TheInsandOutsof
BarrierOptions:Part2
Elaamtrsr DSRMANeNp Innl KAM
Euexunr, DERMAN is head of
the Quantitative Strategies
Group at Goldman,Saclu& Co.
in New York.
Inel Kem is a vice presidentin
the Quantitative Strategies
Gtoup at Goldman, Sachs& Co.
in New York.
Srnwc 197
arrier options are extensionsof standard
calls and puts on stocks. Standard calls
and puts have payoffs that depend on
one market level: the strike. Barrier
options have payoffs that depend on two market
levels:the strike and the barrier. Investorscan use
them to gain exposure to (or enhance returns
from) future market scenariosmore complex than
the simple bullish or bearish expectationsembodied in standardoptions. [n addition, their premiums are usually lower than those of standard
options with the samestrike and experience.
In Part 1 of this article (seeDerman and
Kani [1996]), *" explored the basicsof barrier
options, and compared them to standard puts
and calls. In this article, we explore binary
options and other related barrier options in a
similar manner.
The article closeswith an appendix oudining the mathematicsof barrier options. For some
end-users, this will be useful as an ext'ension of
the andytics underlying standardoptions. Others
may simply want to keep the equationshandy as a
back-up reference that completes the analpis of
the barrier options begun in Part 1.
LESSCOMMON BARRIER OPTIONS
There are four less common barrier
options that we describeand analyzenext.
o
A binary up-anil-in call hes no strike and a
DeruvrrrvrsQumrrnrv73
i
single barrier. It pays the holder a one-time
6xed amount, say $1, the first time the stock
price crossesthe barrier from below during
the option's lifetime. If the stock price has not
crossedthe barrier by expiration, the option
expires wortttless.
In the previous section on barrier options
(seePart 1), we assumedthe rebate was dways
zero. For barrier options with non-zero
rebates,you can rnlue the contribution to the
barrier option vdue of a $1 non-deferred cash
rebate associatedwith an up barrier by noting
that it is equal to the vdue of a binary upand-in call.
A binary down-and-input has no strike and a
single barrier. It pays the holder a one-time
fxed amount, say $1, the first time the stock
price crossesthe barrier from above during
the option's lifetime. If the stock price hasnot
crossedthe barrier by expiration, the option
expiresworthless.
Similarly, you can value the contribution
to the barrier oPtion value of a $1 nondeferred cash rebate associatedwith a down
barrier by noting that it is equd to the value
of a binary down-and-in Put.
A cappeilEuropean-stylecall is characterized
by both a strike and a cap barrier above the
strike. It has the same payoff as a standard
call with the same strike and expiration,
except that, if the stock price ever crosses
the barrier from below, the option immediately terminates and the holder receives a
cash payment equal in value to the difference between the barrier and the strike. [f
this cash is received immediately, the capped
call is termed non-deferred.Ifit is receivedon
the date the option would have expired, it is
termed deJened.
put is characterized
A JlooredEuropean-style
by both a strike and e ioor banier below the
strike. It has the same payoff as a standard
put with the same strike and expiration,
except that if the stock price ever crosses
the barrier from above, the option immediately terminates and the holder receives a
cash payment equal in value to the difference between the strike and the barrier.
The cash vdue can similarly be deferred or
non-deferred.
74 Tnr INseNo Orm or Bmxsn OmoNs:PARI2
We analyzesome examplesof theseoptions
nelrt using the sameformat as in Part 1. For each
option, the first chart describes the value of the
option, and the second and third caPturethe dela
and gamma, respectively.The assumptionsbehind
these evaluations are the same as in Part 1, and
summarizedbelow:
Saike:
Barrier:
Rebate:
100
80if down,120if up
0
Dividend Yield:
Volatiliry:
Risk-Free Rate
5.0% (annually compounded)
20Yoper year
l0.Q% (annudly compounded)
In Panel A of Exhibit 1, as the stock rises
toward the 120 barrier, the up-and-in binary
call vdue increasestoward its constant payoff of
$1. Becauseits payoffis somewhat like the delta
of a standard call at expiration (one in the
money, zero otherwise), its value varies with
stock price somewhat like the delta of the standard call option.
ln Panel B, delta increasesfrom zero far
below the barrier, and then drops back to zero
above the barrier. The increaseis more rapid for
shorter expirations.
The cdl owner is long volatility (positive
gamma) below the barrier, and short volatility
near the barrier as delta &ops to zero (Panel C).
Gamma is infinite at the barrier.
As Panel A of Exhibit 2 shows, as the
stock falls toward the 80 barrier, the down-andin binary put value increasestoward its constant
payoff of $1. Its value varies with stock price
somewhat like that of the delta of the standard
put option.
