-
Wilmott magazine 49
ing millions of dollars-can only be learnedthrough real action.
Now, the manual:
BSD trader Solider, welcome to our tradingteam, this is your
first day and I will instructyou about the Black-Scholes
weapon.
New hired Trader Hah, my Professortaught me probability theory,
It calculus,and Malliavin calculus! I know everythingabout
stochastic calculus and how to comeup with the Black-Scholes
formula.
BSD trader Solider, you may know how toconstruct it, but that
doesnt mean you knowa shit about how it operates!
New hired Trader I have used it for real trad-ing. Before my
Ph.D. I was a market maker instock options for a year. Besides, why
do you callme solider? I was hired as an option trader.
BSD trader Solider, you have not been inreal war. In real war
you often end up inextreme situations. Thats when you need toknow
your weapon.
New hired Trader I have read Liars Poker,Hulls book, Wilmott on
Wilmott, TalebsDynamic Hedging, Haugs formula collec-tion. I know
about Delta Bleed and all thatstuff. I dont think you can tell me
muchmore. I have even read Fooled by Ran . . .
BSD trader SHUT UP SOLIDER! If you wantto survive the first six
months on this trading
floor you better listen to me. On this team wedont allow any
mistakes. We are warriors,trained in war!
New hired Trader Yes Sir!
BSD trader Good, lets move on to our busi-ness. today I will
teach you the basics of theBlack-Scholes weapon.
1 Background on the BSM formulaLet me shortly refresh your
memory of the BSMformula
c = Se(br)T N(d1) XerT N(d2)p = XerT N(d2) Se(br)T N(d1),
where
d1 = ln(S/X) + (b + 2/2)T
T,
d2 = d1
T,
and
S = Stock price.X = Strike price of option.r = Risk-free
interest rate.b = Cost-of-carry rate of holding the underlying
security .
T = Time to expiration in years.
Trading options is War! For anoption trader a pricing or
hedgingformula is just like a weapon. Asolider who has perfected
her pis-tol shooting1 can beat a guy with amachine gun that doesnt
know
how to handle it. Similarly, an option traderknowing the ins and
outs of the Black-Scholes-Merton (BSM) formula can beat a trader
using astate-of-the-art stochastic volatility model. Itcomes down
to two rules, just as in war. Rulenumber one: Know your weapon.
Rule numbertwo: Dont forget rule number one. In my ten+year as a
trader I have seen many a BSD2 optiontrader getting confused with
what the computerwas spitting out. They often thought somethingwas
wrong with their computer system/imple-mentation. Nothing was
wrong, however, excepttheir knowledge of their weapon. Before
youmove on to a more complex weapon (like a sto-chastic volatility
model) you should make sureyou know conventional equipment
inside-out. Inthis installment I will not show the nerdy quantshow
to come up with the BSM formula using somenew fancy mathematicsyou
dont need to knowhow to melt metal to use a gun. Neither is it
aguideline on how to trade. It is meant rather likea short manual
of how your weapon works inextreme situations. Real war
(trading)-the pain,the pleasure, the adrenaline of winning and
loos-
THE COLLECTOR:
To this article I got a lot of ideas from the Wilmott forum.
Thanks! And especially thanks to Jrgen Haug and James Ward for
useful comments on this paper.
Know Your
Weapon Part 1
^
Espen Gaarder Haug
-
ESPEN GAARDER HAUG
50 Wilmott magazine
= Volatility of the relative price changeof the underlying stock
price.
N(x) = The cumulative normal distributionfunction .
2 Delta Greeks
2.1 Delta
As you know, the delta is the options sensitivityto small
movements in the underlying assetprice.
call = cS
= e(br)T N(d1) > 0
put = pS
= e(br)T N(d1) < 0
Delta higher than unity I have many times overthe years been
contacted by confused commodi-ty traders claiming something is
wrong withtheir BSM implementation. What they observedwas a spot
delta higher than one.
As we get deep-in-the-money N(d1) approach-es one, but it never
gets higher than one (sinceits a cumulative probability function).
