für Angewandte Analysis und Stochastik
imForschungsverbund Berline.V.
Preprint ISSN 0946 8633
Ecient Computation of Option Price Sensitivities
Using Homogeneity and other Tricks
Oliver Reiÿ 1
, Uwe Wystup 2
submitted: May24th2000
1
Weierstrass-Institutefor
AppliedAnalysisandStochastics
Mohrenstraÿe39
D -10117Berlin
Germany
E-Mail: reiss@wias-berlin.de
URL:http://www.wias-berlin.de/reiss 2
Commerzbank
TreasuryandFinancialProducts
NeueMainzerStraÿe32-36
D-60261FrankfurtamMain
Germany
E-Mail: wystup@mathnance.de
URL:http://www.mathfinance.de
Preprint No. 584
Berlin2000
WIAS
2000 MathematicsSubjectClassication. 91-08,91B28.
Key words and phrases. Calculation of Greeks, Derivatives of option prices, Homogeneity
propertiesofnancialmarkets.
OliverReiÿ ispartiallyaliatedtoDelft University,bysupportofNWONetherlands.
WeierstraÿInstitut für Angewandte Analysisund Stochastik (WIAS)
Mohrenstraÿe 39
D 10117 Berlin
Germany
Fax: +49 302044975
E-Mail (X.400): c=de;a=d400-gw;p=WIAS-BERLIN;s=preprint
E-Mail (Internet): preprint@wias-berlin.de
World WideWeb: http://www.wias-berlin.de/
No front-oce software can survive without providing derivatives of op-
tions prices with respect to underlying market or model parameters, the so
called Greeks. We present a list of common Greeks and exploit homogene-
ityproperties ofnancialmarkets to derive relationshipsbetween Greeksout
of which many are model-independent. We apply the results to European
styleoptions,rainbowoptions,aswellasoptionspricedinHeston'sstochastic
volatility model and avoid exorbitant and time-consuming computations of
derivativeswhicheven strong symboliccalculators failto produce.
1 Introduction
The computation of sensitivities of option prices, the so-called Greeks, is often
cumbersome-both forthe mathematicianand for symboliccalculators. This paper
provides methods to avoid dierentiation as much as possible. Many Greeks are
related among each other. These relations are based on model-independenthomo-
geneity of time and price level of a nancial product on the one hand and model
dependent relationssuchasthe partialdierentialequation thevalue functionmust
satisfy and relations implied by the assumed distribution of the underlying. The
basic marketmodelwe use is the Black-Scholes modelwith stocks paying acontin-
uousdividendyieldandariskless cashbond. This modelsupportsthe homogeneity
properties which are valid in general, but its structure is so simple, that we can
concentrate on the essential statements of this paper. We will also discuss how to
extend our work tomore generalmarketmodels.
We list the commonly used Greeks and their symbols. We do not claim this list
tobe complete, because one can always dene more derivatives of the option price
function.
As special cases we look at the Greeks of European options in the Black-Scholes
modelin one dimension. It turns out, that one only needs to know two Greeks in
orderto calculateallthe otherGreeks withoutdierentiating.
Another interesting example is a European derivative security depending on two
assets. For such rainbow options the analysis of the risk due to changing correla-
tion of the two assets is very important. We will show how this risk is related to
simultaneous changes of the two underlyingsecurities.
Thereare several applications of these homogeneity relations.
2. It produces a robust implementation compared to Greeks via dierence quo-
tients.
3. It allows to check the quality and consistency of Greeks produced by nite-
dierence-, tree- orMonte Carlo methods.
4. Itadmits a computationof Greeks for Monte Carlo based values.
5. It shows relationships between Greeks which wouldn't be noticed merely by
lookingatdierence quotients.
1.1 Notation
S stock price or stock price process
B cash bond, usually with risk free interest rate r
r risk freeinterestrate
q dividend yield(continuously paid)
volatility of one stock, orvolatilitymatrix of several stocks
correlation inthe two-asset marketmodel
t date of evaluation(today)
T date of maturity
=T t time tomaturity ofan option
x stock price at time t
f() payo function
v(x;t;:::) value of an option
k strike of anoption
l level of anoption
v
x
partial derivation of v with respect tox (and analogous)
The standard normaldistribution and density functions are dened by
n(t)
= 1
p
2 e
1
2 t
2
(1)
N(x)
= Z
x
1
n(t)dt (2)
n
2
(x;y;)
=
1
2 p
1
2 exp
x 2
2xy+y 2
2(1 2
)
!
(3)
N
2
(x;y;)
= Z
x
1 Z
y
1 n
2
(u;v;)dudv (4)
See http://www.MathFinance.de/frontoce.html fora sourcecode to compute N
2 .
Delta v
x
Gamma v
xx
Theta v
t
Rho v
r
inthe one-stock model
Rhor
r
v
r
inthe two-stock model
Rhoq
q
v
q
Vega v
Kappa v
correlation sensitivity (two-stock model)
Greeks, not so commonly used:
Leverage
x
v v
x
sometimes , sometimes called gearing
Vomma
0
v
Speed v
xxx
Charm v
xt
Color v
xxt
Cross v
x
Forward Delta F
v
F
Driftless Delta dl
e q
Dual Theta Dual v
T
StrikeDelta k
v
k
StrikeGamma k
v
kk
Level Delta l
v
l
Level Gamma l
v
l l
Beta
12
1
2
two-stock model
2 Fundamental Properties
2.1 Homogeneity of Time
Inmostcasesthe priceoftheoptionisnotafunctionof boththecurrenttimetand
the maturity time T,but ratheronly afunction of the time to maturity =T t
implyingthe relations
=v
t
= v
= v
T
= Dual: (5)
Thisrelationshipextends naturallytothe situationof optionsdepending onseveral
intermediate timessuch as compound orBermuda options.
