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Research ArticleAsymptotic Analysis of Shout Options Close to Expiry
G. Alobaidi1 and R. Mallier2
1 Department of Mathematics, American University of Sharjah, Sharjah, UAE2Department of Mathematics and Statistics, York University, Toronto, ON, Canada M3J 1P3
Correspondence should be addressed to G. Alobaidi; [email protected]
Received 14 November 2013; Accepted 9 January 2014; Published 17 February 2014
Academic Editors: R. V. Roy and E. Skubalska-Rafajlowicz
Copyright ยฉ 2014 G. Alobaidi and R. Mallier. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.
We use an asymptotic expansion to study the behavior of shout options close to expiry. Series solutions are obtained for the locationof the free boundary and the price of the option in that limit.
1. Introduction
Since the seminal work of Black and Scholes [1] and Merton[2] on the pricing of options appeared forty years ago, therehas been a dramatic growth in both the role and complexityof financial contracts. The worldโs first organized optionsexchange, the Chicago Board of Options Exchange (CBOE),opened in 1973, the same year as [1, 2] appeared in print, andtrading volumes for the standard options traded on exchangessuch as the CBOE exploded in the late 1970โs and early 1980โs.Around the same time as the growth in standard options,financial institutions began to look for alternative forms ofoptions, termed exotic options, both to meet their needs interms of reallocating risk and also to increase their business.These exotics, which are usually traded over-the-counter(OTC), became very popular in the late 1980โs and early 1990โs,with their users including big corporations, financial institu-tions, fund managers, and private bankers.
One such exotic, which is the topic of the current study, isa shout option [3, 4].This option has the feature that it allowsan investor to receive a portion of the pay-off prior to expirywhile still retaining the right to profit from further upsides.In order to use this feature, the investor must shout, meaningexercise the option, at a time of his choosing, and this leadsto an optimization problemwherein the investor must decidethe best time at which to shout, which in turn leads to afree boundary problem, with the free boundary dividing theregion where it is optimal to shout from that where it is not.In practice, shouting should of course only take place on thefree boundary.
This sort of free boundary problem is of course commonin the pricing of options with American-style early optionfeatures, and this aspect of vanilla American options hasbeen studied extensively in, for example, the recent studiesof [5โ13], although American-style exotics have receivedsomewhat less attention. In the present study, we will use atechnique developed by Tao [14โ22] for the free boundaryproblems arising inmelting and solidification; such problemsare termed Stefan problems. Tao used a series expansion intime to find the location of the moving surface of separationbetween two phases of a material, and, in almost all of thecases he studied, he found that the location of the interfacewas proportional to ๐1/2, ๐ being the time from when thetwo phases were first put in contact. Although, like all equityoptions, shout options obey the Black-Scholes-Merton partialdifferential equation [1, 2], it is straightforward to use achange of variables [8, 23] to transform this into the heat con-duction equation studied by Tao, along with a nonhomoge-neous term, and, once this transformation has beenmade, it isstraightforward to apply Taoโs method. This approach hasbeen taken for vanilla American options in the past [5, 6, 8,12].
Before starting our analysis, which is presented inSection 2, we should first mention earlier work on shoutoptions, much of which has been numerical, although as withother options involving a free boundary and choice on thepart of an investor, some standard numerical techniques suchas the forward-looking Monte Carlo method are problematicbecause of difficulties handling the optimization component
Hindawi Publishing CorporationISRN Applied MathematicsVolume 2014, Article ID 920385, 8 pageshttp://dx.doi.org/10.1155/2014/920385
2 ISRN Applied Mathematics
of shout options. In [24], a Greenโs function approach wasused. With this approach, it was assumed that early exercisecould only occur on a limited number of fixed times ๐ก < ๐ก
1<
๐ก2
< โ โ โ < ๐ก๐โ1
< ๐ก๐
= ๐, so that the option was treatedas Bermudan-style or semi-American rather than American-style, and then the value of the option at time ๐ก
๐was used
to compute the value at time ๐ก๐โ1
, which in turn was usedto compute the value at time ๐ก
๐โ2and so on. The value at
time ๐ก๐โ1
was computed by using an integral involving theproduct of the Greenโs function with the value at time ๐ก
๐,
with this integral being evaluated numerically.More standardnumerical methods, such as finite differences, have also beenapplied to shout options [25]. One analytical study was [4]which used partial Laplace transforms to study the freeboundary.
