Ecient Computation of Option Price Sensitivities
Using Homogeneity and other Tricks
Oliver Rei
Weierstra-Institute for Applied Analysis and Sto chastics
Mohrenstrasse 39
10117 Berlin
GERMANY
Uwe Wystup
Commerzbank Treasury and Financial Pro ducts
Neue Mainzer Strasse 32{36
60261 Frankfurt am Main
GERMANY
http://www.mathfinance.de
Preliminary version, January 25, 1999
Abstract
No front-oce software can survive without providing derivatives of options prices
with resp ect to underlying market or mo del parameters, the so called Greeks. We
present a list of common Greeks and exploit homogeneity prop erties of nancial
markets to derive relationships b etween Greeks out of which many are mo del-
indep endent. We apply our results to Europ ean style options, rainbow options,
path-dep endent options as well as options priced in Heston's sto chastic volatility
mo del and show shortcuts to avoid exorbitant and time-consuming computations
of derivatives whicheven strong symb olic calculators fail to pro duce.
partially aliated to Delft University,by supp ort of NWO Netherlands 1
2 Rei, O. and Wystup, U.
1 Intro duction
Based on homogeneity of time and price level of a nancial pro duct we can derive
relations for the options sensitivities, the so-called \Greeks". The basic market
mo del we use is the Black-Scholes mo del with sto cks paying a continuous dividend
yield and a riskless cash b ond. This mo del supp orts the homogeneity prop erties
which are valid in general, but its structure is so simple, that we can concentrate
on the essential statements of this pap er. We will also discuss how to extend out
work to more general market mo dels.
We list the commonly used Greeks and their notations. We do not claim this list
to b e complete, b ecause one can always de ne more derivatives of the option price
function.
First of all we analyze the homogeneity of a nancial market, consisting of one sto ck
and one cash b ond. The techniques we present are easily expanded to the higher
dimensional case as we show later on.
In addition we lo ok at the Greeks of Europ ean options in the Black-Scholes mo del.
Essentially it turns out, that one only needs to knowtwo Greeks in order to calculate
all the other Greeks without di erentiating.
Another interesting example is a Europ ean derivative security dep ending on two
assets. For such rainbow options the analysis of the risk due to changing correlation
of the two assets is very imp ortant. We will showhow this risk is related to
simultaneous changes of the two underlying securities.
There are several applications of these homogeneity relations.
1. It helps saving time in computing derivatives.
2. It pro duces a robust implementation compared to Greeks via di erence quo-
tients.
3. It allows to check the quality and consistency of Greeks pro duced by nite-
di erence-, tree- or Monte Carlo metho ds.
4. It admits a computation of Greeks for Monte Carlo based values.
5. It shows relationships b etween Greeks whichwouldn't b e noticed merely by
lo oking at di erence quotients.
Option Price Sensitivities 3
1.1 Notation
S sto ck price or sto ck price pro cess
B cash b ond, usually with risk free interest rate r
r risk free interest rate
q dividend yield continuously paid
volatility of one sto ck, or volatility matrix of several sto cks
correlation in the two-asset market mo del
t date of evaluation \to day"
T date of maturity
= T t maturity of an option
x sto ck price at time t
f payo function
v x;t;::: value of an option
k strike of an option
l level of an option
v partial derivation of v with resp ect to x and analogous
x
The standard normal distribution and density functions are de ned by
2
1
1
t
2
p
1 e n t =
2
Z
x
N x = n t dt 2
1
2 2
1 x 2xy + y
p
n x; y ; = exp 3
2
2
2
2 1
2 1
Z Z
x y
N x; y ; = n u; v ; du dv 4
2 2
1 1
See http://www.MathFinance.de/frontoce.html for a source co de to compute N . 2
4 Rei, O. and Wystup, U.
1.2 The Greeks
Delta v
x
Gamma v
xx
Theta v
t
Rho v in the one-sto ck mo del
r
Rhor v in the two-sto ck mo del
r r
Rho q v
q q
Vega v
Kappa v correlation sensitivity two-sto ck mo del
Greeks, not so commonly used:
x
Leverage v sometimes , sometimes called \gearing"
x
v
0
Vomma v
Sp eed v
xxx
Charm v
x
Color v
xx
F
Forward Delta v
F
dl q
Driftless Delta e
Dual Theta Dual v
T
k
Strike Delta v
k
k
Strike Gamma v
kk
l
Level Delta v
l
l
Level Gamma v
ll
1
two-sto ck mo del Beta
12
2
2 Fundamental Prop erties
2.1 Homogeneity of Time
In most cases the price of the option is not a function of b oth the current time
t and the maturity time T , but rather only a function of the time to maturity
= T t implying the relations
=v = v = v = D ual 5
t T
This relationship extends naturally to the situation of options dep ending on several
intermediate times such as comp ound or Bermuda options.
2.2 Scale-Invariance of Time
Wemaywant to measure time in units other than years in which case interest rates
and volatilities, which are normally quoted on an annual basis, must b e changed
according to the following rules for all a>0.