We present theprincipleof the scale-invarianceof time inthis section,becausethis
principle holds in general. In a market model parameters may be quoted on an
annual basis. We illustrate this idea in a Black-Scholes framework, in which the
volatility is such a modelparameter. The same idea can easily be applied toother
marketmodels.
Wemay wanttomeasure time inunitsotherthan years inwhichcase interestrates
and volatilities, which are normally quoted on an annual basis, must be changed
according tothe followingrules for all a>0.
!
a
r ! ar
q ! aq
!
p
a (6)
The option'svalue must be invariant underthis rescaling, i.e.,
v(x;;r;q;;:::)=v(x;
a
;ar;aq;
p
a;:::) (7)
We dierentiate this equationwith respect to a and obtain fora =1
0=+r+q
q +
1
2
; (8)
a general relation between the Greeks theta, rho, rhoq and vega. Based on the
relation
v(x
1
;:::;x
n
;;r;q
1
;:::;q
n
;
11
;:::;
nn )=
v(x
1
;:::;x
n
;
a
;ar;aq
1
;:::;aq
n
; p
a
11
;:::;
p
a
nn
) (9)
weobtain
Theorem 1 (scale invariance of time)
0=+r+ n
X
i=1 q
i
q
i +
1
2 n
X
i;j=1
ij
ij
; (10)
where
ij
denotes the dierentiation of v with respect to
ij .
2.3 Scale Invariance of Prices
Thegeneralidea isthatvalue ofsecurities maybemeasuredinadierentunit, just
like values of European stocks are now measured in Euro instead of in-currencies.
Option contracts usually depend on strikes and barrier levels. Rescaling can have
types of homogeneity classes. Letv(x;k) be the value function of an option, where
xis the spot (or a vector of spots) and k the strike orbarrier or a vector of strikes
orbarriers. Let a be apositivereal number.
Denition 1 (homogeneity classes) We calla value functionk-homogeneous of
degree n iffor all a>0
v(ax;ak) = a n
v(x;k): (11)
We callan options whose value function is strike-homogeneousof degree 1 a strike-
denedoption andsimilarly an optionwhose valuefunction islevel-homogeneousof
degree 0 a level-dened option.
The value function of a European call or put option with strike K is then K-
homogeneous of degree 1, a digital option which pays a xed amount if the stock
priceis higher than a level L isL-homogeneous of degree 0. The path-independent
barrier call option paying (S k) +
I
fS>Kg
is (k;K)-homogeneous of degree 1. A
power call with cap paying min(C;((S K) +
) 2
) has a homogeneity structure of
v(aS;aK;a 2
C)=a 2
v(S;K;C).
Weshowhowsuchascaleinvariancecanbeusedtodeterminesomerelationsamong
the Greeks. We explain this with two examples. In the rst example we analyze a
strike-denedoptionandinthe secondoneweconcentrateonaleveldened option.
The generalizationtooptions withsome more parameters likethe mentionedpath-
independent barrier call or power-call can easily be done. For the barrier call one
can use the results from the multi-dimensionalstrike-dened option (26) and (27).
2.3.1 Strike-Delta and Strike-Gamma
Fora strike-dened value function we havefor all a;b>0
abv(x;k) = v(abx;abk): (12)
We dierentiate with respect to a and get for a=1
bv(x;k) = bxv
x
(bx;bk)+bkv
k
(bx;bk): (13)
We nowdierentiate with respect tob get for b=1
v(x;k) = xv
x +xv
xx
x+xv
xk k+kv
k +kv
kx x+kv
kk
k (14)
= x+x 2
+2xkv
xk +k
k
+k 2 k
: (15)
Ifwe evaluate equation(13) at b=1 we get
v =x+k k
: (16)
k
= xv
kx +
k
+k k
; (17)
kxv
kx
= k
2 k
: (18)
Togetherwith equation (15) we conclude
x 2
=k 2 k
: (19)
2.3.2 Level-Delta and Level-Gamma
Fora level-dened value functionwe have forall a;b>0
v(x;l) = v(abx;abl): (20)
We dierentiate with respect to a and get at a=1
0 = v
x
(bx;bl)bx+v
l
(bx;bl)bl: (21)
Ifwe set b =1we get the relation
x+
l
l=0: (22)
Now wedierentiate equation (21) with respect tob and get atb =1
0 = v
xx x
2
+2v
xl xl+v
l l l
2
: (23)
One the other hand we can dierentiate the relation between delta and level-delta
with respect tol and get
v
xl x+l
l
+ l
= 0: (24)
Togetherwith equation (23) we conclude
x 2
+x =l 2 l
+l l
: (25)
In generalwe obtain
Theorem 2 (price homogeneity)
v = n
X
i=1 x
i
i +
m
X
j=1 k
j
k
j
(26)
n
X
i;j=1 x
i x
j ij
= m
X
i;j=1 k
i k
j k
ij
(27)
for strike-dened options and
0= n
X
i=1 x
i
i +
m
X
j=1 l
j
l
j
(28)
n
X
i;j=1 x
i x
j ij +
n
X
i=1 x
i
i
= m
X
i;j=1 l
i l
j l
ij +
m
X
i=1 l
i
l
i
(29)
for level-dened options.
