valuing options on baskets of stocks and forecasting the...

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QSt

of Stocks andolatility Skews

Group

Co.

04

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uantitativerategies

Valuing Options on BasketsForecasting the Shape of V

Joe Z. Zou

Quantitative Strategies

Goldman Sachs &

New York, NY 100

al uncertainty.

one-parameter

lity distributions.

nded power utility

s justified by

spread.

ark to market.

of 44

QuantitativeStrategies

Outline

• Extract risk-neutral distribution under maxim

• Estimating risk-neutral distributions from an

family of distance measures between probabi

• Deriving risk-neutral distributions using exte

functions.

• Is the implied volatility skews of index option

historical data?

• Ranking equity options using strike-adjusted

• Valuing options on basket of stocks.

• Forecast the shape of smile and end-of-day m

• Summary

References:

proach toce, 51, 1996.

latility Skewtegies Research

justed Spread: AEquity Options”.Research Notes,

rlo Valuation of Pathtility Smile”. J. of

of 44


M. Stutzer, “A Simple Nonparameteric ApDerivative Security Valuation” J. of Finan

Emanuel Derman and Joe Zou, “Is the VoFair?” Goldman Sachs Quantitative StraNotes, 1997.

Joe Zou and Emanuel Derman, “Strike-adNew Metric For Estimating The Value of Goldman Sachs Quantitative Strategies

1999.

Joe Zou and Emanuel Derman, “Monte CaDependent Options On Indexes with a VolaFinancial Engineering, V6, 1997.

o

QuaStr

&P 500 options. (a) Pre-

f 44

ntitativeategies

Motivation

Representative implied volatility skews of Scrash. (b) Post-crash.

0.95 0.975 1 1.025 1.05

Strike/Index

14

16

18

20

Vo

latil

ity

0.95 0.975 1 1.025 1.05

Strike/Index

14

16

18

20

Vo

latil

ity

(a)

(b)

from realized

is an investor tovides the best

se to gauge their

ace for an illiquid

% out-of-the-money putsket of bank stocks. Thetocks: Bank One, Chase,

C

curve when one ofnged?

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Can the volatility skew be extracted historical returns?

For a given stock or stock index, howknow which strike and expiration provalue?

What metric can options investors uestimated excess return?

What is the appropriate volatility surfbasket?Suppose an investor is interested in buying a 10and selling a 10% out-of-the-money call on a babasket consists of equal number of shares of 5 sJP Morgan, Wells Fargo and Bank of America.

Bank Basket = ONE + CMB + JPM + WFC + BA

Can we forecast the shape of a skewthe options’ implied volatility has cha

Distribution

ution and the correspond-

turn (%)0 20

ST( ) STd

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Option Prices Implied Distribution

Implied Volatility Skew Skewed

(S&P 500 index option implied three-month distribing implied volatility skew as of 3/10/99)

••••

••

••

••

••

••

••

••

•

••

••

Strike Level

Impl

ied

Vola

tility

(%)

1000 1100 1200 1300 1400 1500

2025

3035

Index Re

Prob

abilit

y (%

)

-40 -20

02

46

CK T, S ΣK T,,( ) e r T t–( )– Max ST K 0,–( )Q∫=

series.

return histogramion of the stock

nt in thejth bin byod estimate of the

of 44


Empirical Distribution

• Step 1: get realized stock price time

• Step 2: estimate the rolling T-periodand calculate the empirical distributreturns

If R falls in thejth bin, increase the couone. The final count in the bin is a goprobabilityP(Rj).

Rt

St

St T–

---------- log=

turn bins.

j

tribution

of 44

QuantitativeStrategies R = log(

St

St-T

)

T-period Returns

Probability ∝ "bean counts" in the re

∑ P(j) = 1

Histogram and empirical dis

o

QuaStr

l Distribution

tion to estimate the

f 44

ntitativeategies

S&P 500 Three-Month Empirica

Can we use the empirical distriburisk-neutral distribution?

vements,

ulls) and

tion!

be far less certainmovement.

=50%

-p=50%

of 44


Uncertainty and Market Equilibrium

Consider two binomial distributions:

• X is far more predictable than Y,

• X is not stable (more bulls than bears),

• Y is most uncertain about the future market mo

• Y is more stable (equal number of bears and b

• Y is likely to be the market equilibrium distribu

The equilibrium distribution tends toabout the direction of future market

p=99%

1-p=1%

p

1

X Y

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Information and Probability

• Probability measures the uncertainty about a single random event

• Entropy measures the uncertainty of a collection of random events.