! a
Option Price Sensitivities 5
r ! ar
q ! aq
p
a 6 !
The option's value must b e invariant under this rescaling, i.e.,
p
;ar;aq; v x; ; r;q;;::: =v x; a;::: 7
a
We di erentiate this equation with resp ect to a and obtain for a =1
1
; 8 0= + r + q +
q
2
a general relation b etween the Greeks theta, rho, rhoq and vega. It extends naturally
to the case of multiple assets and multiple intermediate dates.
2.3 Scale Invariance of Prices
The general idea is that value of securities may b e measured in a di erent unit, just
likevalues of Europ ean sto cks are now measured in Euro instead of in-currencies.
Option contracts usually dep end on strikes and barrier levels. Rescaling can have
di erent e ects on the value of the option. Essentially wemay consider the following
typ es of homogeneity classes. Let v x; k b e the value function of an option, where
x is the sp ot or a vector of sp ots and k the strike or barrier or a vector of strikes
or barriers. Let a b e a p ositive real numb er.
De nition 1 homogeneity classes We cal l a value function k -homogeneous
of degree n if for al l a
n
v ax; ak = a v x; k : 9
The value function of a Europ ean call or put option with strike K is then K -
homogeneous of degree 1, a p ower call with strike K and cap C is b oth K -
homogeneous of degree 2 and C -homogeneous of degree 0, a double barrier call
with strike K and barriers B = L; H isK -homogeneous of degree 1 and B -
homogeneous of degree 0. We will call an option strike-de ned, if there is just one
strike k and the value function is k -homogeneous of degree 1 and level-de ned if
there is just one level l and the value function is l -homogeneous of degree 0, e.g. a
digital cash call.
2.3.1 Strike-Delta and Strike-Gamma
For a strike-de ned value function wehave for all a; b > 0
abv x; k = v abx; abk : 10
6 Rei, O. and Wystup, U.
We di erentiate with resp ect to a and get for a =1
bv x; k = bxv bx; bk + bk v bx; bk : 11
x k
Wenow di erentiate with resp ect to b get for b =1
v x; k = xv + xv x + xv k + kv + kv x + kv k 12
x xx xk k kx kk
2 k 2 k
= x+ x + 2xk v + k + k : 13
xk
If weevaluate equation 11 at b =1 we get
k
v = x+ k : 14
We di erentiate this equation with resp ect to k and obtain
k k k
= xv + + k ; 15
kx
2 k
kxv = k : 16
kx
Together with equation 13 we conclude
2 2 k
x =k : 17
2.3.2 Level-Delta and Level-Gamma
For a level-de ned value function wehave for all a; b > 0
v x; l = v abx; abl : 18
We di erentiate with resp ect to a and get at a =1
0 = v bx; bl bx + v bx; bl bl : 19
x l
If we set b =1 we get the relation
l
x + l =0: 20
Nowwe di erentiate equation 19 with resp ect to b and get at b =1
2 2
0 = v x +2v xl + v l : 21
xx xl ll
One the other hand we can di erentiate the relation b etween delta and level-delta
with resp ect to l and get
l l
v x + l + = 0: 22
xl
Together with equation 21 we conclude
2 2 l l
x +x= l + l : 23
Option Price Sensitivities 7
3 Europ ean Options in the Black-Scholes
Mo del
In this section we analyze the Europ ean claim in the Black-Scholes mo del
dS = S [ r q dt + dW ]: 24
t t t
This situation has a simple structure which allows us to obtain even more relations
for the Greeks. We assume that the sto ckpays a continuous dividend yield with
rate q . It can also b e seen as a foreign exchange rate and q denotes the risk free
interest rate in the foreign currency.
3.1 Scale Invariance of Prices
A Europ ean option is describ ed by its payo function and the price of this option
in the Black-Scholes mo del is given by
r
v x; = e E [f S j S 0 = x]: 25
The exp ectation is taken under the risk-neutral measure P .Nowwe de ne a set
x
parameterized bya of Europ ean options related to this option bya payo f
a
p ositive real number a.We denote their value function by
S
r
u x; ; a = e E f S 0 = x : 26
a
x
Wenow di erentiate the equation u ; ;1 = u x; ; a with resp ect to a and get
a
x S S 1
r 0
; ;1 = e E f xu S 0 = x : 27
x
2 2
a a a a
If we set a = 1 and use v x; =u x; ; 1 , we obtain the relation
0 r
x= E [f S S e j S 0 = x] 28
Remark 1 In this calculation we used, that the payo function is di erentiable.
This condition can berelaxed by the assumption, that there exists a sequence of dif-
ferentiable functions f , which converges P almost surely pointwise to the function
n
f. For instance, f can have a nite number of jumps and becontinuous between
them. This argumentation is always valid, whenever we di erentiate the payo
function under the expectation.
3.1.1 Interest Rate Risk
We di erentiate 25 with resp ect to r and obtain:
r
v = e E [f S j X 0 = x]
r
r 0
+e E [f S S j S 0 = x] 29
0 r
= v + E [f S S e j S 0 = x] 30
8 Rei, O. and Wystup, U.
Comparing this with equation 28 we nd the following relation for the Greeks
= v x : 31
This result is not surprising, since v x is the amount of money one has to invest
in the cash b ond if one hedges the option by delta hedge. On the other hand the
interest rate risk of a zero-coup on b ond is minus the amount of money invested
times the duration.