Model
We start with relations among Greeks for European claims in the n-dimensional
Black-Scholes model
dS
i
(t) = S
i
(t)[(r q
i
)dt+
i dW
i
(t)]; i=1;:::;n (30)
Cov(W
i (t);W
j
(t)) =
ij
t; (31)
wherer isthe risk-freerate, q
i
the dividendrate ofasset i orforeign interest rateof
exchange rate i,
i
the volatility of asset i and (W
1
;:::;W
n
) a standard Brownian
motion (under the risk-neutral measure) with correlation matrix . Let v denote
today's value of the payo f(S
1
(T);:::;S
n
(T)) at maturity T. Then it is known
that v satisesthe Black-Scholes partial dierentialequation
0 = v
rv+ n
X
i=1 x
i (r q
i )v
xi +
1
2 n
X
i;j=1 (Æ
T
)
ij x
i x
j v
xixj
: (32)
3.1 Relations among Greeks Based on the Log-Normal Dis-
tribution
The value function v has a representation given by the n-fold integral
v =e r
Z
f
:::;S
i (0)e
i p
x
i +
i
;:::
g(~x;)d~x; (33)
where
i
=r q
i 1
2
2
i
and g(~x;) is the n-variate standard normal density with
correlationmatrix . Since we donot want to assumedierentiabilityof the payo
f, but we know that the transition density g is dierentiable, we dene a change
the variables y
i
=S
i (0)e
i p
xi+i
, which leads to
v =e r
Z
f(:::;y
i
;:::)g ln
y
i
S
i (0)
i
i p
;
!
d~y
Q
y
i
i p
: (34)
3.1.1 Properties of the Normal Distribution
We collect some properties of the multivariate normal density function g. We sup-
pose that the vector X of n random variables with means zero and unit variances
hasanonsingularnormalmultivariatedistributionwithprobabilitydensityfunction
g(x
1
;:::;x
n
;c
11
;:::;c
nn
)=(2) 1
2 n
jCj 1
2
exp
1
2 x
T
Cx
: (35)
Here C is the inverse of the covariance matrix of X, which is denoted by . Then
thefollowingidentitypublished in[3]canbeproved easilyby writingthe density in
termsof its characteristic function.
@g
@
ij
=
@ 2
g
@x
i
@x
j
: (36)
In the two-dimensional case this reads as
@n
2
(x;y;)
@
=
@ 2
n
2
(x;y;)
@x@y
; (37)
whichcanbeextendedreadilytothecorrespondingcumulativedistributionfunction,
i.e.,
@N
2
(x;y;)
@
=
@ 2
N
2
(x;y;)
@x@y
=n
2
(x;y;): (38)
3.1.2 Correlation Risk and Cross-Gamma
Usingthe abbreviation g
jk
=
@ 2
g
@x
j
@x
k
the cross-gamma and correlation riskare
@ 2
v
@S
j (0)@S
k (0)
= e r
1
S
j (0)S
k (0)
j
k
Z
f(:::;y
i
;:::)g
jk d~y
Q
y
i
i p
; (39)
@v
@
jk
= e r
Z
f(:::;y
i
;:::)g
jk
d~y
Q
y
i
i p
: (40)
InvokingPlackett's identity (36) saying that g
jk
=g
jk
leads to
Theorem 4 (cross-gamma-correlation-riskrelationship)
@v
@
jk
= S
j (0)S
k (0)
j
k
@ 2
v
@S
j (0)@S
k (0)
: (41)
3.1.3 Interest Rate Risk and Delta
A similarcomputation yields
Theorem 5 (delta-rho relationship)
@v
@q
j
= S
j (0)
@v
@S
j (0)
; (42)
@v
@r
=
0
@
v n
X
j=1 S
j (0)
@v
@S
j (0)
1
A
: (43)
The rst and second derivativeof the density g satisfy
g
j
= g
n
X
i=1 x
i C
ij
; (44)
g
jk
= g n
X
i=1 x
i C
ij n
X
i=1 x
i C
ik gC
kj
: (45)
Forthe j-th vega we nd thus
j
@v
@
j
= e r
Z
fg n
X
i=1 x
i C
ij x
j 1
!
d~y
Q
y
i
i p
; (46)
x
j
= ln
y
i
S
i (0)
(r q
i +
1
2
2
i )
i p
=x
j
j p
; (47)
where we omit the arguments of f and g to simplify the notation. For the cross
gammaswe derive
j
k S
j (0)S
k (0)
@ 2
v
@S
j (0)@S
k (0)
=e r
Z
fgB
jk d~y
Q
y
i
i p
; (48)
B
jk
= n
X
i=1 x
i C
ij n
X
i=1 x
i C
ik C
kj n
X
i=1 x
i C
ij
k p
Æ
jk
: (49)
We now multiply by
jk
, sum over k, remember that is the inverse matrix of C
and obtain
n
X
k=1
jk
j
k S
j (0)S
k (0)
@ 2
v
@S
j (0)@S
k (0)
=e r
Z
f gD
j
d~y
Q
y
i
i p
; (50)
D
j
= n
X
i=1 x
i C
ij x
j 1
n
X
i=1 x
i C
ij x
j +
n
X
i=1 x
i C
ij x
j
: (51)
In summarywe obtain
Theorem 6 (gamma-vega relationship)
j
@v
@
j
= n
X
k=1
jk
j
k S
j (0)S
k (0)
@ 2
v
@S
j (0)@S
k (0)
: (52)
In dimension one the gamma-vega and delta-rho relationships are also mentioned
in[4]. Shawshows therethat v
S 2
(t)v
S(t)S(t)
satisesthe Black-Scholes partial
dierentialequationand ishenceidenticallyzerofor path-independent options. We
note that the gamma-vega relationship does not hold for barrier options, simply
because gammaand vega are not equalatthe barrier.