Consider a stock whose next move may be up or down:

Information conveyed by an up move:

Information conveyed by a down move:

If p=1, then an up-move conveys no information at all!

If everyone expected the stock to go up, and it actuallymoves down, the outcome is more informative.

I up( ) plog–=

I down( ) 1 p–( )log–=

stock

up probability p

down probability 1-p

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Entropy and Probability Distribution

It is the expected amount of information of all possibleoutcomes

This entropy is maximized ifp=50%:

The risk-neutral distribution we are seeking containsmore states than the simple up and down states and thusis more complicated than the example. But the basicidea is similar.

S p( ) p p 1 p–( ) 1 p–( )log+log[ ]–=

Maximum Entropy Maximum Uncertainty⇔

Extract Risk-Neutral Distribution by

ing the forward

d condition:

d will change thest prejudicialition.

Rj( )Q Rj( )P Rj( )--------------

log

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Minimizing Relative Entropy

Maintain maximum uncertainty while satisfycondition.

Minimize S(P,Q) subject to the forwar

The minimum relative entropy methoshape of the prior distribution in theleaway so as to satisfy the forward cond

Relative-Entropy Function:S P Q,( ) Qj

∑=

S0er S0e

Rj

j∑ Q Rj( )=

lem is:

y solving the forward con-

λ– ST( )xp

rT

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The solution to the minimization prob

where the constant can be found numerically bstraint

whereP is a given prior distribution.

Qλ S0 0 ST T,;,( )P S0 0 S;, T T,( )

P S( ) λ– S( )exp Sd∫---------------------------------------------- e=

λ

Qλ S0 0 ST T,;,( )ST STd∫ S0e=

QuaStr

stanceDistributions.

tric Theory of Learningper, 1995.

with parameter

its as and

pδq1 δ––)-------------------- sd

δ 0→ δ 1→

pq---

log sd

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ntitativeategies

An One-parameter Family of DiMeasures Between Probability

Reference: Huaiyu Zhu, “Bayesian GeomeAlgorithms”, Santa Fe Institute Working pa

Consider the followingInformation Deviation

The deviation and are defined as lim

δ 0 1,( )∈

Sδ P Q,( ) δp 1 δ–( )q+δ 1 δ–(

----------------------------------∫=

S0 S1

S1 P Q,( ) S0 Q P,( ) p∫= =

QuaStr

as the properties of

utral constraint yields

Q

Q)P)

c1 δ( )s] 1 δ⁄–

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ntitativeategies

It’s straightforward to show that hsquare distances:

Minimizing , subject to the risk-ne

Sδ P Q,( )

Sδ P Q,( ) 0≥Sδ P Q,( ) 0 P ≡⇔≡

Sδ aP aQ,( ) aSδ P,(=

Sδ P Q,( ) S1 δ– Q,(=

Sδ P Q,( )

Qδ s s0( ) P s s0( ) c0 δ( ) +[=

QuaStr

lve the following

ral distribution using ann a class of extended power

corresponding to thes to the log-utility. It canined using the utilityth .

s 1=

s0er=

1 γ–

1–

γ 1 δ⁄=

of 44

ntitativeategies

where the constants and for a givenδ soconstraints:

In the next section, we derive the risk-neutderivatives asset allocation model based outility functions: (See Robert C. Merton)

where , , and if (exponential utility). The limit as leadbe shown that the risk-neutral density obtafunction with exponentγ, is the same asQδ wi

c0 c1

P s s0( ) c0 c1s+[ ] 1 δ⁄– d∫sP s|s0( ) c0 c1s+[ ] 1 δ⁄– sd∫

U W( ) γ1 γ–----------- a

bγ---W+

=

γ 0≠ b 0> a 1= γ ∞→γ 1→

Derivatives Asset Allocation and Risk-Neutral Power Utility

through a conditional

er E has a price

tor, and

o riskless bonds assets.at is invested in.

lth will be

)

Q S t E T,;,( )

of 44


Distributions Based on ExtendedFunctions

Consider an economy in equilibrium:• A representative investor with initial wealthW0• The investor has a market view expressed

density

• An Arrow-Debreu security with parametgiven by

where is the discount fac

is the risk-neutral density.• Let α be the portion of wealth allocated t

and (1-α) be the portion allocated to risky• Let be the portion of the (1-α) th

Arrow-Debreu security with parameter E• At the end of periodT, the total investor wea

P S0 0 ST T,;,( )