For aEuropean call option on a non-dividend-payingstock the delta
is equal to N(d1), so the delta cannever go higher than one. For
other options thedelta term will be multiplied by e(br)T . If
thisterm is larger than one and we are deep-in-the-money we can get
deltas considerable higherthan one. This occurs if the
cost-of-carry is largerthan the interest rate, or if interest rates
are neg-ative. Figure 1 illustrates the delta of a calloption. As
expected the delta reaches aboveunity when time to maturity is
large and theoption is deep-in-the-money.
2.2 Delta mirror strikes and assetFor a put and call to have the
same absolute deltavalue we can find the delta symmetric strikes
as
Xp = S2
Xce(2b+
2 )T , Xc = S2
Xpe(2b+
2 )T .
That is
c(S, Xc, T, r, b, ) = p(S, S2
Xce(2b+
2 )T , T, r, b, ).
where Xc is the strike of the call and Xp is thestrike of a put.
These relationships are useful to
determine strikes for delta neutral option strate-gies,
especially for strangles, straddles, and but-terflies. The weakness
of this approach is that itworks only for a symmetric volatility
smile. Inpractice, however, you often only need an approx-imately
delta neutral strangle. Moreover, volatili-ty smiles often are more
or less symmetric in thecurrency markets.
In the special case of a straddle-symmetric-delta-strike,
described by Wystrup (1999), the for-mulas above can be simplified
further to
Xc = Xp = Se(b+ 2/2)T .
Related to this relationship is the
straddle-symmetric-asset-price. Given the identical strikesfor a
put and call, for what asset price will theyhave the same absolute
delta value? The answer is
S = Xe(b 2/2)T .
At this strike and delta-symmetric-asset-price thedelta is e
(br)T
2 for a call, and e(br)T
2 for a put. Onlyfor options on non-dividend paying stocks3 (b =
r)can we simultaneously have an absolute delta of
0.5 (50%) for a put and a call. Interestingly, thedelta
symmetric strike also is the strike given theasset price where the
gamma and vega are at theirmaximums, ceteris paribus. The maximal
gammaand vega,4 as well as the delta neutral strikes, arenot
at-the-money forward as I have noticedassumed by many traders.
Moreover, an in-the-money put can naturally have absolute
deltalower than 50% while an out-of-the-money callcan have delta
higher than 50%.
For an option that is at the straddle-symmetric-delta-strike the
generalized BSM formula can besimplified to
c = Se(br)T
2 XerT N(T),
and
p = XerT N(T) Se(br)T
2.
At this point the option value will not changebased on changes
in cost of carry (dividend yieldetc). This is as expected as we
have to adjust thestrike accordingly.
0
150
300
450
600
0306
09012
015018
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Days to maturity
Asset price
X = 100, r = 5%, b = 30%, = 25%,
Figure 1. Spot Delta
-
Wilmott magazine 51
^
2.3 Strike from delta
In several OTC (over-the-counter) markets optionsare quoted by
delta rather than strike. This is acommon quotation method in, for
example, theOTC currency options market, where one typicallyasks
for a delta and expects the sales person toreturn a price (in terms
of volatility or pips) as wellas the strike, given a spot
reference. In these casesone needs to find the strike that
corresponds to agiven delta. Several option software systems
solvesthis numerically using Newton-Raphson or bisec-tion. This is
actually not necessary, however. Usingan inverted cumulative normal
distribution N1()the strike can be derived from the delta
analytical-ly as described by Wystrup (1999). For a call option
Xc = S exp[N1(ce(rb)T)
T + (b + 2/2)T],
and for a put we have
Xp = S exp[N1(pe(rb)T)
T + (b + 2/2)T].
To get a robust and accurate implementation ofthis formula it is
necessary to use an accurateapproximation of the inverse cumulative
nor-mal distribution. I have used the algorithm ofMoro (1995) with
good results.
2.4 DdeltaDvol and DvegaDvol
DdeltaDvol:
which mathematically is thesame as DvegaDspot: vega
S , a.k.a. Vanna,5 shows
approximately how much your delta will changefor a small change
in the volatility, as well ashow much your vega will change with a
smallchange in the asset price:
DdeltaDvol = cS
= pS
= e(br)T d2
n(d1),
where n(x) is the standard normal density
n(x) = 12
ex2/2.