4.1 Results for European Claims in the Black-Scholes Model
We listseveral relationsfor European options.
0 = +r+q
q +
1
2
scale invarianceof time (53)
v = x+k
k
price homogeneity and strikes (54)
x 2
= k 2 k
price homogeneity and strikes (55)
x = l
l
price homogeneity and levels (56)
x 2
+x = l 2 l
+l l
price homogeneity and levels (57)
= (v x) delta-rho relationship (58)
+
q
= v rates symmetry (59)
rv = +(r q)x+ 1
2
2
x 2
Black-Scholes PDE (60)
qv = +(q r)k k
+ 1
2
2
k 2 k
dual Black-Scholes (strike) (61)
rv = +(q r+ 2
)l l
+ 1
2
2
l 2 l
dual Black-Scholes (level) (62)
q
= x delta-rho relationship (63)
= k
k
combination of (63) and (54) (64)
= x
2
gamma-vega relationship (65)
An interpretation of equation (65) can be found in[6]. We would like to point out
thatthisrelationshipisbasedonafactconcerningthenormaldistributionfunction,
namelydening
n(t;)
= 1
p
2 2
e t
2
2 2
; (66)
N(x;)
= Z
x
1
n(t;)dt; (67)
one can verify that
@ 2
xx
N(x;)=@
N(x;): (68)
There are surely more relations one can prove, but the next theorem will give a
deeper insight intothe relationsof the Greeks.
Theorem 7 If the price and two Greeks g
1
;g
2
of a European option are given with
g
1
2 G
1
=f;
k
; l
;;
q
g; (69)
g
2
2 G
2
=f ; k
; l
;;g; (70)
then all the other Greeks (2 G
1 [ G
2
) can be calculated. Furthermore, if and
another Greek from G
2
is given, it is alsopossible, to determine all other Greeks.
(61) to(63) are conclusions. To get an overview overall these relations,we listthe
appearance of each Greek in all these relations. WithX or O we denote, that the
marked Greek appears in the relation. The relations marked with X show, that
thereisarelationbetween Greeks of G
1
and G
2
and theO shows, thatthis relation
concerns onlythe Greeks of one set.
Greeks2G
1
Greeks2G
2
equation v k
l
q
k l
(53) X X X X
(54) O O O
(55) O O
(56) O O
(57) X X X X
(58) O O O
(59) O O O
(60) X X X X
(61) X X X X
(62) X X X X
(64) O O
(65) O O
(63) O O
Letusnowassumethe optionprice andone Greek fromthe set G
1
are given. Then
a look at the table shows that all Greeks of the set G
1
can be evaluated. If all
Greeks of the set G
1
are known and additionally one Greek of the set G
2
is given,
allother Greeks can be determined. One the other hand, only eight equations are
independent, so the knowledge of two Greeks is also the minimum knowledge one
needs todetermine allten Greeks. This isthe proof of the rst statement.
If and another Greek from G
2
is given, then it is always possible to determine
oneGreek ofthe set G
1
and one appliesthe partof this theorem already proved. If
; k
or l
isgiven, one can use one of the Black-Scholes equations(60) to (62). If
vega is given, one can use (65) to get .
We conclude this sectionwith anexample. In the special case of plain vanilla calls
and puts in a foreign exchange marketall relations for the Greeks presented above
are valid. These formulas are wellknown and can be found in [7].
4.2 A Path-Independent Barrier Call
4.2.1 Value
The payo of a path-independentdown-and-out barrier callisgiven by
f(S
T
;k;K) = (S k) +
I
fS
T
>Kg
(71)
payocanbewrittenas(S
T k)I
fS
T
>Kg
. WeclaimthatkandK arestrikes,because
this optionhas the scaling behaviorf(aS
T
;ak;aK)=af(S
T
;k;K). Intuitively one
would callK a level; but we dened alevel by its scalingbehavior in section2.3.2,
which is not valid in this case. Therefore the path-independent barrier call is an
examplefor a strike-dened option.
Usingthe abbreviation
d
= ln(
S
0
K
)+(r q) 1
2
2
p
2
; (72)
the value of a path-independent down-and-out barriercall isgiven by
v(S
0
;k;K) = e r
Z
1
K
s k
s p
2 2
exp
(ln(
s
S
0
) (r q) + 1
2
2
) 2
2 2
!
ds
= S
0 e
q
N(d
+ ) ke
r
N(d ): (73)
We now want to calculateall Greeks of this option. We show that Theorem 7 can
beused toorganizethe calculation of the Greeks.
