z

n

w

n n

S e K w dw

t

w

Z

n

w

n n

S e K w dw

t

w

n n

I I I

with w log KS The rst piece has the same form as BlackScholes

t n n

r n

nt

e I BS

See The second piece weevaluate by rep eated application of integration by

parts

r n

nt

I Kw w e BS K w

n n

n n

x n

This uses a prop erty of derivatives of the normal density lim e x

x

The third piece is

i h

r n

nt

w e BS K w I Kw w

n

n n n

The call price is therefore

n n

r n

nt

I I C e I

nt

n n

BS

h i

n

r n r n

nt nt

e Kw w e K w

n n

n

i h

n

r n r n

nt nt

K w e e Kw w w

n

n n

This formula is exact given the GramCharlier expansion Equation of the

prop osition results from i substitution of w d ii application of the

n

arbitrage condition

r r n

n nt n n

nt n n n

iii substitution using the identity

r n r n

n

nt

S e d Ke d

t n

and iv elimination of terms involving and which are extremely small for

n n

options of common maturities

Prop osition Consider the BlackScholes formula as a function of implied volatil

ity v A linear approximation around the p oint v is

n n n

r n r n

n

nt

C S e dv Ke dv v

nt t

r n r n r n

n

nt nt

S e d Ke d S e dv

t n n n t n

We equate this to the GramCharlier price equation apply and eliminate

terms of p ower two and higher in The result is equation

n

A Sto chastic Volatility

We derive moments of np erio d logchanges in the exchange rate for an arbitrary

sto chastic volatility mo del as describ ed in Section

For n unconditional moments and cumulants follow from the derivatives of

the moment generating function Taking exp ectations with resp ect to rst

s x E exps sz

t t

t

s s s

z z z E exp s

t z t

t t

where E denotes the exp ectation with resp ect to z The moments of x come from

z

derivatives evaluated at s They imply cumulants see Section

x

t

x E z

x E z

x Var z E z

The standard indicators of skewness and kurtosis are

E z

x

E z

E z Var z

x

E z E z

as stated in equation

Conditional higher moments follow from similar metho ds If fz g and f g are

t t

indep endent as we assumed the moment generating function is

n

X

n

s x E exps z

tj tj

tj

j

n

X

s s s

E exp s z z z

z t tj

tj

tj

j

where E now denotes the conditional exp ectation with resp ect to the z s Condi

z

tional cumulants are

n

X

n

x

tj n

j

n

X

n

x E z

t tj

j

n

X

n

x E z

t

tj

j

n n

X X

n

z z E x Var

tj t t

tj

j j

This leads to equation

A Digital Options

The GramCharlier expansion leads to relatively simple expressions for digital op

tions In the riskneutral framework of Section the price is

Z

r n

nt

D e f xdx

nt

logKS

t

n

If x is normal the BlackScholes case the price is

r n

nt

D e d

nt n

For the GramCharlier expansion the price is

r n

nt

D e d

nt n

n n

r n

nt

d d d e d

n n n n

If we use to compute implied volatility the analog to Prop osition is

n n

v d d d d

n n n n n

Note that the skewness and kurtosis terms arent multiplied by as they are in

n

equation For that reason they can have larger eects

References

Abken Peter Dilip Madan and Sailesh Ramamurtie Estimation of risk

neutral and statistical densities by hermite p olynomial approximation With

an application to euro dollar futures options manuscript Federal Reserve

Bank of Atlanta June

AitSahalia Yacine and Andrew Lo Nonparametric estimation of state

price densities implicit in nancial asset prices manuscript Universityof

Chicago June

AkgirayVedat and Georey Bo oth Mixed diusionjump pro cess mo deling

of exchange rate movements Review of Economics and Statistics

Bates David a Jumps and sto chastic volatility Exchange rate pro cesses

implicit in Deutsche mark options Review of Financial Studies

Bates David b Dollar jump fears Distributional abnormalities

implicit in currency futures data Journal of International Money and Fi

nance

Baillie Richard and Tim Bollerslev The message in daily exchange rates

A conditional variance tale Journal of Business and Economic Statistics

Black Fischer and Myron Scholes The pricing of options and corp orate

liabilities Journal of Political Economy

Black Fischer Fact and fantasy in the use of options Financial Analysts

Journal JulyAugust

Bo durtha James and Georges Courtadon The Pricing of Foreign Currency

Options Monograph Series in and Economics New York

Salomon Center

Brenner Menachem and YoungHo Eom Noarbitrage option pricing New

ork evidence on the validity of the martingale prop erty manuscript New Y

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Campa Jose and Kevin Chang Testing the exp ectations hyp othesis on the

term structure of volatilities Journal of Finance

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Campa Jose Kevin Chang and Rob ert Reider Implied exchange rate dis

tributions Evidence from OTC option markets manuscript New York Uni

versity August

Das Sanjiv and Rangara jan Sundaram Taming the skew Higherorder

moments in mo deling asset price pro cesses in nance NBER Working Pap er

No

Dieb old Francis Empirical Modeling of Exchange Rate Dynamics NewYork

SpringerVerlag

Drost Feike Theo Nijman and Bas Werker Estimation and testing in mo d

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Business and Economic Statistics