One fine day in the dealing room my risk manag-er asked me to
get into his office. He asked mewhy I had a big outright position
in some stockindex futures-I was supposed to do arbitragetrading.
That was strange as I believed I was deltaneutral: long call
options hedged with short
index futures. I knew the options I had were farout-of-the-money
and that their DdeltaDvol wasvery high. So I immediately asked what
volatili-ty the risk management used to calculate theirdelta. As
expected, the volatility in the risk-man-agement-system was
considerable below the mar-ket and again was leading to a very low
delta forthe options. This example is just to illustrate howa
feeling of your DdeltaDvol can be useful. If youhave a high
DdeltaDvol the volatility you use tocompute your deltas becomes
very important.6
Figure 2 illustrates the DdeltaDvol. As we cansee the DdeltaDvol
can assume positive and neg-ative values. DdeltaDvol attains its
maximalvalue at
SL = XebT
T
4+T 2/2,
and attains its minimal value when
SU = XebT+
T
4+T 2/2.
Similarly, given the asset price, options withstrikes XL have
maximum negative DdeltaDvol at
XL = SebT
T
4+T 2/2,
and options with strike XU have maximum posi-tive DdeltaDvol
when
XU = SebT+
T
4+T 2/2.
One naturally can ask if these measures have anymeaning? Black
and Scholes assumed constantvolatility, or at most deterministic
volatility.Despite being theoretically inconsistent it mightwell be
a good approximation. How good anapproximation it is I leave up to
you to find out ordiscuss at the Wilmott forum, www.wilmott.com.
Formore practical information about DvegaDspot orVanna see Webb
(1999).
2.5 DdeltaDtime, Charm
DdeltatDtime, a.k.a. Charm (Garman 1992) orDelta Bleed (a term
used in the excellent book byTaleb 1997), is deltas sensitivity to
changes intime,
cT
= e(br)T[
n(d1)
(b
T d2
2T
)
+ (b r)N(d1)] 0,
0
110
220
330
50658095
110
125
140
0.015
0.01
0.005
0
0.005
0.01
0.015
Days to maturity
Asset price
X = 100, r = 5%, b = 0%, = 20%,
Figure 2. DdeltaDvol
-
ESPEN GAARDER HAUG
52 Wilmott magazine
and
pT
= e(br)T[
n(d1)
(b
T d2
2T
)
(b r)N(d1)] 0.
This Greek gives an indication of what happenswith delta when we
move closer to maturity.Figure 3 illustrates the Charm value for
differentvalues of the underlying asset and different timeto
maturity.
As Nassim Taleb points out one can have bothforward and backward
bleed. He also points outthe importance of taking into account
howexpected changes in volatility over the giventime period will
affect delta. I am sure most read-ers already have his book in
their collection (ifnot, order it now!). I will therefore not
repeat allhis excellent points here.
All partial derivatives with respect to timehave the advantage
over other Greeks in that weknow which direction time will move.
Moreover,we know that time moves at a constant rate. Thisis in
contrast, for example, to the spot price,volatility, or interest
rate.7
2.6 Elasticity
The elasticity of an option, a.k.a. the option lever-age, omega,
or lambda, is the sensitivity in per-cent to a percent movement in
the underlyingasset price. It is given by
call = call Scall
= e(br)T N(d1) Scall
> 1
put = put Sput
= e(br)T N(d1) Sput
< 0
The options elasticity is a useful measure on itsown, as well as
to estimate the volatility, beta,and expected return from an
option.
Option volatility The option volatility o can beapproximated
using the option elasticity. Thevolatility of an option over a
short period of timeis approximately equal to the elasticity of
theoption multiplied by the stock volatility ..8
o ||.
Option Beta The elasticity also is useful to com-pute the
options beta. If asset prices follow geo-metric Brownian motions
the continuous-time
capital asset pricing model of Merton (1971)holds. Expected
asset returns then satisfy theCAPM equation
E[return] = r + E[rm r]i
where r is the risk free rate, rm is the return on themarket
portfolio, and i is the beta of the asset. Todetermine the expected
return of an option weneed the options beta. The beta of a call is
givenby (see for instance Jarrow and Rudd 1983)
c = Scall
cS,
where S is the underlying stock beta. For a putthe beta is
p = Sput
pS.