3.1.2 Rates Symmetry
Di erentiating equation 25 with resp ect to the dividend yield q leads to
r 0
v = e E [f S S jS 0 = x]; 32
q
and a comparison of this with equation 30 to homogeneity relation:
+ = v: 33
q
3.2 Black-Scholes PDE
Lemma 1 For al l European contingent claims the fol lowing equation Black-Scholes
PDE holds.
1
2 2
x v =0 34 v rv + r q xv +
xx x
2
Remark 2 From this lemma we obtain the fol lowing relation for the Greeks.
1
2 2
x : 35 rv =+ r q x+
2
3.2.1 Dual Black-Scholes PDEs
One can combine the general results for the dual Greeks with the Black-Scholes
PDE and obtains the dual Black-Scholes equations:
1
k 2 2 k
qv = + q r k + k 36
2
1
2 l 2 2 l
rv = + q r + l + l 37
2
3.3 Results for Europ ean Claims in the Black-Scholes Mo del
We found a lot of relations for Europ ean options. Of course, the general relations
hold and additionally we proved some more relations for Europ ean options in the
Option Price Sensitivities 9
Black-Scholes mo del. We list all these relations now.
1
0 = +r + q + scale invariance of time 38
q
2
k
v = x+k log-price homogeneity and strikes 39
2 2 k
x = k log-price homogeneity and strikes 40
l
x = l log-price homogeneity and levels 41
2 2 l l
x + x = l + l log-price homogeneity and levels 42
= v x Interest rate risk 43
+ = v rates symmetry 44
q
1
2 2
x Black-Scholes PDE 45 rv = + r q x+
2
1
k 2 2 k
qv = + q r k + k Dual Black-Scholes 46
2
1
2 l 2 2 l
rv = + q r + l + l Dual Black-Scholes 47
2
Lemma 2 From the relations above we conclude
k
= k ; 48
= x: 49
q
2
= x ; 50
Pro of. Equation 48 follows from 39 and 43 . From 43 and 44 one eas-
ily obtains 49 . The pro of of 50 starts with 38 and equation 45 times .
Eliminating yields
1 1
2 2
r + q + = rv + r q x+ x 51
q
2 2
Using 43 to get rid of xwe get:
1 1
2 2
= q qv + x 52 q +
q
2 2
All terms containing q vanish b ecause of 44 which establishes equation 50 .
An interpretation of equation 50 can b e found in [6]. Wewould like to p oint out
that this relationship is based on a fact concerning the normal distribution function,
namely de ning
2
t
1
2
2
p
; 53 n t; = e
2
2
Z
x
N x; = n t; dt; 54
1
one can verify that
2
N x; = @ N x; : 55 @
xx
10 Rei, O. and Wystup, U.
There are surely more relations one can prove, but the next theorem will givea
deep er insightinto the relations of the Greeks.
Theorem 1 If the price and two Greeks g ;g of a European option are given with
1 2
k l
g 2 G = f; ; ;; g; 56
1 1 q
k l
g 2 G = f ; ; ; ; g; 57
2 2
then al l the other Greeks 2 G [ G can becalculated. Furthermore, if and
1 2
another Greek from G is given, it is also possible, to determine al l other Greeks.
2
Pro of. The relations 38 to 45 are indep endent of each other. The relations
46 to 49 are conclusions. To get a overview over all these relations, we list the
app earance of each Greek in all these relations. With X or O we denote, that the
marked Greek app ears in the relation. The relations marked with X show, that
there is a relation b etween Greeks of G and G and the O shows, that this relation
1 2
concerns only the Greeks of one set.
Gr eek s 2 G Gr eek s 2 G
1 2
k l k l
equation v
q
38 X X X X
39 O O O
40 O O
41 O O
42 X X X X
43 O O O
44 O O O
45 X X X X
46 X X X X
47 X X X X
48 O O
50 O O
49 O O
Let us now assume the option price and one Greek from the set G are given. Then
1
a lo ok at the table shows that all Greeks of the set G can b e evaluated. If all
1
Greeks of the set G are known and additionally one Greek of the set G is given,
1 2
all other Greeks can b e determined. One the other hand, only eight equations are
indep endent, so the knowledge of two Greeks is also the minimum knowledge one
needs to determine all ten Greeks. This is the pro of of the rst statement.
If and another Greek from G is given, then it is always p ossible to determine
2
one Greek of the set G and one applies the part of this theorem already proved.
1
k l
If ; or is given, one can use one of the Black-Scholes equations 45 to 47 .
If vega is given, one can use 50 to get .
Option Price Sensitivities 11
4 Examples for the One-Dimensional Case
4.1 Vanilla Calls and Puts in a Foreign Exchange Market
In the sp ecial case of plain vanilla calls and puts in a foreign exchange market all
relations for the Greeks presented ab ove are valid. These formulas are well known
and can b e found in [7].