Finally in this section, we should mention that, in addi-tion to shout options themselves, the shout feature can also befound embedded in several other financial contracts, someof which are offered to retail investors. One such contractis a segregated fund, sold by life insurance companies inCanada, which allows investors to lock in their profits priorto maturity. Some of these contracts have multiple shoutingopportunities, although in this analysis, we assume that theholder can shout only once so that there is only one freeboundary whose location must be optimized: with multipleshouting opportunities, there would be multiple free bound-aries.
2. Analysis
As with any equity option, the price ๐(๐, ๐ก) of a shout optionis governed by the Black-Scholes-Merton partial differentialequation (PDE) [1, 2]
๐๐
๐๐ก
+
๐2๐2
2
๐2๐
๐๐2+ (๐ โ ๐ท) ๐
๐๐
๐๐
โ ๐๐ = 0, (1)
where ๐ is the price of the underlying and ๐ก < ๐ is thetime, with ๐ being the expiry time. The parameters in thisequation are the risk-free rate, ๐, the dividend yield, ๐ท, andthe volatility, ๐, all of which are assumed constant here.Merton [26] observed that this same PDE (1) governs theprice of many different securities, and it is the boundary andinitial conditions which differentiate the securities, not thePDE. For the shout options considered here, the pay-off for anoption held to maturity without shouting is max(๐ โ ๐ธ, 0) fora call andmax(๐ธโ๐, 0) for a put, where๐ธ is the original strikeprice of the shout option; these pay-offs are the same as forvanilla European and American options. In addition, a shoutoption gives the holder the right to cash in some of the gainsprior to expiry, and a shout call can be exchanged at any timefor the excess of the current stock price ๐ over the strike price๐ธ together with a European call with a new strike price equalto the current stock price, provided the stock price is greaterthan the original exercise price. Obviously, the price of sucha European call can be written down using the Black-Scholesoption pricing formula, whichmeans that, upon shouting, the
holder of a call receives a package consisting of cash togetherwith a European call with a total value of
๐๐(๐, ๐) = ๐ โ ๐ธ +
๐๐โ๐ท๐
2
erfc[โ(๐ โ ๐ท + ๐
2/2)โ๐
โ2๐
]
โ
๐๐โ๐๐
2
erfc[โ(๐ โ ๐ท โ ๐
2/2)โ๐
โ2๐
] .
(2)
Obviously, this leads to the constraint that the value of ashout option cannot be less than the proceeds from shoutingimmediately, so that, for the call, ๐ โฅ ๐
๐for ๐ โฅ ๐ธ, where
๐๐is the pay-off from shouting. Similarly, a shout put can be
exchanged at any time for the deficit of the current stock price๐ below the strike price ๐ธ together with a European put witha new strike price equal to the current stock price, providedthe stock price is less than the original exercise price. Uponshouting, therefore, the holder of a put receives a packageconsisting of cash together with a European put with a totalvalue of
๐๐(๐, ๐) = ๐ธ โ ๐ โ
๐
2
๐โ๐ท๐ erfc[
(๐ โ ๐ท + ๐2/2)โ๐
โ2๐
]
+
๐
2
๐โ๐๐ erfc[
(๐ โ ๐ท โ ๐2/2)โ๐
โ2๐
] ,
(3)
and for a put we have the constraint that ๐ โฅ ๐๐for ๐ โค
๐ธ. In both (2) and (3), erfc denotes the complementary errorfunction.
As with American options, the possibility of โshoutingโleads to a free boundary where it is optimal to shout. Severalproperties of this free boundary are known. Firstly, we knowthe value of the option at the free boundary, namely,๐
๐, given
by (2) and (3) above, and also the value of the optionโs delta,or derivative of its value with respect to the stock price, at thefree boundary, where it is equal to (๐๐
๐/๐๐), which for a call
is
๐๐(๐, ๐) + ๐ธ
๐
, (4)
while for a put it is
๐๐(๐, ๐) โ ๐ธ
๐
. (5)
The condition on the delta, (๐๐/๐๐), comes from requiringthat the delta is continuous across the boundary and isessentially the โhigh contactโ or โsmooth-pastingโ condition,which was first proposed by Samuelson [27] for Americanoptions.