Garman Mark and Steven Kohlhagen Foreign currency option values

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Economic Review

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Kim Sang jo on and Neil Shephard Sto chastic volatility Optimal likelihood

inference and comparison with ARCH mo dels manuscript Nueld College

Oxford March

Knight John and Stephen Satchell Pricing derivatives written on assets with

arbitrary skewness and kurtosis manuscript Trinity College Cambridge

November

Kolassa John Series Approximation Methods in Statistics New York Springer

Verlag

Longsta Francis Option pricing and the martingale restriction Review of

Financial Studies

Madan Dilip and Frank Milne Contingent claims valued and hedged by

pricing and investing in a basis

Malz Allan Optionbased estimates of the probability distribution of ex

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Melino Angelo and Stuart Turnbull Pricing foreign currency options with

sto chastic volatility Journal of Econometrics

Merton Rob ert Option pricing when underlying sto ck prices are discontin

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Rosenb erg Joshua Pricing multivariate contingent claims using estimated

riskneutral density functions manuscript New York UniversityNovem

ber

Rubinstein Mark The valuation of uncertain income streams and the pricing

of options Bel l Journal of Economics

Taylor Stephen and Xinzhong Xu The magnitude of implied volatility

smiles Theory and empirical evidence for exchange rates Review of Futures

Markets

Zhu Yingzi Three Essays in Mathematical Finance Do ctoral dissertation

ork UniversityMay submitted to the Department of Mathematics New Y

Table

Prop erties of Daily Exchange Rate Changes

Canadian German Japanese

Statistic Dollar Mark Yen

Mean Annualized

Std Deviation Annualized

Skewness

Kurtosis

Statistics p ertain to daily depreciation rates x log S log S computed from

t t t

dollar prices S of other currencies The data are from Datastream and cover business

days b etween January and May observations Means and

standard deviations have b een multiplied by and resp ectivelyto

convert them to annual units where is the average numb er of business days

in the years

Table

Prop erties of AttheMoney Volatilities

Canadian German Japanese

StatisticMaturity Dollar Mark Yen

Mean

Median

Standard Deviation

Auto correlation

Volatilities are quotes from a large international bank expressed as annual p er

centages January to January daily observations See Zhu

Table

GramCharlier Approximations to JumpDiusions

Parameters Estimates

Example

A Benchmark

B Jump Intensity

Low

High

C Volatility

Low

High

D Maturity

Three Months

Six Months

E Kurtosis

Low

High

F Skewness

Low

Medium

High

The table rep orts parameters of jumpdiusion mo dels annualized standard devia

tions and of nonjump and jump comp onents and estimates of skewness and

kurtosis and implied by GramCharlier approximations to them based on

equation Errors in the approximation are indicated by dierences b etween es

timates of skewness and kurtosis and their values in the corresp onding jump mo del

For the b enchmark case annualized volatility annualized

jump intensity interest rates are zero and the maturity of the option is one

month Exceptions are noted in the rst column

Table

Conditional Moments Implied byVolatility Smiles

Maturity Standard Deviation Skewness Kurtosis

Overnight

Month

Months

Months

Months

Months

Months

Months

Entries are moments inferred from the volatility smiles in Figure as outlined in

Section The standard deviation is rep orted in annual units Skewness and

n

kurtosis are the standard measures and

n n

Figure

An Example of a Volatility Smile

10

9.8

9.6

9.4

9.2

9

8.8 Implied Volatility (Percent Per Year) 8.6

8.4

8.2 −1.5 −1 −0.5 0 0.5 1 1.5

Moneyness d

The line plots implied volatility against moneyness for onemonth overthecounter

Deutschemark options on April Moneyness d is dened by equation

Figure

Median Implied Volatilityby Maturity

1

0.98

0.96

0.94

0.92

0.9

0.88

0.86

Median Implied Volatility (12−Month=1) 0.84

0.82

0.8 0 2 4 6 8 10 12

Maturity in Months

Lines represent median implied volatility quotes for atthemoney options for the

Canadian dollar dotted line German mark dashed line and Japanese yen dash

dotted line The lines are scaled to aid comparison For each currency median

volatility has b een divided by its value at months

Figure

BlackScholes and GramCharlier Call Prices

5

4.5

4

3.5

3

2.5

2

1.5

Ratio of Call Price to Underlying (Percent) 1

0.5

0 −1.5 −1 −0.5 0 0.5 1 1.5

Moneyness d

Each line represents the relation b etween the call price and moneyness d for an

underlying distribution of logchanges in the sp ot exchange rate The solid line is the