π S t E T,;,( ) DQ S t E T,;,(=

D 1 1 r f+( )⁄=

ω E( )dE

Pag

, the supply andurity will also risk-neutral densitye. Therefore, to

ust solve the asset

)r E ST( ) Ed ]

δ ST E–( )Q S t E T,;,( )

------------------------------- 1–

f

e 19 of 44


where

and

• If the allocation is changeddemand for the Arrow-Debreu secchange and thus the shape of thefunction will also changachieve a market equilibrium, we m

WT ST( ) W0 1 αr f 1 α–( ) ω E(∫+ +[=

r E ST( )π ST T E T,;,( ) π St t E T,;,( )–

π St t E T,;,( )---------------------------------------------------------------------≡

D---=

Er E( )Q St t E T,;,( )d∫ r=

ω E( )dE

Q St t E T,;,( )

QuaStr

ing the expected

E= 1d

E= 1d

S0 1 r f+( )

of 44

ntitativeategies

allocation problem by maximizutility :

subject to

U WT( )

Max Ep U WT( )[ ]{ }α ω E( ) Q E( ),,{ }

budget constraint: ω E( )∫normalization: Q E( )∫

forward constraint: Q E( )E E=d∫

QuaStr

problem

S0 t ST T,;,( )

α) ω E( )DQ E( )------------------

)r f

----------

of 44

ntitativeategies

Solving the optimization

where

and

Q S0 t S;, T T,( )U' WT ST( )( )EP U' WT( )[ ]------------------------------P=

WT E( ) W0 α 1 r f+( ) 1 –(+=

EP U' WT( )[ ] λ1

W0 1 α–(--------------------=

QuaStr

l utility:

λ1

1 α– )---------------

E

r f

f

--------

th b 0>

c0 c1E–( )p

of 44

ntitativeategies

We now specialize in exponentia

The final results:

EP U' WT( )r E ST( )[ ]W0(----------=

ω E( )Q E( )-------------

r f

1 r f+--------------

λ2

λ1

-----λ3

λ1

-----+=

λ2

λ1

-----λ3

λ1

-----S0 1 r f+( )+1 +

r------=

U WT( ) bWT–( ) wiexp–=

Q St t E T,;,( ) P St t E T,;,( )ex=

be determinedrd price con-

parameter, b,depend on thestraints. Thethe represen-

It is essentialn be indepen-n!

T

of 44


where the constantc0 and c1 are toby the normalization and forwastraints

and they are independent of theof the utility function. They onlyprior distribution P, and the conparameter, b, is characteristic oftative investor’s risk aversion.that the risk-neutral distributiodent of the investor’s risk aversio

Q S0 0 ST T,;,( )ST STd∫ S0er=

Q S0 0 ST T,;,( ) STd∫ 1=

r’s allocation ofmeter E, ω(E),

his solution tot the risk-neu-

is ofimum relativer the extendedse the general-roper choice of

s above).

Q St t ST T,;,( )

of 44


Finally, the representative investoArrow-Debreu security with paracan be calculated using c0 and c1.

The most important feature of tthe asset allocation problem is thatral probability density functionthe same form given by the minentropy approach. This is true fopower utility, provided that we uized relative entropy with the pthe parameter

δ=1/γ (the proof follows the same a

of 44


Is the Volatility Skews of Index OptionsJustified by Historical Data?

• Justified: Fair risk-neutral expected value using theempirical distribution as a prior.

mean historical returnriskless return

empirical distribution

of 44


Calculate the expectation of option’s payoff atexpiration

Discounting the expectation, and extracting theimplied volatility from the Black-Scholes formula.

Repeating this for all strikes and maturities, weextract an implied volatility surface from thehistorical return distribution.

ET CK S[ ] max SeRj K 0,–( )Q Rj( )

j∑=

e rT– ET CK S[ ] BS S K T r dΣK T,, , , , ,( )=

SPX Pre-crash and Post-crash Distributions

dex Return (%)0 20

3% 7.8%

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Index Return (%)

Pro

babi

lity

(%)

-20 0 20

01

23

4

In

Pro

babi

lity

(%)

-20

01

23

45

6

mean =1.8%std. div. = 7.3%

mean = 3.std. div. =

QuaStr

/18/98)

of 44

ntitativeategies

FTSE “Fair” Skew (9/30/98 - 12

00 index and DAX,ope of the skews

n vary dramatically over

98 for three month options.

elta put-25 delta call)

air” Spread5p-25c)

.0%

.5%

.0%

of 44


Examples

• Applications of our model to S&P 5and FTSE-100 index show thatthe slare approximately fair!

Note: The absolute levels of implied volatility catime, the slope of the skew is relatively stable.