For a beta neutral option strategy the expectedreturn should be
the same as the risk-free-rate (atleast in theory).
Option Sharpe ratios As the leverage does notchange the Sharp
(1966) ratio, the Sharpe ratioof an option will be the same as that
of theunderlying stock,
o ro
= S r
.
where o is the return of the option, and S isthe return of the
underlying stock. This rela-tionship indicates the limited
usefulness of theSharpe ratio as a risk-return measure foroptions
(?). Shorting a lot of deep out-of-the-money options will likely
give you a niceSharpe ratio, but you are almost guaranteed toblow
up one day (with probability one if youlive long enough). An
interesting question hereis if you should use the same volatility
for allstrikes. For instance deep-out-of-the-moneystock options
typically trade for much higherimplied volatility than at-the-money
options.Using the volatility smile when computingSharpe ratios for
deep out-of-the-moneyoptions also possibly can make the Sharperatio
work better for options. McDonald (2002)offers a more detailed
discussion of optionSharpe ratios.
10
43
76
109
5063758
8100113125138150
5
4
3
2
1
0
1
2
3
4
5
Days to maturityAsset price
X = 100, r = 5%, b = 0%, = 30%,
h
Figure 3. Charm
-
Wilmott magazine 53
^
3 Gamma Greeks3.1 Gamma
Gamma is the deltas sensitivity to small move-ments in the
underlying asset price. Gamma isidentical for put and call options,
ceteris paribus,and is given by
call ,put = 2c
S2=
2p
S2= n(d1)e
(br)T
S
T> 0
This is the standard gamma measure given inmost text books (Haug
1997, Hull 2000, Wilmott2000).
3.2 Maximal gamma and the illu-sions of risk
One day in the trading room of a former employ-er of mine, one
of the BSD traders suddenly gotworried over his gamma. He had a
long dateddeep-out-of-the money call. The stock price hadbeen
falling, and the further the out-of-the-money the option went the
lower the gamma heexpected. As with many option traders hebelieved
the gamma was largest approximatelyat-the-money-forward. Looking at
his Bloombergscreen, however, the further out of the moneythe call
went the higher his gamma got. AnotherBSD was coming over, and they
both tried tocome up with an explanation for this. Was
theresomething wrong with Bloomberg?
In my own home-built system I often wasplaying around with 3 and
4-dimensionalcharts of the option Greeks, and I already knewthat
gamma doesnt attain its maximum at-the-money forward (4 dimensions?
a dynamic 3-dimensional graph). I didnt know exactlywhere it
attained its maximum, however.Instead of joining the BSD
discussion, I did afew computations in Mathematica. A few min-utes
later, after double checking my calcula-tions, I handed over an
equation to the BSDtraders showing exactly where the BSM gammawould
be at its maximum.
How good is the rule of thumb that gamma islargest for
at-the-money or at-the-money-forwardoptions? Given a strike price
and time to maturi-ty, the gamma is at maximum when the assetprice
is9
S = Xe(b32/2)T .
Given the asset price and time to maturity,gamma is maximal when
the strike is
X = Se(b+2/2)T .
Confused option traders are bad enough, con-fused
risk-management is a pain in the behind.Several large investment
firms impose risk limitson how much gamma you can have. In the
equitymarket it is common to use the standard text-book approach to
compute gamma, as shownabove. Putting on a long term call (put)
optionthat later is deep-out-of-the money (in-the-money) can blow
up the gamma risk limits, evenif you actually have close to zero
gamma risk.The high gamma risk for long dated deep-out-of-the-money
options typically is only an illusion.This illusion of risk can be
avoided by looking atpercentage changes in the underlying
asset(gammaP), as is typically done for FX options.
Saddle Gamma Alexander (Sasha) Adamchukwas the first to make me
aware of the fact thatgamma has a saddle point.10 The saddle point
isattained for the time
TS = 12( 2 + b) ,
and at asset price
S = Xe(b32/2)TS .
The gamma at this point is given by
S = (S , TS) =e(br)T
e
b 2+ 1
X
Many traders get surprised by this feature ofgamma-that gamma is
not necessary decreas-ing with longer time to maturity. The
maximumgamma for a given strike price is first decreasinguntil the
saddle gamma point, then increasingagain, given that we follow the
edge of the maxi-mal gamma asset price.