4.2.2 Greeks
Delta. Since dierentiation cannot be avoided entirely, we choose the derivative
with respect to k,which isobviously
v
k
= e
r
N(d ): (74)
Next we dierentiate the integral representation of v with respect to K and
obtain
v
K
= e r
k K
K p
2 2
exp
0
@ (ln(
K
S
0
) (r q) + 1
2
2
) 2
2 2
1
A
=
k K
K 1
p
2
e
r
n(d ): (75)
In Theorem 7 we had assumed only one strike. In our example we have two
strikes, and therefore we need two Greeks from the set G
1
to determine all
other Greeks of this set. >From the price homogeneity we know that the
relation
v = S
0 v
S
0 +kv
k +Kv
K
(76)
holds,whence we obtainfor the spot delta
v
S
0
= e q
N(d
+ )+
K k
S
0 1
p
2
e
r
n(d ): (77)
v
r
= ke r
N(d )+
K k
p
2
e
r
n(d ); (78)
v
q
= S
0 e
q
N(d
+
)
K k
p
2
e
r
n(d ): (79)
Gamma. We have calculated all Greeks in G
1
. To determine some other Greeks
withoutdierentiationweneedatleastoneGreekofthesetG
2
. Inthetheorem
above we assumed, that the option will be described by one strike, but the
optionwe analyzedepends ontwostrikes. Sowe haveto dierentiate trice to
get alldual gammas.
v
kk
= 0 (80)
v
kK
= 1
K 1
p
2
e
r
n(d ) (81)
v
KK
= k
K 2
e r
p
2
n(d )+
k K
K 2
e r
2
n(d )d (82)
The extensionof (55) to the case of one stock and two strikes isthe equation
(27)with n=1and m =2. In our example this relationis given by
S 2
0
= k 2 kk
+2kK kK
+K 2 KK
: (83)
>From this relation, which follows from the homogeneity of v, we obtain for
the spotgamma withoutdierentiation
v
S
0 S
0
= ke
r
S 2
0 p
2
n(d )+
k K
S 2
0
e r
2
n(d )d : (84)
Vega. >From (65) we get
v
= p
ke r
n(d ) (K k)e r
1
n(d )d : (85)
Theta. >From the scale invariance of time (53) we obtain
v
t
= v
= rke r
N(d )+qS
0 e
q
N(d
+ )
(r q)
K k
p
2
e
r
n(d )
2 p
ke
r
n(d )
+ 1
2
(K k)e r
n(d )d (86)
Black-Scholes Model
5.1 Pricing of a European Option
Rainbow options are nancial instruments which depend on several risky assets.
Many of them are very sensitive to changes of correlation. We call kappa () the
derivativeof the option value v with respect to the correlation .
Thecomputationaleorttocomputethe kappaishard,eveninasimpleframework,
but in the Black-Scholes model with two stocks and one cash bond we can use the
cross-gamma-correlation-riskrelationshipwhich can be used easilytond kappa.
Letthe stock price processes S
1
and S
2
be described by
ln S
1 ()
S
1 (0)
= (r q
1 1
2
2
1 ) +
1 W
1
; (87)
ln S
2 ()
S
2 (0)
= (r q
2 1
2
2
2 ) +
2 W
1
+
2 q
1
2
W 2
: (88)
W 1
andW 2
are twoindependentBrownianmotionsundertheriskneutralmeasure.
Theprobabilitydensityforthe distributionofS
1
()isdenotedbyh
1
(x)andisgiven
by the log-normaldensity
h
1
(x) =
1
q
2 2
1
1
x exp
A 2
2 2
1
!
; (89)
A
= ln x
S
1 (0)
!
r +q
1 +
1
2
2
1
: (90)
The equationfor the second stock priceprocess can be writtenas
ln S
2 ()
S
2 (0)
= (r q
2 1
2
2
2 ) +
2
1 ln
S
1 ()
S
1 (0)
!
(r q
1 1
2
2
1 )
!
+
2 q
1
2
W 2
: (91)
The conditionaldistribution of S
2
() given S
1
() is thuslog-normalwith density
h
2j1
(yjx) =
1
y q
2 2
2
(1
2
) exp
B 2
2 2
2
(1
2
)
!
; (92)
B
=
"
ln y
S
2 (0)
!
r+q
2 +
1
2
2
2
2
1 A
#
: (93)
The joint distributionof S
1
()and S
2
() isgiven by the productof h
1
and h
2
h(x;y) = h
1 (x)h
2j1
(yjx): (94)
1 2
v = e r
1
Z
0 1
Z
0
h(x;y)f(x;y)dxdy: (95)
Thisintegralhas exactlythe structure of the integralsstudiedin section3.1. Using
theresultsprovidedabove,onecancollectseveralrelationshipsforthe Greeksinthe
two-dimensional case. Additional, the fundamental symmetry scale invariance of
timeisvalidtoo. BecauseweconcentrateonEuropeanoptions,thetwodimensional
Black-Scholes-PDE also holds.
5.2 Relations among the Greeks
We specialize the relationships among the Greeks found in n dimensions. Some
resultsare
0 =
q
1 +S
1 (0)
1
; (96)
0 =
q2 +S
2 (0)
2
; (97)
0 = q
1
q
1 +q
2
q
2 +
1
2
1
1 +
1
2
2
2 +r
r
+; (98)
0 = rv+(r q
1 )S
1 (0)
1
+(r q
2 )S
2 (0)
2
+ 1
2
2
1 S
1 (0)
2
11
+
1
2 S
1 (0)S
2 (0)
12 +
1
2
2
2 S
2 (0)
2
22
; (99)
=
1
2 S
1 (0)S
2 (0)
12
; (100)
0 =
1
1 +
2
1 S
1 (0)
2
11
; (101)
0 =
2
2 +
2
2 S
2 (0)
2
22
; (102)
0 =
1
1
2
2
2
1 S
1 (0)
2
11 +
2
2 S
2 (0)
2
22
; (103)
r
= (v S
1 (0)
1 S
2 (0)
2
); (104)
0 = v +
q1 +
q2 +
r
: (105)
Of course one can get more relations by combining some relations above. The
relationswehavechosen topresentare eithersimilartothe one-dimensionalcase or
have anothernatural interpretation.