* three-year data from October 95 to September

Size of skew: compare actual data with model results. (25 d

Index Normal Spread*(25p -25c)

Recent Spread(25p-25c)

“F(2

SPX 4-7% 14% 6

DAX 3-6% 10% 3

FTSE 2-6%. 10% 4

s.

riod, we obtain theis empirical returnf the stock.

a statistical prior totral distribution bydifference betweenrisk-neutral distri-

d price of the stock.ed in this way the

or RNHD.

cted values of stan-d convert these val-

e denote the Black-e price is computedated fair option vol-

of 44


Strike-Adjusted Spread

The SAS of a stock option is calculated as follow

1.First, choosing some historically relevant pedistribution of stock returns over time T. Thdistribution characterizes the past behavior o

2.We use the empirical return distribution asprovide us with an estimate of the risk-neuminimizing the entropy associated with thethe distributions, subject to ensuring that thebution is consistent with the current forwarWe call this risk-neutral distribution obtainrisk-neutralized historical distribution,

3.We then use the RNHD to calculate the expedard options of all strikes for expiration T, anues to Black-Scholes implied volatilities. WScholes implied volatility of an option whosfrom this distribution as . This is our estimatility.

ΣH

QuaStr

ation T, whose mar-the strike-adjusted

ichness of the

S for which the risk-ther constrained to

t-the-money options.distribution the at-d historical distri-justed spread com-

, is a

t strikes, assumingd implied volatility

K T,( )

SASATM K T,( )

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ntitativeategies

4.For an option with strike K and expirket implied volatility is ,spread in volatility is defined as

This spread is a measure of the current roption based on historical returns.

5. We often use a modified version of SAneutralized historical distribution is furreproduce the current market value of aWe call this (additionally constrained)the-money adjusted, risk-neutralizebution, or RNHDATM. The strike-ad

puted using this distribution, denoted

measure of the relative value of differenthat, by definition, at-the-money-forwaris fair.

Σ K T,( )

SAS K T,( ) Σ K T,( ) ΣH–=

QuaStr

Optionsine which strikes

dards.

0

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ntitativeategies

Use to Rank Equityon the same underlyer, in order to determprovide the best value by historical stan

SASATM K T,( )|K SF= =

SASATM K T,( )

QuaStr

Strike Level1250 1300 1350 1400 1450

500 index options on May 18,

ber 17, 1999. Both fair andd to match at the money,turns from May 1987 to May

(b)

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ntitativeategies

Strike Level

Vo

latil

ity (

%)

1200 1250 1300 1350 1400 1450

22

24

26

28

30

32

"fair skew" market skew

SA

S (

%)

1200

-0.5

0.0

0.5

1.0

FIGURE 1. (a) Fair and market skews for S&P1999. (b) SASATM for the same options.The options considered expire on Septemmarket implied volatilities are constraineforward. The RNHD is constructed using re1999, including the 1987 crash.

(a)

QuaStr

trike Level300 1350 1400 1450

ptions on May 18,

th fair and marketney, forward. Theay 1999, thereby

of 44

ntitativeategies

Strike Level

Vol

atili

ty (

%)

1200 1250 1300 1350 1400 1450

2426

2830


SS

AS

(%

)

1200 1250 1

02

46

FIGURE 2. (a) Fair and market skews for S&P 500 index o1999. (b) SASATM for the same options.The options considered expire on September 17, 1999. Boimplied volatilities are constrained to match at the moRNHD is constructed using returns from May 1988 to Mexcluding the 1987 crash.

(a) (b)

QuaStr

Strike Level1300 1350 1400 1450

Strike Level1300 1350 1400 1450

r September 17, 1999o the crash-inclusivespond to the crash-

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ntitativeategies

Strike Level

Volat

ility (%

)

1200 1250 1300 1350 1400 1450

1820

2224

2628

30


SAS (

%)

1200 1250

-2-1

01

Strike Level

Volat

ility (%

)

1200 1250 1300 1350 1400 1450

2022

2426

28 "fair skew" market skew

SAS (

%)

1200 12500

12

34

5

FIGURE 3. Re-evaluated SASATM on June 21, 1999 foS&P 500 options. The top two figures correspond tdistributions of Figure 1.; the bottom two correexclusive distributions of Figure 2..

Pag

d is the correlation

log-normal process asoes the basket.

ith the market volatility.

s of the basket.

ρij σiσ j

e 36 of 44


Valuing Options on Basket of Stocks

• Estimate Basket Volatility - the old way

where, is the weight of stock i in the basket, an

between stock i and stock j.