Figure 4 shows the saddle gamma. The saddlepoint is between the
two gamma mountaintops. This graph also illustrates one of the
biglimitations in the textbook gamma definition,which is actually
in use by many option systemsand traders. The gamma increases
dramaticallywhen we have long time to maturity and theasset price
is close to zero. How can the gammabe larger than for an option
closer to at-the-money? Is the real gamma risk that big? No, thisis
in most cases simply an illusion, due to the
10
487
964
1441
0357
010514
0175
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Days to maturity
Asset price
X = 100, r = 5%, b = 5%, = 80%,
Figure 4. SaddleGamma
-
ESPEN GAARDER HAUG
54 Wilmott magazine
above unmotivated definition of gamma.Gamma is typically defined
as the change indelta for a one unit change in the asset price.When
the asset price is close to zero a one unitchange is naturally
enormous in percent of theasset price. In this case it is also
highly unlikelythat the asset price will increase by one dollar
inan instant. In other words, the gamma measure-ment should be
reformulated, as many optionsystems already have done. It makes far
moresense to look at percentage moves in the under-lying than unit
moves. To compare gamma riskfrom different underlyings one should
alsoadjust for the volatility in the underlying.
3.3 GammaP
As already mentioned, there are several prob-lems with the
traditional gamma definition. Abetter measure is to look at
percentage changesin delta for percentage changes in the
underly-ing,11 for example: a one percent point change
inunderlying. With this definition we get for bothputs and calls
(gamma Percent)
P = S100
> 0 (1)
GammaP attains a maximum at an asset price of
SP = Xe(b2/2)T
Alternatively, given the asset price the maximalP occurs at
strike
XP = Se(b+2/2)T .
Interestingly, this also is where we have a strad-dle symmetric
asset price as well as maximalgamma. This implies that a delta
neutral strad-dle has maximal P . In most circumstancesgoing from
measuring the gamma risk as Pinstead of gamma we avoid the illusion
of a highgamma risk when the option is far out-of-the-money and the
asset price is low. Figure 5 is anillustration of this, using the
same parameters asin Figure 4.
If the cost-of-carry is very high it is still possi-ble to
experience high P for deep-out-of-the-money call options with a low
asset price and along time to maturity. This is because a high
cost-of-carry can make the ratio of a deep-out-of-the
money call to the spot close to the at-the-money-forward. At
this point the spot-delta will be closeto 50% and so the P will be
large. This is not anillusion of gamma risk, but a reality. Figure
6shows P with the same parameters as in Figure 5,with cost-of-carry
of 60%.
To makes things even more complicated thehigh P we can have for
deep-out-of-the-money
calls (in-the-money puts) is only the case whenwe are dealing
with spot gammaP (change inspot delta). We can avoid this by
looking atfuture/forward gammaP. However if you hedgewith spot,
then spot gammaP is the relevantmetric. Only if you hedge with the
future/for-ward the forward gammaP is the relevant met-ric. The
forward gammaP we have when the
10
408
805
1203
1600
0306
09012
015018
0
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Days to maturity
Asset price
X = 100, r = 5%, b = 5%, = 80%,
Figure 5. GammaP
20
494
968
1442
0357
010514
0175
0
0.005
0.01
0.015
0.02
0.025
X = 100, r = 5%, b = 60%, = 80%,
Days to maturity
Asset price
Figure 6. SaddleGammaP
-
Wilmott magazine 55
^
cost-of-carry is set to zero, and the underlyingasset is the
futures price.
3.4 Gamma-symmetry
Given the same strike the gamma is identical forboth put and
call options. Although this equalitybreaks down when the strikes
differ, there is auseful put and call gamma symmetry. The put-call
symmetry of Bates (1991) and Carr and Bowie(1994) is given by
c(S, X, T, r, b, ) = XSebT
p(S,(SebT )2
X, T, r, b, )
This put-call value symmetry yields the gammasymmetry, however
the gamma symmetry is moregeneral as it is independent of wether
the optionis a put or call, for example, it could be two calls,two
puts, or a put and a call.