Delta decreasesfrom zero far above the
barrier, and then rises back to zero below the
barrier (Panel B). The increaseis more rapid for
shorter expirations.
The down-and-in binary Put owner is
long volatiliry (positive gamma) above the barrier (Panel C). For some binary Puts, though not
this one, the put holder can also be short
volatility nearer the barrier. Gamma is infinite
at the barricr.
Panel A of Exhibit 3 shows that, for stock
prices far below the barrier, it is unlikely that the
barrier will be crossed, and the value of the
Srnwc 1997
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capped call is close to that of a standardcall. Closer to 120, the call risessteadilyin value toward is
maximum payoffof 20.
Delta increasesat low stock prices (Panel
B). Closer to the 120 barrier, delta actually
decreases
becauseof the limited upside.
Spnwc1997
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The cdl owner is long volatiliry below the
barrier and short volatiliry nearer the barrier
(PanelC).
As Panel A of Exhibit 4 shows, for stock
prices far above the barrieq it is unlikely that the
barrier will be crossed, and the value of the
Denrvrrrvrs
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floored put is close to that of a standardput. Closer to 80, the pud rises steadily in vdue toward its
maximum payoffqf 20.
Delta decreasesfrom zero at high stock
prices to larger negative values at lower stock
prices @anel B). Close to the 80 barrier, delta has
76 TirsltvservpOrmor BennnnOrroxs:Penr2
to
rb*9rlo.
r10
magnitude greater than one, the maximum
dlowed for a sandard put.
The put owner is long volatility above the
barrier for this particular put, though some
floored puts can havenegative gamma close to the
barrier (PanelC).
Srnwc197
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floored put is close to that of a standardput. Closer to 80, the put rises steadily in vdue toward its
maximum payoffo,f 20.
Delta decreasesfrom zero at high stock
prices to larger negative values at lower stock
prices (PanelB). Close to the 80 barrier, delta has
75 TneINsANDOursor BennnnOrnoNs:
Penr2
3
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rbd(0.t
magnitude greater than one, the maximum
allowed for a stan&rd put.
The put owner is long volatility above the
barrier for this particular put, though some
floored puts can have negativegamma close to the
barrier @anelC).
Spnwc1997
CDo(S, K, o, t; b) - CBS(S,K, o, t)
ArrsNDx
Tnr MarnrMATIcs oF BARRIEROPTIoNS
(4)
forS > > b
In this section we present a more detailed
mathematical description of barrier options and derive
'We assumeno fee
some of their analytical properties.
for borrowing stock.
Dowru-exo-OurCerr OrtoNs
We denote the spot stock price by S, the strike
price by K, and the barrier by b. Assume that S > b
(otherwise the option has already terminated and has
zero value). For the moment also assumethat K > b,
and that the option rebate R is zero. The price of the
down-and-out call option is given by
The value of the call is given by the B-S
formula when the stock price is much greater than the
barrier level. This is completely expected, as the stock
price has very small probability of ever crossing the
barrier level. Therefore, the option is unlikely to
terminate early by knockout, and will pay its terminal
payoffto the investor at expiration.
It is interesting to examine the behavior of a
down-and-out call option when the stock price is close
to the barrier level. In this region, it is not too dificult
to show that one has the following approximation for
the option price:
C oo(S ,K ,o,T;b) =
Coo(S, K, o, t;b; =
cur0'/S, K, o, !) (1)
Css(S,K, o, t) -
e(2se-eN(x,
- r)curts,K,o,t)) (s)
- [3
[:)'"'""-'
where i is the continuous cost of carry rate for the
stock given by the difference between the risk-free rate
and the continuous dividend yield.
In Equation (1), CBS(S,K, o, t) is the BlackScholes (B-S) expressionfor the European-style cdl
option with strike K, volatility o, and time to expiration t. The second term is the "barrier" term, which is
necessaryto guarantee that the value of the option
given by Equation (1) is zero at the knockout barrier.