4.2 Comp ound Options
A compound option is an option on an option, i.e., a call on a call or a call on a put
or a put on a call or a put on a put. For comparison you may consult [1].
4.2.1 abbreviations
K : strike of the underlying option
k : strike of the comp ound option
: +1 -1 is the underlying option is a call put
! : +1 -1 is the comp ound option is a call put
r q
=
2
W + t
t
S = S e : price of the underlying at time t
t 0
S
t
ln +
K
p
d =
S
0
ln + T
K
p
d =
T
q r
Vanilla S ;K;;r;q; ; = S e N d Ke N d
t t
+
p
tx+ t
H x =Vanilla S e ;K;;r;q; ; k
0
X : the unique numb er satisfying H X =0
p
p
T + td X
p
e =
The value of a comp ound option is given by
v S ;K;k;T;t;;r;q; ;! 58
0
Z
y =+1
h i
p
+
rt ty + t
! Vanilla S e = e n y dy ;K;;r;q; ; k
0
y = 1
!
r
p
t
qT
t ; d ; ! ! X = ! S e N
+ 0 2
T
!
r
t
rT
! K e N !X; d ; !
2
T
rt
!ke N ! X :
12 Rei, O. and Wystup, U.
Delta The value function v is K; k -homogeneous of degree 1, whence v has the
representation
@v @v @v
v = S + K + k ; 59
0
@S @K @k
0
and the deltas can b e just read o , e.g.,
!
r
p
t @v
qT
t ; d ; ! : 60 = ! e N ! X
+ 2
@S T
0
Gamma. To compute gamma, we need to know the derivativeofX , whichis
actually a function X = X S ;K;k;T;t;;r;q; . Although we can not
0
determine this function explicitly,we can compute its derivatives explicitly
using implicit di erentiation. We obtain
p
@H
q tX + t
e e N d
1 @X
@S
+
0
p
p
; 61 = = =
p
@H
tX + t q
@S
S t
e N d S e t
0
0
0
+
@x
and hence for gamma
qT 2
p
e @ v 1 !
p p
= n X + n d N ! e : 62 t N d
+
+
2
@S S
t
T
0
0
Theta. To nd theta we use the Black-Scholes PDE and get
!
r
p
t
qT
t ; d ; ! 63 = !qS e N ! X
+ 0 2
T
!
r
t
rT
!rK e N !X; d ; !
2
T
rt
!rke N ! X
p
1 !
qT
p p
n X n d N ! e : S e t N d +
+ 0
+
2
t
T
Intuitively,we deduce a formula for v by just omitting the terms involving
t and obtain
!
r
p
@v t
qT
= !qS e N ! X t ; d ; ! 64
0 2 +
@ T
!
r
t
rT
!X; d ; ! !rK e N
2
T
!
qT
p
S e n d N ! e :
0 +
2
T
Wenow extend the scale invariance of time equation 38 to
r q
v v v v ; 65 =
r q
t t t 2t
Option Price Sensitivities 13
compare this formula with 63 and can instantly read o vega and rho.
Vega.
h i
p
p p
@v
qT
= S e tn X t N d Tn d N ! e : 66 + !
0 +
+
@
Rho.
!
r
@v t
rT rt
= ! T K e N !X; d ; ! + !tke N ! X 67
2
@r T
!
r
p
@v t
qT
= ! T S e N ! X t ; d ; ! : 68
0 2 +
@q T
5 A Europ ean Claim in the Two-dimensional Black-
Scholes Mo del
5.1 Pricing of a Europ ean Option
Rainbow options are nancial instruments which dep end on several risky assets.
Many of them are very sensitivetochanges of correlation. we call kappa the
derivative of the option value v with resp ect to the correlation .
The computational e ort to compute the kappa is hard, even in a simple framework,
but in the Black-Scholes mo del with two sto cks and one cash b ond we nd a cross-
gamma-correlation-risk relationship which can b e used easily to nd kappa.
Let the sto ck price pro cesses S and S b e describ ed by
1 2
1 S
1
1 1 2
; 69 + W = r q ln
1
1
S 0 2
1
p
S 1
2
2 2 2 1
2
ln 1 W : 70 = r q + W +
2 2
2
S 0 2
2
1 2
W and W are two indep endent Brownian motions under the risk neutral measure.