(S, X, T, r, b, ) = XSebT
(S,(SebT )2
X, T, r, b, ).
Interestingly, the put-call symmetry also gives usvega and
cost-of-carry symmetries, and in thecase of zero cost-of-carry also
theta and rho sym-metry. Delta symmetry, however, is not
obtained.
3.5 DgammaDvol, Zomma
DgammaDvol, a.k.a. Zomma, is the sensitivity ofgamma with
respect to changes in impliedvolatility. In my view, DgammaDvol is
one of themore important Greeks for options trading. It isgiven
by
DgammaDvol call ,put =
= (
d1d2 1
) 0.
For the gammaP we have DgammaPDvol
DgammaPDvol call ,put = P(
d1d2 1
) 0
where is the text book Gamma of the option.For practical
purposes, where one typically
wants to look at DgammaDvol for a one unitvolatility change, for
example from 30% to 31%,one should divide the DGammaDVol by
100.Moreover, DgammaDvol and DgammaPDvol arenegative for asset
prices between SL and SU and
positive outside this interval, where
SL = XebT
T
4+T 2/2,
SU = XebT+
T
4+T 2/2
For a given asset price the DgammaDvol andDgammaPDvol are
negative for strikes between
XL = SebT
T
4+T 2/2,
XU = SebT+
T
4+T 2/2,
and positive for strikes above XU or below XL ,ceteris paribus.
In practice, these points willchange with other variables and
parameters.These levels should, therefore, be consideredgood
approximations at best.
In general you want positive DgammaDvol-especially if you dont
need to pay for it (f latvolatility smile). In this respect
DgammaDvolactually offers a lot of intuition for how stochas-tic
volatility should affect the BSM values (?).Figure 7 illustrates
this point. The DgammaDvolis positive for deep-out-of-the-money
options,outside the SL and SU interval. For at-the moneyoptions and
slightly in- or out-of the moneyoptions the DgammaDvol is negative.
If thevolatility is stochastic and uncorrelated with theasset price
then this offers a good indication forwhich strikes you should use
higher/lowervolatility when deciding on your volatility smile.
In the case of volatility correlated with the assetprice this
naturally becomes more complicated.
3.6 DgammaDspot, Speed
I have heard rumors about how being on speedcan help see higher
dimensions that are ignoredor hidden for most people. It should be
of littlesurprise that in the world of options the thirdderivative
of the option price with respect tospot, known as Speed, is ignored
by most people.Judging from his book, Nassim Taleb is also a fanof
higher order Greeks. There he mentionsGreeks of up to seventh
order.
Speed was probably first mentioned by Garman(1992),12 for the
generalized BSM formula we get
3c
S3=
(1 + d1
T
)S
A high Speed value indicates that the gamma isvery sensitive to
moves in the underlying asset.Academics typically claim that third
or higherorder Greeks are of no use. For an optiontrader, on the
other hand, it can definitelymake sense to have a sense of an
optionsSpeed. Interestingly, Speed is used by Fouque,Papanicolaou,
and Sircar (2000) as a part of astochastic volatility model
adjustment. More tothe point, Speed is useful when gamma is at
its
10
133
255
5065809
5
110125140
0.3
0.25
0.2
0.15
0.1
0.05
0
0.05
0.1
0.15
Days to maturity
Asset price
X = 100, r = 5%, b = 0%, = 30%,
Figure 7. DgammaDvol
-
ESPEN GAARDER HAUG
56 Wilmott magazine
maximum with respect to the asset price.Figure 8 shows the graph
of Speed with respectto the asset price and time to maturity.
For P we have an even simpler expression forSpeed, that is
SpeedP (Speed for percentagegamma)
SpeedP = d1100
T.
3.7 DgammaDtime, Colour
The change in gamma with respect to smallchanges in time to
maturity, DGammaDtimea.k.a. GammaTheta or Colour (Garman 1992),
isgiven by (assuming we get closer to maturity):
T
= e(br)T n(d1)
S
T
(r b + bd1
T+ 1 d1d2
2T
)
= (
r b + bd1
T+ 1 d1d2
2T
) 0
Divide by 365 to get the sensitivity for a one daymove. In
practice one typically also takes intoaccount the expected change
in volatility withrespect to time. If you, for example, on Friday
arewondering how your gamma will be on Mondayyou typically also
will assume a higher impliedvolatility on Monday morning. For P we
haveDgammaPDtime
PT
= P(
r b + bd1
T+ 1 d1d2
2T
) 0
Figure 9 illustrates the DgammaDtime of anoption with respect to
varying asset price andtime to maturity.