That is, when (S = b),
cDo(S = b, K, o, t; b) = 0
(2)
Equation (1) resemblesthe formula for a call
spread.One can make this resemblancemore complete
by transforming it to give:*
Coo(S,K,o,t;b) =
Cns(S,K, o, t)
(6)
It is clear that the option price vanishesas the
stock price approachesthe barrier level (e + 0). Equation
(5) has an interesting form in the limit where volatfity or
time to expiration is very large. For S - b and o large,
cDo(S,K, o, r; b) - eSe+
cR=
(3)
For stock prices much larger than the barrier
>>
(S
b), the conrribution of the second term in Equation (1) is negligible, and so we have:
(7)
If the holder of the down-and-out option
receivesa rebate R on knockout, this adds the following vdue C* to the premium:
x
- 1012(izo'?)+t
tsj
css(S,KS2/b2, o, g
SPR!.IG 1997
Here S = (1 * e)b,wheres > 0 is assumed
to
be smallso that S is closeto b. 6 is the dividendvield
on the stock,and
+
"l(i)'N(21)(3)"*,r,,]*,
where
,, =
-,r-' + (i - !o\'t""f,u
['"*(:)
(e)
DenrvervnsQuanrnnrv77
,, =
. o-' + (i ]o'ft"'")ru
[t"r(f)
(10)
Here U(S., t, o) denotes the Black-Scholes
lognormal distribution function for the final stock
price at time ? corresponding to the volarility parameter o and having the following
expression:
n, =
+ (i - (2,o2
]o'fl''fro'
[G ]"1
U(Sil t, o) =
(11)
7
n , = [ n - f,o\* Qro'
W*
t' l
+ (i - )o2121t/2lro2
z)
(r2)
and where we have r., n = ofi.
In deriving Equation (8), we assume that the
- that is, it is received by the
rebate is non-deJerred
optionholder as soon as the stock price crossesthe
barrier level. Occasionally, however, this payoff is
ilefmed until the designated expiration of the contract;
in that case,the optionholder immediately receivesthe
rebate value discounted from the designated option
expiration to the crossing time. In this ilefenedpayment
siruation Equations (9)-(12) are modified to read:
(13)
,,=['"r(l)
.,'- t"',,f,"
(14)
(1s)
hr=o
h r =(2 (i -)o 1 )ro '
(16)
The justification for Equation (1) begins wiih
the knockout barrier distribution function for the 6nd
stock price S" at time t conditional upon the initial
value S of the stock price, a barrier at b, and a stock
volatiliry o. This distribution function u(S' t, o; b lS)
is given by the following expressionfor dl S > b:
u(S'r, o;ul$ = tI(St,r, ol$
*
- [l)""""-'
lsj
=0forSSb
(18)
The knockout distribution function of Equation (17) vanishesat the knockout barrier. Fint assume
that the rebate R = 0. The value of the knockout call
is then given by:
C(S,K, t, o; b) =
u(St,r' o; bls)MAX(S- K, o)dst
(19)
MAX(K,b)
It is a simple matter to perform the integration in
Equation (19) and so derive Equation (1).
The addition ofthe non-zero rebate values to
the computation proceeds with the construction of
the probability distribution function for first-time
absorption in the time interval t to tr + At. This is
the first passagetime distribution function. This
density function is readily constructed as the time
derivative of the total absorption probabiliry function and is given by:
= -*i"O't,o;ulgds. (20)
c(r,o;ulsl
lf non-deferred, the contribution of the rebate
to the value of the down-and-out call option C* is
then computed simply asfollows:
t.
cn. = RJqt,o;ulge-"at
(2r)
0
For deferred payoG, the rebate contribution to
the vdue of the down-and-out call option is given by:
u{.s ..r."| {l rorS > b
t'
exP x
- (i ]o\q'rtzo'zrl]
[O"r,r,/s)
I
-,,-1"\,f,,
',=[.{:)
well-known
(r7)
cR = R"-*Jc6,o;uls)at
Q2)
0
OmoNs:Panr2
78 TnEIrvseNpOursor Benruen
Srnnc 197
The integrations in Equations (21) and (22) tan
be
readily performed using Equations (17) and (20).
So far, we have assumedthat the strike is above
the barrier, that is, K > b. In genenl, the
strike of a
down-and-out option may be below the barrier
level
and we musr account for this condition by using
the
correc limits of integration as in Equation (19).
The
correct solution for any srike is given by
the value of an up-and-out call option under
the
assumption
that b > S, K is givenby:
Cuo(S,K, o, T;b) =
sr&(N(xr) - N(xJ) Ke-*(N(x, - v) - N(x, - v)) _
Coo(S, K, o, t; b) =
1, y2i /o2+l
t""[ij
se-6'N1x; - Ke-nN1x - v) -
o'*'*,r,
,r.-"[:)
-
*"-"(!)""
*
N(Y-v))+C1.
+
o,{ffr)- N(Yz))
'-'*,",
".-"[3)
- v) - N(Yz- v)) + ci
(27)
(23)
where
where
=
. ( . i")4," eB)
",
f.{i)
=
" f'"'[6) . ('. :o)4,ve4)
"' =f'"{;). ['. +44'v Qs)
, =f'"{*). ('.:o)"f,v (30)
DowN-eNp-In Cnr. Optrorrrs
A standardcall option is clearlyequivdentto a
long positionin a down-and-ourcall oprionanda long
position in a down-and-in cdl option, eachwith the
samestrike and expirationas the standardcall. Their
payoffsare the sameunder all stock price scenarios.