1
The probability density for the distribution of S is denoted by h x and is given
1
by the log normal density:
2
1 1 A
p
h x = exp ; 71
1
2
2
x 2
2
1
1
1 x
1 2
r + q + : 72 A = ln
1
S 0 2
1
The equation for the second sto ck price pro cess can b e written as
S 1 S 1
2 2 1
2 2 1 2
ln = r q + ln r q
2 1
S 0 2 S 0 2
2 1 1
p
2
2
: 73 + 1 W
2
14 Rei, O. and Wystup, U.
The conditional distribution of S given S isthus log-normal with density
2 1
2
B 1
p
exp ; 74 h S jS =x y =
2 2 1
2
2
2
2
2 1
y 2 1
2
2
y 1
2
2 2
A : 75 B = ln r + q +
2
S 0 2
2 1
The joint distribution of S and S is given by the pro duct of h and h
1 2 1 2
h x; y = h x h S j S =x y : 76
1 2 2 1
A Europ ean option with maturity and payo f S ;S will b e priced by
1 2
1 1
Z Z
r
h x; y f x; y dxdy : 77 v = e
0 0
5.2 Calculation of the Greeks
From the general pricing of a Europ ean claim in a two-dimensional Black-Scholes
market we can compute all the derivatives of the option price. We suppress the
arguments of the functions and we assume, that we are allowed to di erentiate
~
under the integral. Therefore, we only have to di erentiate the function h x; y =
r
e h x; y with resp ect to the parameters resulting into the equations
2
2
~
@ h = E + E E ; 78
4 1 6
2 2
1 1
1
1
2
~
@ h = E + E ; 79
2 5
S 0
1
S 0 S 0
1 1 1
1
~
E ; 80 h = @
5
S 0
2
S 0
2
2 1
2
~
E E + E + E E ; 81 @ h =
1 3 1 2 4 2 5
1
2
1 1
1
1 2
~
@ h = E E + E E ; 82
1 6 2 5 4
2
2 2 1
2
~
1
@ h = E E ; 83
2 5
q
1
~
2
@ h = E ; 84
5
q
2
~
h = E E + @ E ; 85
1 2 r 5
1
1 1 1 1
1 2
~
h = r + @ E E + r + q + E E
1 3 2 6
1
2
1 1
2
2 2 1 2
+ r + q + r + q + E ; 86
5
2 1
2 2 1
Option Price Sensitivities 15
1 2 1
~
E + E E h = @ @
1 2 3
S 0 S 0
1 1
2 2
2 2 2 2
1 S 0 S 0 S 0
1 1 1
1 1
2
2 2
E E +
4 5
3
2 2
S 0 S 0
1 1 1
1
2
2
E ; 87
6
2
2 2
1 S 0
1
1
1
~
@ @ h = E E
1 4
S 0 S 0
1 2
2
2
1 S 0 S 0 S 0 S 0
1 2 1 2 1 2
1
2
+ E ; 88
6
2
1 S 0 S 0
1 2 1 2
1 1
~
E + E h = @ @
1 5
S 0 S 0
2 2
2
2 2 2
1 S 0 S 0
2 2
2
2
E ; 89
6
2
2 2
1 S 0
2
2
where the terms E are de ned by
i
~
E = h; 90
1
2A
~
= h E ; 91
2
2
2
1
2
A
~
; 92 E = h
3
2
2
1
2BA
~
E ; 93 = h
4
2
2
2 1
2
2B
~
E = h ; 94
5
2
2
2 1
2
2
B
~
: 95 E = h
6
2
2
2 1
2
5.3 Conclusions
To obtain several relations b etween the Greeks one only has to do some linear
6
algebra in R . Some results are
0 = 1 + S 0 96
1 1
q
2
+ S 0 97 0 =
2 2
q
1 1
1 2
1 2
0 = q + + r + 98 + q +
1 1 2 2 r
q q
2 2
1 2
0 = rv + r q S 0 + r q S 0
1 1 2 2
1 1
2 2 2 2
S 0 + S 0 S 0 + S 0 99 +
1 11 1 2 1 2 12 2 22
1 2
2 2
= S 0 S 0 100
1 2 1 2 12
16 Rei, O. and Wystup, U.
2 2
0 = + S 0 101
1 1 1 11
1
2 2
0 = + S 0 102
2 2 2 22
2
2 2 2 2
0 = S 0 + S 0 103
1 1 2 2 1 11 2 22
1 2
= v S 0 S 0 104
r 1 1 2 2
0 = v + 1 + 2 + 105
r
q q
Of course one can get more relations by combining some relations ab ove. The
relations wehavechosen to present are either similar to the one-dimensional case
or have another natural interpretation.
96 and 97 . These relations are a justi cation for the rough way to deal
with dividends. One subtracts the dividends from the actual sp ot price and
prices the option with this price and without dividends. This relation is not
e ected by the two-dimensionality of the problem.
98 . This is the two-dimensional version of the general invariance under
time scaling.
99 . This is the Black-Scholes di erential equation. This relation must hold,
b ecause we concentrated on Europ ean claims. It turns out, that the dynamic
of an option price is describ ed by the market mo del and that the price of the
option is de ned as a b oundary problem.
100 . The Greek , the change of the option price b ecause of a change of
the correlation was our motivation to do this calculation. One would exp ect
a relationship b etween and , but it is remarkable, that this relationship
12
has such a simple structure.
101 and 102 . Notice that one can determine the only by the knowledge
of some derivatives with resp ect to parameters which concern only one sto ck.
Of course, there is no di erence b etween the rst and the second sto ck. These
relations are valid in the one-dimensional case with 0.
103 . This is an extension of the vega-gamma relation as derived in the
one-dimensional case, see 50 .