Secondly, we know that the location of the free boundaryat expiry ๐ = 0 is ๐
๐(0) = ๐ธ, which can be deduced intuitively
because the pay-off for early exercise is so sweet for shoutoptions. In our terms, ๐
๐(๐ธ) = 0. We also know that the
optimal exercise boundary moves upwards (or at worst is
ISRN Applied Mathematics 3
flat) as we move away from the expiration date for a call anddownwards (or again at worst is flat) for a put.
To analyze this equation, we will use an expansion whichis essentially along the lines of those used by Tao [14โ22].An approach very similar to this has previously been appliedto American options [5, 6, 8, 12]. To apply Taoโs method tothe Black-Scholes-Merton PDE (1), it is necessary to make achange of variables to transform (1) into a more standard dif-fusion equation together with a forcing term.Wewill proceedalong the same lines as [5, 6, 8, 12] and make the change ofvariables ๐ = ๐ธ๐๐ฅ, ๐ก = ๐ โ 2๐/๐2, and ๐(๐, ๐ก) = ๐
๐+ ๐ธV(๐ฅ, ๐),
which leads us to the diffusion-like PDE๐V๐๐
=
๐2V
๐๐ฅ2+ ๐2
๐V๐๐ฅ
โ ๐1V + ๐ (๐ฅ, ๐) , (6)
where ๐1= 2๐/๐
2 and ๐2= 2(๐โ๐ท)/๐
2โ1 and the nonhomo-
geneous term for the call is
๐ (๐ฅ, ๐) = ๐1+ (1 + ๐
2โ ๐1) ๐๐ฅ
โ
๐๐ฅโ๐1๐
2
ร (
๐โ๐2
2๐/4
โ๐๐
+ (๐2+ 1) erfc [โ๐2
โ๐
2
]) ,
(7)
while for the put it is
๐ (๐ฅ, ๐) = โ ๐1โ (1 + ๐
2โ ๐1) ๐๐ฅ
โ
๐๐ฅโ๐1๐
2
ร (
๐โ๐2
2๐/4
โ๐๐
โ (๐2+ 1) erfc [๐2
โ๐
2
]) .
(8)
Equation (6) is valid for ๐ > 0 and must be solved togetherwith the payoff at expiry, ๐ = 0, which is V(๐ฅ, 0) = max(1 โ๐๐ฅ, 0) for a call and V(๐ฅ, 0) = max(๐๐ฅ โ1, 0) for a put, while on
the free boundary, we have
V =๐V๐๐ฅ
= 0. (9)
At expiry the free boundary starts at ๐ = ๐ธ or equivalently๐ฅ = 0. In the analysis that follows, strictly speaking (6) is validonly where it is valid to hold the option, so that at expiry, wecan only impose the initial condition on ๐ฅ โค 0 for the call andon ๐ฅ โฅ 0 for the put.
To tackle (6) and associated boundary and initial condi-tions, we will follow Tao [14โ22] and seek a series solution ofthe form
V (๐ฅ, ๐) =โ
โ
๐=1
๐๐/2
๐๐(๐) , (10)
where ๐ = ๐ฅ/2โ๐ is a similarity variable, while we assumethat the free boundary is located at ๐ฅ = ๐ฅ
๐(๐) which we also
write as a series as follows:
๐ฅ๐(๐) =
โ
โ
๐=1
๐ฅ๐๐๐/2
. (11)
In our analysis, we substitute the assumed form for V(๐ฅ, ๐)(10) in the PDE (6) and group powers of ๐. To abbreviate thepresentation, we introduce the operator
๐ฟ๐โก
1
4
๐2
๐๐2+
1
2
๐
๐๐
โ
๐
2
