valuing options on baskets of stocks and forecasting the shape of
TRANSCRIPT
Page
QSt
of Stocks andolatility Skews
Group
Co.
04
1 of 44
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.
Page 2 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
Page 3 of 44
QuantitativeStrategies
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.
Page 4 o
QuaStr
&P 500 options. (a) Pre-
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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?
Page 5 of 44
QuantitativeStrategies
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
Page 6 of 44
QuantitativeStrategies
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
Page 7 of 44
QuantitativeStrategies
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
Page 8 of 44
QuantitativeStrategies R = log(
St
St-T
)
T-period Returns
Probability ∝ "bean counts" in the re
∑ P(j) = 1
Histogram and empirical dis
Page 9 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%
Page 10 of 44
QuantitativeStrategies
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
Page 11 of 44
QuantitativeStrategies
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
Page 12 of 44
QuantitativeStrategies
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
Page 13 of 44
QuantitativeStrategies
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
Page 14 of 44
QuantitativeStrategies
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=
Page 15
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∫= =
Page 16
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( ) +[=
Page 17
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⁄=
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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,;,( )
Page 18 of 44
QuantitativeStrategies
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
QuantitativeStrategies
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,;,( )
Page 20
QuaStr
ing the expected
E= 1d
E= 1d
S0 1 r f+( )
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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∫
Page 21
QuaStr
problem
S0 t ST T,;,( )
α) ω E( )DQ E( )------------------
)r f
----------
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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–(--------------------=
Page 22
QuaStr
l utility:
λ1
1 α– )---------------
E
r f
f
--------
th b 0>
c0 c1E–( )p
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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
Page 23 of 44
QuantitativeStrategies
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,;,( )
Page 24 of 44
QuantitativeStrategies
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
Page 25 of 44
QuantitativeStrategies
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
Page 26 of 44
QuantitativeStrategies
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%
Page 27 of 44
QuantitativeStrategies
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. =
Page 28
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%
Page 29 of 44
QuantitativeStrategies
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-
Page 30 of 44
QuantitativeStrategies
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
Page 31
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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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–=
Page 32
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,( )
Page 33
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)
Page 34
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
"fair skew" market skew
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)
Page 35
QuaStr
Strike Level1300 1350 1400 1450
Strike Level1300 1350 1400 1450
r September 17, 1999o the crash-inclusivespond to the crash-
of 44
ntitativeategies
Strike Level
Volat
ility (%
)
1200 1250 1300 1350 1400 1450
1820
2224
2628
30
"fair skew" market skew
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
QuantitativeStrategies
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+----------------
Page 37 of 44
QuantitativeStrategies
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--------------------------------------=
Page 38
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[=
Page 39
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.
Page 40
QuaStr
••
110 120
for thefrom the risk-e 1987 to June
of 44
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
Page 41 of 44
QuantitativeStrategies
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
-------------------------------
Page 42 of 44
QuantitativeStrategies
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
•
Page 43 of 44
QuantitativeStrategies
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
••
••
Page 44 of 44
QuantitativeStrategies
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.