104 . The interest rate risk is well known to b e the negative pro duct of
duration and the amount of money invested. The term in the parentheses is
exactly the amount of money one would havetoinvest in the cash b ond in
order to delta-hedge the option.
105 . This relation is the two-dimensional rates symmetry, an extension of
equation 44 .
5.4 Cross-Gamma and Correlation Risk
The simplicity of equation 100 is based on a well-known relationship of multivari-
ate normal distribution functions published in [4], whichsays the following. We
Option Price Sensitivities 17
supp ose that the vector X of n random variables with means zero and unit vari-
ances has a nonsingular normal multivariate distribution with probability density
function
1 1 1
T n
2 2
x Cx : 106 x ;:::;x ; c ;:::;c = 2 exp jCj
n 1 n 11 nn
2
Here C is the inverse of the covariance matrix of X , which is denoted by R, and
has elements f g. Then the following identity can b e proved easily by writing the
ij
density in terms of its characteristic function.
2
@ @
n n
= : 107
@ @x @x
ij i j
In the two-dimensional case this reads as
2
@ n x; y ; @n x; y ;
2 2
= ; 108
@ @x@y
which can b e extended readily to the corresp onding cumulative distribution func-
tion, i.e.,
2
@ N x; y ; @ N x; y ;
2 2
= = n x; y ; : 109
2
@ @x@y
6 Examples in the Two-Dimensional Case
6.1 Europ ean Options on the Minimum/Maximum of Two
Assets
We consider the payo
h i
+
1 2
: 110 min S ; S K
T T
This is a Europ ean put or call on the minimum = +1 or maximum = 1 of
1 2
the two assets S and S with strike K . As usual, the binary variable takes
T T
the value +1 for a call and 1 for a put. Its value function has b een published
in [5] and can b e written as
1 2
1 2
v t; S ;S ;K;T;q ;q ;r; ; ;; ;
1 2
t t
h
1
1
q
= S e N d ; d ;
2 1 3 1
t
2
2
q
+ S e N d ; d ;
2 2 4 2
t
p p
1
r
Ke + N d ; d ; ; 111
2 1 1 2 2
2
2 2 2
2 ; 112 + =
1 2
2 1
18 Rei, O. and Wystup, U.
2 1
; 113 =
1
1 2
= ; 114
2
1
1
1 2
=K + r q + ln S
t 1
2
p
; 115 = d
1
1
2
1
2 2
ln S =K + r q +
2 t
2
p
; 116 d =
2
2
2 1
1
1 2 2
ln S =S + q q
t t
2
p
d = ; 117
3
1 2
1
2 1 2
ln S =S + q q
t t
2
p
d = : 118
4
6.1.1 Greeks
Space homogeneity implies that
@v @v @v
2 1
+ S + K : 119 v = S
t t
1 2
@K
@S @S
t t
Using this equation we can immediately write down the deltas
1
@v
q
= e N d ; d ; ; 120
2 1 3 1
1
@S
t
2
@v
q
= e N d ; d ; ; 121
2 2 4 2
2
@S
t
p p
@v 1
r
= e + N d ; d ; : 122
2 1 1 2 2
@K 2
Computing the gammas is actually the only situation where di erentiation is needed.
We use the identities
!
@ y x
p
N x; y ; = n x N ; 123
2
2
@x
1
!
x y @
p
N x; y ; = n y N ; 124
2
2
@y
1
and obtain
" !
1
2 q
@ v d d e
3 1 1
p
n d N 125 =
1
p
1 1
2
2
1 1
@ S S
2
t t
Option Price Sensitivities 19
!
d d
1 3 1
p
n d N ;
3
2
1
2
! "
2
2 q
@ v d d e
4 2 2
p
126 n d N =
2
p
2 2
2
2
2
1
@ S S
1
t t
!
d d
2 4 2
p
n d N ;
4
2
1
1
!
1
2 q
d d @ v e
1 3 1
p
: 127 n d N =
3
p
1 2 2
2
1
@S @S S
2
t t t
The sensitivity with resp ect to correlation is directly related to the cross-gamma
2
@v @ v
1 2
: 128 = S S
1 2
t t
1 2
@
@S @S
t t
We refer to 101 and 102 to get the following formulas for the vegas,
1
2 2
v v + S
1 1
t 1
@v
S S
t t
= 129
@
1 1
! "
p
1
d d
1 3 1
1
q
p
130 e = S n d N
1 3
t
2
1
2
!
d d
3 1 1
p
+ n d N ;
1
2
1
2
2
2 2
S v v +
2 2
2 t
@v
S S
t t
= 131
@
2 2
" !
p
2
d d
2 4 2
2
q
p
= S n d N 132 e
2 4
t
2
1
1
!
d d
4 2 2
p
+ n d N :
2
2
1
1
Lo oking at 96 and 97 the rhos are given by
@v @v
1
; 133 = S
t
1
1
@q
@S
t
@v @v
2
; 134 = S
t
2
2
@q
@S
t
@v @v
= K : 135
@r @K
20 Rei, O. and Wystup, U.
Among the various ways to compute theta one may use the one based on 98 .