. (12)
2.1. The Call. For the call, at the first few orders, we find thefollowing equations for the various ๐
๐:
๐ฟ1๐1=
1
2โ๐
,
๐ฟ2๐2= โ
๐2
2
๐
1โ
๐2+ 1
2
+
๐
โ๐
,
๐ฟ3๐3= โ
๐2
2
๐
2โ ๐1๐1+ (2๐1โ ๐2โ 1) ๐
+
๐2โ (๐1/2) + (3๐
2
2/8) + (๐
2/2)
โ๐
,
๐ฟ4๐4= โ
๐2
2
๐
3โ ๐1๐2+ (2๐1โ 1 โ ๐
2) ๐2
โ
1
2
๐1(๐2+ 1)
+
(2๐3/3) + (๐
2โ ๐1+ (3๐2
2/4)) ๐
โ๐
,
(13)
with similar equations for the higher orders, and it isstraightforward to write the solutions to (13) which satisfy theinitial condition that V(๐ฅ, 0) = max(1 โ ๐๐ฅ, 0) for ๐ฅ โค 0 orequivalently ๐
๐โ โ(2๐)
๐/๐! as ๐ โ โโ. The solutions for
the first few orders are
๐1= โ2๐ โ
1
โ๐
+ ๐ถ1[
๐โ๐2
โ๐
+ ๐ erfc (โ๐)] ,
๐2= โ 2๐
2โ
2๐
โ๐
โ
๐2+ 1
2
+ ๐ถ2[
2๐๐โ๐2
โ๐
+ (2๐2+ 1) erfc (โ๐)]
โ ๐ถ1๐2[
๐๐โ๐2
โ๐
+ ๐2 erfc (โ๐)] ,
๐3=
โ8 + 12๐1โ 12๐
2โ 3๐2
2โ 24๐2
12โ๐
โ
๐
3
(4๐2+ 3๐2+ 3)
+ ๐ถ3[(1 + ๐
2)
๐โ๐2
โ๐
+ (๐3+
3๐
2
) erfc (โ๐)]
4 ISRN Applied Mathematics
+ 2๐ถ2๐2[
๐โ๐2
โ๐
+ ๐ erfc (โ๐)]
โ ๐ถ1[(๐1+
3
4
๐2
2)
๐โ๐2
โ๐
+ ๐ (๐1+ ๐2
2) erfc (โ๐)] ,
๐4=
๐ (โ8๐2+ 12๐
1โ 12๐
2โ 8 โ 3๐
3
2)
6โ๐
โ
2๐4
3
โ (๐2+ 1) ๐
2โ
(๐2+ 1)2
4
+
๐1(๐2+ 1)
2
+ ๐ถ4[
[
(5๐ + 2๐3) ๐โ๐2
3โ๐
+ (
4
3
๐4+ 4๐2+ 1) erfc (โ๐)]
]
โ ๐ถ3๐2[
[
(๐ + ๐3) ๐โ๐2
โ๐
+ (
3
2
๐4+ ๐2) erfc (โ๐)]
]
+ ๐ถ2[
[
((4๐1โ 5๐2
2) ๐ + (4๐
1โ 2๐2
2) ๐3) ๐โ๐2
3โ๐
+ (
2
3
(2๐1โ ๐2
2) ๐4+ 2 (๐
1โ ๐2
2๐2))
ร erfc (โ๐)]
]
+ ๐ถ1๐2[((
3
4
๐2
2+ 1) ๐ +
๐2
2๐3
3
)
๐โ๐2
โ๐
+ ((๐2
2+ ๐1) ๐2+
๐2
2๐4
3
) erfc (โ๐) ] ,
(14)
where the ๐ถ๐are constants that must be found by applying
the conditions (9) at the free boundary.To apply the conditions (9) at the free boundary, we
reconstitute the series (10) using the expressions (14) forthe ๐๐and then substitute the assumed form (11) for ๐ฅ
๐(๐)
and again group powers of ๐. The solution of the resultingequations will yield the coefficients ๐ฅ
๐and ๐ถ
๐.
Proceeding in thismanner, at leading order, we obtain thepair of equations for ๐ฅ
1and ๐ถ
1:
๐ถ1[
๐โ๐ฅ2
1/4
โ๐
+
๐ฅ1
2
erfc(โ๐ฅ12
)] โ ๐ฅ1โ
1
โ๐
= 0,
๐ถ1
2
erfc(โ๐ฅ12
) โ 1 = 0,
(15)
so that ๐ฅ1must satisfy the equation
erfc(โ๐ฅ12
) = 2๐โ๐ฅ2
1/4, (16)
with ๐ถ1then given by
๐ถ1=
2
erfc (โ๐ฅ1/2)
= ๐๐ฅ2
1/4. (17)
These expressions ((16) and (17)) are similar to but notidentical to their counterparts for the American call with๐ > ๐ท given in [5, 8]; the analysis of the American call with๐ โค ๐ท is rather different and involves logarithms. As with[5, 8], (16) and (17) must be solved numerically, and we find
๐ฅ1= 1.030396155,
๐ถ1= 1.303990345.