Consequendy,
.
CDr(S,K, o, t; b) =
C(S,K, o, t) - CDo(S,K, o, f;b)
(26)
In contrastto the holder of knockout options,
the holder of knock-in optionswill alwap beneEtfrom
increasesin volatility (and is *rus long volatiliry) for dl
stock prices.This is intuitively clear,asany increasein
volatility will increaseboth the knock-in probabilityand
the expectedpayoffifthe option is knockedin.
Up'rulp-Orrr Cllr Ortorus
Using the samenotation as in previouscases,
Snnntc197
.('.+o),),,
"=[''*(:)
(31)
The rebate-ter- C R in this case is a slighdy
modified form of C* for down-and-out options:
p2)
ci.="lt:I N(-zr)
. (3)"*n,,1
Ur-exp-Ir.r Cen Onnor*s
A standardcall option is alsoidenticdly equivalent to an up-and-in call combinedwith an up-andout call, for zero rebate. Thus the following
relationshiphol&:
Cu(S, K, o, T;b) =
C(S,K, o,1) - Cuo(S,K, o, r; b)
(33)
79
DnnrvarrwsQue*rrruv
The holder of an up-and-in option will always
beneft from increasesin the market volatility. Because
up-and-out call options may possess
negativekappa
values,Equation (33) showsthat the sensitiviryto
volatility of up-and-iri call options may be considirably larger than that ofa standardcall option.
Up-ero-Our
Dowru-ato-Our Pnr Ornons
It is relatively simple to make the proper modi_
fication in the up-and-out cdl formulation to derive
the formula for a down-and-out put:
Poo(S,K, o, f;b) =
Pur OynoNs
Ke-(1N1-X, + v) - N(-Xz + v)) Up-and-out puts are similar to down-and-out
cdls. Therefore,using our earliernotation,we simply
give the find result for an up-and-ourput option with
barrierat b, assumingthat *re conditionb > S is satisfied:
s;&6N1-x,) - N(-xz)) 1,12i /o2-1
Kt"l: I
Aut-", + v) - N(-yz + v)) +
\r,/
Puo(S,K, o, t;b) =
(N(-Yr)-N(-Y2))+cR
Ke-nN1-x + v) - se-6'N1-xy -
'.-"(:)
f*'-"[3)"'"""
(36)
N(-Y + v) -
,,-"(:)"'
*(-u]+ ci.
(34)
The X and Y in this equation are obtained
from Eguations (24) and (25) by replacing MAX with
MIN on the right-hand side. The rebate contribution
C'* is the same as in Equation (32). The proof of this
formula is similar to the case of down-and-out call
options. For large barrier values this formula approaches the value ofa standardput.
Furthermore, for small volatilities the value of
this option is given by Ke{ - Se{t. Also, for barrier
values very close to the spot price (at 6xed volatiliry),
the only contribution to the put value is due to the
rebate term and is simply equal to R (or Re{ if the
rebate is defened).
Up-exo-IH Pur Ornoxs
A zero rebate up-and-in put combined with a
zero rebate up-and-out put is equivalent to a standard put. Thus:
PUI(S, K 6, t; b) =
P(S,K, o, t) - Puo(S,K o, t; b1
P.*nr2
80 tna tr.rsru'rpOursor BlnnnnOmor.ts:
(35)
with variablesXr, X2, Yl, Y2 as defined by Equations
(28)-(31) and with the rebate tern asin Equation (8).
Dowt r-.lNp-In Pur Ornons
A zero rebate down-and-in put combined with
a zero rebate down-and-out put is equivalent to a
sandard put. Thus the following relationship holds:
PDr(s,
K, o, t; b) =
P(S,K, o, t) - PDo(S,
K, o, t; b)
(37)
ENDNOTES
The authors are grateful to Ken Iseya and Yoshi
Kobashi, who encouraged them to write this article after
many stimulating convenations about barrier options. Tbey
also thank Deniz Ergener, who helped obtain the results
described here, and Alex Bergier and Barbara Dunn, who
patiendy read the manuscript and made many helpful and
clari$ing suggestiors.
=
'Use the scaling property CBS(IS, l,K, o, t)
K, o, r).
l,cBs(s,
REFERENCE
Derman, Emanuel, and lraj Kani. "The Ins and Outs of
Barrier Optiom: Part 1." DerivativesQuarterly,Winrcr 96'
pp. 55-67.
Srnmc197