h i
1 @v
1 2
2 1
2 1
+ rv + + q v q v : 136 = v v +
r
q q
1 2
@t 2 2
6.2 Outside Barrier Options
The payo of an outside barrier option is
+
[ S T K ] II 137
1
fmin S t > B g
2
0tT
This is a Europ ean put or call with strike K and a kno ck-out barrier H in a second
asset, called the outer asset. As usual, the binary variable takes the value +1 for
a call and 1 for a put and the binary variable takes the value +1 for a lower
barrier and 1 for an upp er barrier. We abbreviate
i
= r q : 138
i
This is an example of a path-dep endent rainbow option, a case which is not covered
by the previous theory.However, it is illuminating to see how some of the ideas
are still successfully applicable. The value provided by Heynen and Kat has b een
published in [3].
1
q T
V = S 0 e N d ; e ;
0 1 2 1 1
1
2 + ln H=S 0
2 1 2 2
0 q T 0
; ; e S 0 e exp N d
1 2
1 1
2
2
rT
K e N d ; e ;
2 2 2
2 ln H=S 0
2 2
rT 0 0
+ K e exp N d ; e ; ; 139
2
2 2
2
2
2
ln S 0 =K + + T
1 1
1
p
; 140 d =
1
T
1
p
d = d T; 141
2 1 1
2 ln H=S 0
2
0
p
d ; 142 = d +
1
1
T
2
2 ln H=S 0
2
0
p
; 143 = d + d
2
2
T
2
ln H=S 0 + T
2 2 1 2
p
e = ; 144
1
T
2
p
e = e + T; 145
2 1 1
2 ln H=S 0
2
0
p
e = e ; 146
1
1
T
2
2 ln H=S 0
2
0
p
e : 147 = e
2
2
T 2
Option Price Sensitivities 21
6.2.1 Greeks
Homogeneity in price tells us that
@V @V
0 0
+ K : 148 V = S 0
0 1
@S 0 @K
1
This allows us to read o the delta inner sp ot
1
@V
0
q T
= e N d ; e ; 149
2 1 1
@S 0
1
1
2 + ln H=S 0
2 1 2 2
0 0 q T
N d ; e ; e exp
2
1 1
2
2
and the dual delta inner strike
@V
0
rT
= e N d ; e ; 150
2 2 2
@K
2 ln H=S 0
2 2
rT 0 0
+ e exp N d ; e ; :
2
2 2
2
2
7 Generalization to Higher Dimensions and other
Market Mo dels
7.1 Multidimensional Black-Scholes Mo del
7.1.1 Scale Invariance of Time
The general idea presented in the one dimensional case is valid in higher dimensions
to o. Therefore wehave got the relation, which hold for all a>0:
v x ; :::; x ; ;r;q ; :::; q ; ; :::; =
1 n 1 n 11 nn
p p
v x ; :::; x ; a ; :::; a 151 ;ar;aq ; :::; aq ;
1 n 11 nn 1 n
a
We di erentiate with resp ect to a and evaluate at a =1:
n n
X X
1
0= + r + q 152 +
i q ij ij
i
2
i=1 i;j =1
denotes the di erentiation of v with resp ect to .
ij ij
7.1.2 Price Homogeneity
Equations 39 and 40 easily extend to
n m
X X
k
v = x + k 153
i i j
j
i=1 j =1
22 Rei, O. and Wystup, U.
n m
X X
k
x x = k k 154
i j ij i j
ij
i;j =1 i;j =1
for strike-de ned options, and equations 41 and 42 to
n m
X X
l
0= x + l 155
i i j
j
i=1 j =1
n n m m
X X X X
l l
x x + x = l l + l 156
i j ij i i i j i
ij i
i;j =1 i=1 i;j =1 i=1
for level-de ned options.
7.1.3 Europ ean Options in the Black-Scholes Mo del
The previous homogeneity based relations are mo del-indep endent. In the Black-
Scholes mo del wemay furthermore invoke the multidimensional Black-Scholes PDE
n n
X X
1
T
x x @ @ v 157 0 = v rv + x r q @ v +
ij i j x x i i x
i j i
2
i;j =1 i=1
to compute Greeks.
7.2 Beyond Black-Scholes
Up to nowwe illustrated our ideas in the Black-Scholes mo del and in some parts
we used sp eci c prop erties of this mo del. Nevertheless there are some prop erties,
which are so fundamental, that they should hold in any realistic market mo del.
These fundamental prop erties are the homogeneity of time, the scale invariance
of time and the scale invariance of prices. For every market mo del one uses, one
should check, if the mo del ful lls these prop erties.
An example for a market mo del with non-deterministic volatility is Heston's sto chas-
tic volatility mo del [2].
In this more general framework one needs to clarify the notion of vega, rho etc. A
change of volatility could mean a change of the entire underlying volatility pro cess.
If the pricing formula dep ends on input parameters such as initial volatility,volatil-
ityofvolatility, mean reversion of volatility, then one can consider derivatives with
resp ect to such parameters. It turns out that our strategy to compute Greeks can
still b e applied successfully in a sto chastic volatility mo del.