Problems:

• Component stocks usually do not follow thethe implied volatility skews show. Neither d

• The correlations between stocks can vary w

• No obvious way of estimating volatility skew

σB2 wi

2σi2 2 wiwj

i j<∑

j∑+

i∑=

wi ρij

Correlations and the Market Volatility

be linked to the market

d is the market

latility.

σm2 ε2

2+----------------

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Consider two stocks whose returns mayreturn via CAPM:

where are the “tracking errors”, an

return. It’s easy to see the correlation

increases as a function of the market vo

r 1 β1r m ε1+=

r 2 β2r m ε2+=

ε1 ε2,( ) rm

ρ r 1 r 2,( ) β1β2σm2

β12σm

2 ε12+ β2

2--------------------------------------=

QuaStr

w way

opean options on a

bility distributionest rate.

ted bypy using theasket as a prior.

ff at T |S t, ]

of 44

ntitativeategies

Basket Volatility Skews - the ne

We are interested in valuing Eurbasket whose spot price isS.

whereQ is the risk-neutral probawhose mean is the riskless inter

The distribution Q is calculaminimizing the relative entroempirical distribution of the b

CK T, S t,( ) e r T t–( )– EQ Payo[=

QuaStr

the beginning, the0% OTM call andol. points!

d skew of BKX by our model. Therket is almostr model, even

olatility is off by

of 44

ntitativeategies

Basket Option Examples

• For the bank basket shown at volatility spread between the 1the 10% OTM put is almost 7 v

• We compare the market implieindex with the skew calculatedsize of skew seen from the maidentical to that predicted by outhough the actual level of the vroughly 5 points.

QuaStr

••

110 120

for thefrom the risk-e 1987 to June

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ntitativeategies

•

•

•

Strike Level

Vol

atili

ty (%

)

80 90 100

2022

2426

2830

32

The estimated fair three-month implied volatility skew basket of five bank stocks listed in the text, estimated neutralized historical distribution using returns from Jun1999

A Change in A

ns on a skew curveded away?esence of stale

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Forecast the Shape of the Skew FromSingle Option Price

• How to adjust quotes for other optiogiven that one of the options has tra

• End-of-day mark to market in the proption prices.

tion, is the

ew option price for

T S0 0, ) STd

Q̃

T λ2 f j ST( )– ]f j S( )] Sd

-------------------------------

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whereQ is the original implied distribu

forecasted distribution, and is the nstrikeKj.

where .

Min S Q Q̃,( ) EQ̃

Q̃ S( )Q S( )------------

log=

S. T. Q̃ ST( )ST STd∫ S0er f T

=

C̃ K j T,( ) er f T–

Max ST K j 0,–[ ]Q̃ ST,(∫=

C̃

Q̃ ST T S0 0,;,( )Q ST T S;, 0 0,( ) λ1– S[exp

Q S( ) λ– 1S λ2–[exp∫------------------------------------------------------------=

f j S( ) Max S Kj 0,–( )=

ity skewscurve has changed

•

120

••

•

120

•

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Example: Updated distribution and volatilafter one volatility on the skew

•••••••••••••••••••••••••••••••••••••••••••••••••••

••••

••••••••••••••••

••••••••••••

••••••••••••••••

••••••••

•

•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••

Index LevelP

roba

bilit

y (%

)

50 100 150

0.0

0.5

1.0

1.5

2.0

2.5

•

•

•

•

Strike Level

Impl

ied

Vol

atili

ty (%

)

80 90 100 110

2025

30

•

•

•

•

X

X

(a) (b)

••

••

•

Imp

lied

Vo

latil

ity (

%)

15

20

25

30

••••

•

••

••

•

Strike Level

80 90 100 110 120

15

20

25

30

••

••

•

••

••

••

••

••

••

80 90 100 110

••

••

of 44


Summary:

• For the pre-crash period, our method produces noappreciable skew. For the post-crash period, the modelproduces significant skew that is comparable with theobserved market data.

• Strike-adjusted spread as a gauge of the relative richnessof equity options.

• Particularly useful for valuing OTC options on singlestock or on a basket of stocks.

• The method may be used to forecast the change of smilewhen some of the options have traded away.

• It may be helpful for volatility traders to mark to marketat the end of trading days with stale option prices.

• An equilibrium asset allocation model of Arrow-Debreusecurities with a class of utility functions yields the samerisk-neutraldistributions as those obtained byminimizing a class of generalized relative entropyfunction.

valuing options on baskets of stocks and forecasting the...

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