BlackScholes formula equation The dashed and dashdotted lines incorp orate

resp ectivelyskewness and kurtosis into the GramCharlier call

n n

price formula equation In each case wehave set n one month

annually and r r

n nt

nt

Figure

Exact and Approximate Volatility Smiles for GramCharlier Call Prices

0.12

0.115

0.11

0.105 Implied Volatility v 0.1

0.095

0.09 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2

Moneyness d

Each line represents the relation b etween annualized volatility and moneyness d of

an option The solid horizontal line represents the standard deviation whichis

n

constant The dashed parab ola represents approximate implied volatilities from the

GramCharlier formula in equation of Prop osition The solid curve represents

exact implied volatilities based on the GramCharlier call price formula equation

Both use n one month

n n n

annually and r r

nt

nt

Figure

Eects of Skewness and Kurtosis on GramCharlier Call Prices

0.12

0.115

0.11

0.105 Implied Volatility v 0.1

0.095

0.09 −1.5 −1 −0.5 0 0.5 1 1.5

Moneyness d

The lines graph approximate annualized implied volatility against moneyness d for

dierentvalues of higher moments in the distribution of logchanges in the sp ot

exchange rate The horizontal solid line represents BlackScholes The smile curved

solid line has p ositive kurtosis but no skewness The skew

n n

dashed line has negativeskewness and p ositive kurtosis

n n

Both are based on the approximation In each case n one month

annually and r r

n nt

nt

Figure

Deltas for BlackScholes and GramCharlier Call Prices

1

0.9

0.8

0.7

0.6

0.5 Delta

0.4

0.3

0.2

0.1

0 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2

Moneyness d

Each line represents the relation b etween the and moneyness d for an underlying

distribution of logchanges in the sp ot exchange rate The solid line is BlackScholes

dd The dashed and dashdotted lines incorp orate resp ectivelyskewness

and kurtosis into the GramCharlier equation Other

n n

parameter values are the same as Figure

Figure

GramCharlier Volatility Smiles for Two Maturities

0.12

0.115

0.11

0.105 Implied Volatility v 0.1

0.095

0.09 −1.5 −1 −0.5 0 0.5 1 1.5

Moneyness d

The lines representvolatility smiles for two dierent maturities The horizontal

solid line represents BlackScholes The curved lines have p ositive kurtosis

n

n for n one month solid line and n

three months dashed line Both are based on the approximation Other

parameters are annually and r r

n nt

nt

Figure

Kurtosis in Currency Prices Over Dierent Time Horizons

1

0.8

0.6

0.4

Kurtosis Relative to n=1 0.2

0

−0.2 0 5 10 15 20 25 30 35 40 45 50

Time Interval n in Days

Lines represent estimates of kurtosis for time intervals of n days relativeto

n

n The solid line is n and serves as a theoretical b enchmark The other lines

are estimates for the Canadian dollar dotted line German mark dashed line and

Japanese yen dashdotted line all for the same time p erio d covered byTable

Figure

Volatility Smiles for Deutschemark Options on April

11.5

12 Months 11

10.5 6 Months

10

3 Months 9.5

9 1 Month

Implied Volatility (Percent Per Year) 8.5

8

Overnight 7.5 −1.5 −1 −0.5 0 0.5 1 1.5

Moneyness d

Lines describ e volatility smiles for overthecounter Deutschemark options on April

The maturities of the options are noted in the gure

Figure

Conditional Kurtosis with Sto chastic Volatility

3

2.5

2

1.5 Conditional Kurtosis 1

0.5

0 0 2 4 6 8 10 12 14 16 18

Maturity in Months

The gure describ es conditional kurtosis v maturity in the data and in the sto chas

tic volatility mo del of Section Asterisks represent the data the values of im

plied kurtosis rep orted in Table The solid line is the sto chastic volatility mo del

with nonnormal innovations The dashed line represents the same mo del without

sto chastic volatility essentially a pure jump mo del The dashdotted line

represents the mo del with normal innovations a gaussian sto chastic

volatility mo del