(96) and (97). These relations are a justication for the rough way to deal
with dividends. One subtracts the dividends from the actual spot price and
prices the option with this price and without dividends. This relation is not
eectedby the two-dimensionalityof the problem.
(98). This isthe two-dimensionalversionof thegeneral invarianceundertime
scaling.
(99). This is the Black-Scholes dierential equation. This relation must hold,
because weconcentrated on European claims. It turns out, that the dynamic
optionis dened as aboundaryproblem.
(100). This is the cross-gamma-correlation-risk relationship;it is remarkable,
that this relationshiphas such a simple structure.
(101)and(102). Thesearethegamma-vegarelationships. Noticethatonecan
determineonlyby knowledgeofsomederivativeswith respect toparameters
which concern only one stock. Of course, there is no dierence between the
rst and the second stock. These relations are valid in the one-dimensional
case with 0.
(103)follows from(100).
(104). This is the delta-rho relationship. The interest rate risk is well known
to be the negative product of duration and the amount of money invested.
The term in the parentheses is exactly the amount of money one would have
toinvest in the cash bond in order todelta-hedge the option.
(105). This relation is the two-dimensional rates symmetry, an extension of
equation(59). It follows from(104), (96) and (97).
Inthefollowingwetreatoneexampleinfulldetail. Furtherexamplessuchasoutside
barrieroptions and spreadoptions are available in[7].
5.3 European Options on the Minimum/Maximum of Two
Assets
We consider the payo
[(min(S
1
(T);S
2
(T)) K)]
+
: (106)
This isa European put orcallonthe minimum ( =+1) ormaximum ( = 1)of
thetwoassetsS
1
(T)and S
2
(T)withstrikeK. Asusual, thebinary variabletakes
thevalue +1foracalland 1foraput. Itsvalue functionhasbeen publishedin[5]
and can be writtenas
v(t;S
1 (t);S
2
(t);K;T;q
1
;q
2
;r;
1
;
2
;;;) (107)
=
h
S
1 (t)e
q
1
N
2 (d
1
;d
3
;
1 )
+S
2 (t)e
q
2
N
2 (d
2
;d
4
;
2 )
Ke r
1
2
+N
2 ((d
1
1 p
);(d
2
2 p
);)
!#
;
2
=
2
1 +
2
2
2
1
2
; (108)
1
=
2 1
; (109)
2
=
1
2
; (110)
d
1
= ln(S
1
(t)=K)+(r q
1 +
1
2
2
1 )
1 p
; (111)
d
2
= ln(S
2
(t)=K)+(r q
2 +
1
2
2
2 )
2 p
; (112)
d
3
= ln(S
2 (t)=S
1
(t))+(q
1 q
2 1
2
2
)
p
; (113)
d
4
= ln(S
1 (t)=S
2
(t))+(q
2 q
1 1
2
2
)
p
: (114)
5.3.1 Greeks
Delta. Space homogeneity implies that
v =S
1 (t)
@v
@S
1 (t)
+S
2 (t)
@v
@S
2 (t)
+K
@v
@K
: (115)
Usingthisequationoneonlyhastodierentiatetwiceinordertogetalldeltas.
It turns out, that the value function is given in the natural representation,
which ispresented inthe appendix, and one is allowed toread o the deltas:
@v
@S
1 (t)
= e
q
1
N
2 (d
1
;d
3
;
1
); (116)
@v
@S
2 (t)
= e
q
2
N
2 (d
2
;d
4
;
2
); (117)
@v
@K
= e
r
1
2
+N
2 ((d
1
1 p
);(d
2
2 p
);)
:
(118)
Gamma. Computingthegammasisactuallythelastsituationwheredierentiation
isneeded. We use the identities
@
@x N
2
(x;y;) = n(x)N
y x
p
1
2
!
; (119)
@
@y N
2
(x;y;) = n(y)N
x y
p
1
2
!
; (120)
and obtain
@ 2
v
@(S
1 (t))
2
= e
q
1
S
1 (t)
p
1 n(d
1
)N
d
3 d
1
1
2 p
1
2
n(d
3
)N
d
1 d
3
1
2 p
1
2
!#
; (121)
@ 2
v
@(S
2 (t))
2
= e
q2
S
2 (t)
p
"
2 n(d
2
)N
d
4 d
2
2
1 p
1
2
!
n(d
4
)N
d
2 d
4
2
1 p
1
2
!#
; (122)
@ 2
v
@S
1 (t)@S
2 (t)
= e
q
1
S
2 (t)
p
n(d
3
)N
d
1 d
3
1
2 p
1
2
!