4 Numerical GreeksSo far we have looked only at analytical
Greeks. Afrequently used alternative is to use numericalGreeks.
Most first order partial derivatives canbe computed by the
two-sided finite differencemethod
c(S + S, X, T, r, b, ) c(S S, X, T, r, b, )2S
In the case of derivatives with respect to time, weknow what
direction time will move and it ismore accurate (for what is
happening in thereal world) to use a backward derivative
c(S, X, T, r, b, ) c(S, X, T T, r, b, )T
.
Numerical Greeks have several advantages overanalytical ones. If
for instance we have a stickydelta volatility smile then we also
can changethe volatilities accordingly when calculating
thenumerical delta. (We have a sticky delta volatilitysmile when
the shape of the volatility smile
10
115
220
325
5065809
5
11012
5140
0.0006
0.0004
0.0002
0
0.0002
0.0004
0.0006
Days to maturity
Asset price
X = 100, r = 5%, b = 0%, = 30%,
Figure 8. Speed
1035
6084
109
50556065707580859095100
105
110
115
120
125
130
135
140
145
150
1
0.5
0
0.5
1
1.5
Days to maturity
Asset price
X = 100, r = 5%, b = 0%, = 30%,
Figure 9. DgammaDtime
-
Wilmott magazine 57
ESPEN GAARDER HAUG
sticks to the deltas but not to the strike; in otherwords the
volatility for a given strike will moveas the underlying
moves.)
c c(S+S, X, T, r, b, 1)c(SS, X, T, r, b, 2)2S
Numerical Greeks are moreover model inde-pendent, while the
analytical Greeks presentedabove are specific to the BSM model.
For gamma and other second derivatives, 2 f
x2 ,(for example DvegaDvol) we can use the centralfinite
difference method
c(S + S, . . .) 2c(S, . . .) + c(S S, . . .)S2
If you are very close to maturity (a few hours) andyou are
approximately at-the-money the analyticalgamma can approach
infinity, which is naturallyan illusion of your real risk. The
reason is simplythat analytical partial derivatives are
accurateonly for infinite changes, while in practice onesees only
discrete changes. The numerical gammasolves this problem and offers
a more accurategamma in these cases. This is particularly truewhen
it comes to barrier options (Taleb 1997).
For Speed and other third order derivatives, 3 fx3 , we can for
example use the followingapproximation
Speed 1S3
[c(S + 2S, . . .) 3c(S + S, . . .)+ 3c(S, . . .) c(S S, . .
.)].
What about mixed derivatives, fxy , for example
DdeltaDvol and Charm, this can be calculatednumerical by
DdeltaDvol
14S
[c(S + S, . . . , + )c(S + S, . . . , )c(SS, . . . , + )+c(S S,
. . . , )]
In the case of DdeltaDvol one would typicallydivide it by 100 to
get the right notation.
End Part 1BSD trader That is enough for today solider.
New Hired Trader Sir, I learned a few thingstoday. Can I start
trading now?
BSD trader We dont let fresh soldiers playaround with ammunition
(capital) before
they know the basics of a conventionalweapon like the
Black-Scholes formula.New Hired Trader Understood Sir!BSD trader
Next time I will tell you aboutvega-kappa, probability Greeks and
someother stuff. Until then you are Dismissed!Now bring me a double
cheeseburger with alot of fries!New Hired Trader Yes Sir!
1. The author was among the best pistol shooters inNorway.2. If
you dont know the meaning of this expression, BSD,then its high
time you read Michael Lewis Liars Poker.3. And naturally also for
commodity options in the specialcase where cost-of-carry equals
r.4. You have to wait for the next issue of Wilmott Magazinefor the
details on vega.5. I wrote about the importance of this Greek
variable backin 1992. It was my second paper about options, and
myfirst written in English. Well, it got rejected. What could
Iexpect? Most people totally ignored DdeltaDvol at thattime and the
paper has collected dust since then.6. An important question
naturally is what volatility youshould use to compute your deltas.