7.3 Heston's Sto chastic Volatility Mo del
h i
p
1
dt + v t dW ; 158 dS = S
t t
t
p
2
; 159 dv = v dt + v t dW
t t t
Option Price Sensitivities 23
h i
1 2
Cov dW ;dW = dt; 160
t t
S; v ; t = v : 161
The mo del for the variance v is the same as the one used by Cox, Ingersol l and
t
Ross for the short term interest rate. 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 market
price of volatility risk.
Heston provides a closed-form solution for Europ ean vanilla options paying
+
[ S K ] : 162
T
As usual, the binary variable takes the value +1 for a call and 1 for a put, K
the strike in units of the domestic currency, q the risk free rate of asset S , r the
domestic risk free rate and T the expiration time in years.
7.3.1 Abbreviations
a = 163
1
164 u =
1
2
1
u = 165
2
2
b = + 166
1
b = + 167
2
q
2 2 2
d = 'i b 2u 'i ' 168
j j j
b 'i + d
j j
g = 169
j
b 'i d
j j
= T t 170
d
j
b 'i + d 1 e
j j
D ; ' = 171
j
d
2
j
1 g e
j
C ; ' = r q 'i 172
j
d
j
1 g e a
j
b 'i + d 2ln +
j
2 d
j
1 e
x =lnS 173
t
C ;' +D ;' v +i'x
j j
f x; v ; t; ' = e 174
j
Z
1
i'y
1 e f x; v ; ; ' 1
j
d' 175 + < P x; v ; ; y =
j
2 i'
0
Z
1
1
i'y
p x; v ; ; y = < e d' 176 f x; v ; ; '
j j
0
24 Rei, O. and Wystup, U.
1
P = + P ln S ;v ; ;ln K 177
+ 1 t t
2
1
+ P ln S ;v ; ;ln K 178 P =
2 t t
2
This notation is motivated by the fact that the numb ers P are the cumulative
j
distribution functions in the variable y of the log-sp ot price after time starting
at x for some drift . The numb ers p are the resp ective densities.
j
7.3.2 Value
The value function for Europ ean vanilla options is given by
q r
V = e S P Ke P 179
t +
The value function takes the form of the Black-Scholes formula for vanilla options.
The probabilities P corresp ond to N d in the constantvolatility case.
7.3.3 Greeks
The deltas can b e obtained based on the homogeneity of prices.
Sp ot delta.
@V
q
= = e P 180
+
@S
t
Dual delta.
@V
r
= e P 181
@K
Gamma.
q
@V e @x @
= = p ln S ;v ; ;ln K 182 =
1 t t
@S @x @S S
t t t
As in the case of vanilla options we observe that
q r
S e p ln S ;v ; ;ln K =Ke p ln S ;v ; ;ln K ; 183
t 1 t t 2 t t
Rho. Rho is connected to delta via equations 48 and 49 .
@V
r
= K e P ; 184
@r
@V
q
= S e P : 185
t + @q
Option Price Sensitivities 25
Theta. Theta can b e computed using the partial di erential equation for the He-
ston vanilla option
1 1
2
vV + vS V + vSV qV V + r q SV +
vv SS vS t S
2 2
+[ v ]V =0; 186
v
where the derivatives with resp ect to initial variance v must b e evaluated
numerically.
8 Summary
Wehave learned how to employ homogeneity-based metho ds to 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 mo del there
are numerous relations b etween various Greeks which are of fundamental interest.
The metho d helps saving computation time for the mathematician who has to
di erentiate complicated formulas as well as for the computer, b ecause analytical
results for Greeks are usually faster to evaluate than nite di erences involving at
least twice the computation of the option's value. Knowing how the Greeks are
related to each other can sp eed up nite-di erence-, tree-, or Monte Carlo-based
computation of Greeks or lead at least to a qualitycheck. Many of the results
are valid b eyond the Black-Scholes mo del. Most remarkably some relations of the
Greeks are based on prop erties of the normal distribution refreshing the active
interplaybetween mathematics and nancial markets to our very pleasure.
References
[1] GESKE, R. 1979 . The Valuation of Comp ound Options. Journal of Financial
Economics. 7, 63-81.
[2] HESTON, S. 1993 . A Closed-Form Solution for Options with Sto chastic
Volatility with Applications to Bond and Currency Options. The Review of
Financial Studies,Vol. 6, No. 2.
[3] HEYNEN, R. and KAT, H. 1994 . Crossing Barriers. RISK. 7 6 , pp. 46-51.
Risk publications, London.
[4] PLACKETT, R. L. 1954 . A Reduction Formula for Normal Multivariate
Integrals. Biometrika. 41, pp. 351-360.
[5] STULZ, R. 1982 . Options on the Minimum or Maximum of Two Assets.
Journal of Financial Economics. 10, pp. 161-185.
[6] TALEB, N. 1996 . Dynamic Hedging. Wiley, New York.
[7] WYSTUP, U. 1999 . Vanil la Options. Formula Catalogue of
http://www.math nance.de.