: (123)
Kappa. The sensitivity with respect to correlation is directly relatedto the cross-
gamma
@v
@
=
1
2 S
1 (t)S
2 (t)
@ 2
v
@S
1 (t)@S
2 (t)
: (124)
Vega. We referto (101) and (102)to get the following formulas for the vegas,
@v
@
1
= v
+
2
1 (S
1 (t))
2
v
S
1 (t)S
1 (t)
1
(125)
= S
1 (t)e
q
1
p
"
1 n(d
3
)N
d
1 d
3
1
2 p
1
2
!
+n(d
1
)N
d
3 d
1
1
2 p
1
2
!#
; (126)
@v
@
2
= v
+
2
2 (S
2 (t))
2
v
S
2 (t)S
2 (t)
2
(127)
= S
2 (t)e
q
2
p
"
2 n(d
4
)N
d
2 d
4
2
1 p
1
2
!
+n(d
2
)N
d
4 d
2
2
1 p
1
2
!#
: (128)
Rho. Lookingat(96), (97) and (104) the rhos are given by
@v
@q
1
= S
1 (t)
@v
@S
1 (t)
; (129)
@v
@q
2
= S
2 (t)
@v
@S
2 (t)
; (130)
@v
@r
= K
@v
@K
: (131)
on(98).
@v
@t
= 1
q
1 v
q1 +q
2 v
q2 +rv
r +
1
2 v
1 +
2
2 v
2
: (132)
6 Generalization to Higher Dimensions
and other Market Models
6.1 Beyond Black-Scholes
Up to now we illustrated our ideas in the Black-Scholes model and in some parts
we used specic properties of this model. Nevertheless there are some properties,
which are so fundamental, that they should hold in any realistic market model.
These fundamental properties are the homogeneity of time, the scale invariance of
timeandthescaleinvarianceofprices. Foreverymarketmodeloneuses,one should
check, if the modelfulllsthese properties.
Anexample for amarket modelwith non-deterministicvolatilityis Heston'sstoch-
astic volatility model[2].
In this more general framework one needs to clarify the notion of vega. A change
of volatilitycould mean a change of the entire underlying volatility process. If the
pricing formula depends on input parameters such as initialvolatility, volatility of
volatility,meanreversionofvolatility,thenonecanconsiderderivativeswithrespect
to such parameters. It turns out that our strategy to compute Greeks can still be
appliedsuccessfully ina stochastic volatilitymodel.
6.2 Heston's Stochastic Volatility Model
dS
t
= S
t
dt+ q
v(t)dW (1)
t
; (133)
dv
t
= ( v
t
)dt+ q
v(t)dW (2)
t
; (134)
Cov h
dW (1)
t
;dW (2)
t i
= dt; (135)
(S;v;t) = v: (136)
The model for the variance v
t
is the same as the one used by Cox, Ingersoll and
Ross for the short term interest rate, see [1]. We think of > 0 as the long term
variance, of > 0 as the rate of mean-reversion. The quantity (S;v;t) is called
the marketprice of volatilityrisk.
Heston providesa closed-formsolutionfor European vanilla optionspaying
[(S
T
K) ] +
: (137)
the strike in units of the domestic currency, q the risk free rate of asset S, r the
domesticrisk freerate and T the expiration time inyears.
6.2.1 Abbreviations
a
= (138)
u
1
= 1
2
(139)
u
2
= 1
2
(140)
b
1
= + (141)
b
2
= + (142)
d
j
= q
('i b
j )
2
2
(2u
j
'i ' 2
) (143)
g
j
= b
j
'i+d
j
b
j
'i d
j
(144)
= T t (145)
D
j (;')
= b
j
'i+d
j
2
"
1 e d
j
1 g
j e
d
j
#
(146)
C
j (;')
= (r q)'i
+ a
2
(
(b
j
'i+d) 2ln
"
1 g
j e
d
j
1 e d
j
# )
(147)
f
j
(x;v;t;')
= e C
j (;')+D
j
(;')v+i'x
(148)
P
j
(x;v;;y)
= 1
2 +
1
Z
1
0
<
"
e i'y
f
j
(x;v;;')
i'
#
d' (149)
p
j
(x;v;;y)
= 1
Z
1
0
<
h
e i'y
f
j
(x;v;;') i
d' (150)
P
+ ()
=
1
2
+P
1 (lnS
t
;v
t
;;lnK) (151)
P ()
=
1
2
+P
2 (lnS
t
;v
t
;;lnK) (152)
This notation is motivated by the fact that the numbers P
j
are the cumulative
distribution functions (inthe variable y) of the log-spotprice after time starting
atx for some drift. The numbers p
j
are the respective densities.
6.2.2 Value
The value function for European vanillaoptions isgiven by
V = h
e q
S
t P
+
() Ke
r
P () i
(153)
The probabilitiesP
() correspond to N(d
) inthe constant volatilitycase.
6.2.3 Greeks
Weuse thehomogeneityofprices, toobtainthe deltas. Butwemust show, thatthe
priceis given in itsnaturalrepresentation. Sowe use the following strategy.
We assume, that equation (153) gives the natural price representation, which is
dened inappendix A. Under this assumptionwe can read othe deltas, and from
the deltas we derive the gammas. Using Theorem 8 we show that the assumption
of (153)giving the naturalprice representation was correct.
Spot delta.
=
@V
@S
t
=e q
P
+
() (154)
Dual delta.
K
=
@V
@K
= e
r
P () (155)
Gamma. Under the condition, that the deltas are correct, we obtain for the gam-
mas by dierentiation:
Spot Gamma.