I will not give you ananswer to that here, but there has been
discussions on thistopic at www.wilmott.com.7. This is true only
because everybody trading options atMother Earth moves at about the
same speed, and areaffected by approximately the same gravity. In
the future,with huge space stations moving with speeds
significantto that of the speed of light, this will no longer hold
true.See Haug (2003a) and Haug (2003b) for some
possibleconsequences.8. This approximation is used by Bensoussan,
Crouhy, andGalai (1995) for an approximate valuation of
compoundoptions.9. Rubinstein (1990) indicates in a footnote that
this maxi-mum curvature point possibly can explain why the
greatestdemand for calls tend to be just slightly out-of-the
money.10. Described by Adamchuck at the Wilmott
forumwww.wilmott.com February 6, 2002,
http://www.wilmott.com/310/messageview.cfm?catid=4&threa-did=664&highlight_key=y&keyword1=vanna
and evenearlier on his page
http://finmath.com/Chicago/NAFTCORP/Saddle_Gamma.html11. Wystrup
(1999) also describes how this redefinition ofgamma removes the
dependence on the spot level S. Hecalls it traders gamma. This
measure of gamma has for along time been popular, particularly in
the FX market, butis still absent in options text books.12. However
he was too lazy to give us the formula so Ihad to do the boring
derivation myself.
FOOTNOTES & REFERENCES
BATES, D. S. (1991): The Crash of 87: Was It Expected?The
Evidence from Options Markets, Journal of Finance,46(3), 10091044.
BENSOUSSAN, A., M. CROUHY, AND D. GALAI (1995):Black-Scholes
Approximation of Warrant Prices,Advances in Futures and Options
Research, 8, 114. BLACK, F. (1976): The Pricing of
CommodityContracts, Journal of Financial Economics, 3, 167179.
BLACK, F., AND M. SCHOLES (1973): The Pricing ofOptions and
Corporate Liabilities, Journal of PoliticalEconomy, 81, 637654.
CARR, P., AND J. BOWIE (1994): Static Simplicity, RiskMagazine,
7(8). FOUQUE, J., G. PAPANICOLAOU, AND K. R. SIRCAR
(2000):Derivatives in Financial Markets with StochasticVolatility.
Cambridge University Press. GARMAN, M. (1992): Charm School, Risk
Magazine,5(7), 5356. HAUG, E. G. (1997): The Complete Guide To
OptionPricing Formulas. McGraw-Hill, New York. (2003a): Frozen Time
Arbitrage, WilmottMagazine, January. (2003b): The Special and
GeneralRelativitys Implications on Mathematical Finance.,Working
paper, January. HULL, J. (2000): Option, Futures, and
OtherDerivatives. Prentice Hall. JARROW, R., AND A. RUDD (1983):
Option Pricing. Irwin. LEWIS, M. (1992): Liars Poker. Penguin.
MCDONALD, R. L. (2002): Derivatives Markets. AddisonWesley. MERTON,
R. C. (1971): Optimum Consumption andPortfolio Rules in a
Continuous-Time Model, Journal ofEconomic Theory, 3, 373413.
(1973): Theory of Rational OptionPricing, Bell Journal of Economics
and ManagementScience, 4, 141183. MORO, B. (1995): The Full Monte,
Risk Magazine,February. RUBINSTEIN, M. (1990): The Super Trust,
WorkingPaper, www.in-the-money.com. SHARP, W. (1966): Mutual Fund
Performance, Journalof Business, pp. 119138. TALEB, N. (1997):
Dynamic Hedging. Wiley. WEBB, A. (1999): The Sensitivity of Vega,
DerivativesStrategy,
http://www.derivativesstrategy.com/maga-zine/archive/1999/1199fea1.asp,
November, 1619. WILMOTT, P. (2000): Paul Wilmott on
QuantiativeFinance. Wiley. WYSTRUP, U. (1999): Aspects of Symmetry
andDuality of the Black-Scholes Pricing Formula forEuropean Style
Put and Call Options, Working Paper,Sal. Oppenhim jr. & Cie.
W