=
@
@S
t
=
@
@x
@x
@S
t
= e
q
S
t p
1 (lnS
t
;v
t
;;lnK) (156)
Dual Gamma.
K
=
@ K
@K
=
@ K
@y
@y
@K
= e
r
K p
1 (lnS
t
;v
t
;;lnK) (157)
Proof of the natural representation assumption >FromTheorem8weknow,
that our initialguess forthe deltasis correct, if the relation
S 2
t
= K
2 K
(158)
holds. In fact, this equation isgiven by
S
t e
q
p
1 (lnS
t
;v
t
;;lnK)=Ke r
p
2 (lnS
t
;v
t
;;lnK); (159)
and this statement is true. Soour calculation for the deltas and gammas has
been nished.
@V
@r
=Ke r
P (); (160)
@V
@q
= S
t e
q
P
+
(): (161)
Theta. ThetacanbecomputedusingthepartialdierentialequationfortheHeston
vanilla option
V
t
+(r q)SV
S +
1
2 vV
vv +
1
2 vS
2
V
SS
+vSV
vS qV
+[( v) ]V
v
=0; (162)
where the derivatives with respect to initial variance v must be evaluated
numerically.
7 Summary
Wehave learnedhowto employ homogeneity-based methodsto compute analytical
formulas of Greeks for analytically known value functions of options in a one-and
higher-dimensional market. Restricting the view to the Black-Scholes model there
are numerous further relations between various Greeks which are of fundamental
interest. Themethodhelpssavingcomputationtimeforthemathematicianwhohas
todierentiatecomplicatedformulasaswellasforthecomputer, becauseanalytical
results for Greeks are usually faster to evaluate than nite dierences involving at
least twice the computation of the option's value. Knowing how the Greeks are
related among each other can speed up nite-dierence-, tree-, or Monte Carlo-
based computation of Greeks or lead at least to a quality check. Many of the
results are validbeyond the Black-Scholes model. Mostremarkably some relations
oftheGreeksarebasedonpropertiesofthenormaldistributionrefreshingtheactive
interplay between mathematicsand nancialmarkets.
A The Natural Price Representation for Homoge-
neous Options
Weanalyze the followingproblem. Letv(x;k)bethe value ofanoptionand v(x;k)
ishomogeneous of degree1. After Evaluatingthe integral to determine the option-
price,one obtainsthe following formula:
v(x;k) = xf(x;k)+kg(x;k) (163)
One the other hand,weknowfrom the homogeneity of v:
v(x;k) = xv
x
(x;k)+kv
k
(x;k) (164)
x y
answer is: No, not in general.
Because of v being homogeneous of degree 1 we know, that f(x;k),g(x;k),v
x (x;k)
and v
k
(x;k) are homogeneous of degree 0. Therefore we know that f(x;k) has the
representation f(
x
k
) and so on. Introducing the notation u= x
k
we nd from (163)
and (164) that
uf(u)+g(u) = uv
x
(u)+v
y
(u) (165)
We dene h(u)=v
x
(u) f(u). The answer to the question above would be yes, if
and only if h(u)=0 for all u. One can easilyshow, that the function h(u)has the
following properties:
lim
u!0
h(u) = 0 (166)
lim
u!1
h(u) = 0 (167)
h(u) = uf 0
(u)+g 0
(u) (168)
Sowe come tothe followingdenition:
Denition 2 (Natural Representation of Homogeneous Functions)
Let v(x;k) be a homogeneousfunctionof degree 1. Thenthere is a unique represen-
tation
v(x;k) = xf
x
k
+kg
x
k
with (169)
0 = uf 0
(u)+g 0
(u) (170)
We callthis the naturalrepresentation.
Of course, the denition of the natural representation can be extended to higher
dimensions. The question of this section can now be answered more exactly. One
can read o the deltas if and only if the price formula isgiven in its natural repre-
sentation. This statement alsoholdsinhigherdimensions. Forthe two dimensional
case, wesummarize:
Theorem 8 Let v(x;k) be a homogeneousfunction of degree 1. Therepresentation
v(x;k) = xf(x;k)+kg(x;k) (171)
isthe natural representation if and only if
x 2
@
x
f(x;k) = k 2
@
k
g(x;k) (172)
Theorem 9 Let v(x;k) = xf(x;k)+kg(x;k) be the natural representation of a
homogeneous function of degree 1. Then the followingequations hold:
@
x
v(x;k) = f(x;k) (173)
@
k
v(x;k) = g(x;k) (174)
[1] COX, J.C., INGERSOLL, J.E.and ROSS, S.A.(1985). A Theory of the Term
Structure of Interest Rates. Econometrica 53, 385-407.
[2] HESTON, S. (1993). A Closed-Form Solution for Options with Stochastic
Volatility with Applications to Bond and Currency Options. The Review of
FinancialStudies, Vol. 6, No.2.
[3] PLACKETT, R. L. (1954).A Reduction Formulafor Normal MultivariateIn-
tegrals.Biometrika. 41, pp. 351-360.
[4] SHAW, W. (1998). Modelling Financial Derivatives with Mathematica. Cam-
bridgeUniversity Press.
[5] STULZ,R.(1982).OptionsontheMinimumorMaximumofTwoAssets.Jour-
nal of Financial Economics.10, pp. 161-185.
[6] TALEB, N. (1996). DynamicHedging. Wiley, New York.
[7] WYSTUP, U. (1999). Vanilla Options. Formula Catalogue of
http://www.MathFinance.de.