(18)
At the next order on the free boundary, we obtain the pair ofequations
๐ถ2
๐ถ1
[2 +
๐ฅ1
โ๐
+ ๐ฅ2
1] โ
๐2
2
[1 +
๐ฅ1
โ๐
+ ๐ฅ2
1]
โ
1
2
[1 +
2๐ฅ1
โ๐
+ ๐ฅ2
1] = 0,
๐ถ2
๐ถ1
[4๐ฅ1+
1
โ๐
] โ ๐2[2๐ฅ1+
1
โ๐
] โ [2๐ฅ1+
2 โ ๐ฅ2
โ๐
] = 0,
(19)
which have a solution
๐ฅ2=
2๐๐ฅ1(1 + ๐
2) + โ๐ (2 โ ๐ฅ
2
1(๐2+ 2)) โ ๐ฅ
1(๐2+ 2)
โ๐ (2 + ๐ฅ2
1) + ๐ฅ1
= 0.5516261066๐2+ 0.6496056829,
๐ถ2=
๐ถ1
2
[
โ๐ (1 + ๐2) (1 + ๐ฅ
2
1) + ๐ฅ1(2 + ๐
2)
โ๐ (2 + ๐ฅ2
1) + ๐ฅ1
]
= 0.4730258268๐2+ 0.5770676475.
(20)
If we continue the analysis to higher orders, it is straightfor-ward to show that
๐ฅ3= 0.292207785๐
2
2+ 0.893808684๐
2
โ 0.3950836719๐1+ 0.610198033,
๐ฅ4= 0.203979819๐
3
2+ 0.9698633439๐
2
2
+ 1.452965717๐2โ 0.8345732085๐
1๐2
โ 0.9203954739๐1+ 0.6875373101,
๐ฅ5= 0.1638998310๐
4
2+ 1.037819212๐
3
2
+ 2.400660126๐2
2+ 2.386097546๐
2
โ 1.469136768๐1๐2
2โ 3.178694724๐
1๐2
โ 1.768309502๐1+ 0.4855736529๐
2
1
+ 0.8593396200,
(21)
ISRN Applied Mathematics 5
with
๐ถ3= 0.2474229510๐
2
2โ 0.1188463797๐
2
+ 0.5304417776๐1+ 0.7375354641,
๐ถ4= 0.00318404397๐
3
2+ 0.2473130714๐
2
2
+ 0.5280891544๐2โ 0.3593239715๐
1๐2
โ 0.4660110242๐1+ 0.2730318578,
๐ถ5= 0.04939807173๐
4
2โ 0.02503076966๐
3
2
+ 0.1638363804๐2
2โ 0.00645471784๐
2
+ 0.1970541793๐1๐2
2โ 0.08183232660๐
1๐2
+ 0.03734886087๐1+ 0.09802891395๐
2
1
+ 0.1461511036.
(22)
In (10), (14), (18), (20), and (22), we have an expression for thevalue of a shout call close to expiry, with the location of thefree boundary given by (11), (18), (20), and (21).
2.2. The Put. The analysis for the put is very similar to thatfor the call, but with a different nonhomogeneous term anddifferent initial condition. Once again using the operator ๐ฟ
๐
defined in (12), at the first few orders, we find
๐ฟ1๐1=
1
2โ๐
,
๐ฟ2๐2=
๐2
2
๐
1+
๐2+ 1
2
+
๐
โ๐
,
๐ฟ3๐3= โ
๐2
2
๐
2โ ๐1๐1+ (๐2+ 1 โ 2๐
1) ๐
+
๐2โ (๐1/2) + (3๐
2
2/8) + (๐
2/2)
โ๐
,
๐ฟ4๐4= โ
๐2
2
๐
3โ ๐1๐2+ (๐2+ 1 โ 2๐
1) ๐2
+
1
2
๐1(๐2+ 1) +
(๐2โ ๐1+ (3๐2
2/4)) ๐ + (2๐
3/3)
โ๐
.
(23)
Not surprisingly, (23) for the put are very similar to those(13) for the call, differing only in the signs of variousnonhomogeneous terms. Once again, it is straightforward towrite the solutions to (23) which satisfy the initial condition,which for the put is V(๐ฅ, 0) = max(๐๐ฅ โ 1, 0) for ๐ฅ โฅ 0 orequivalently ๐
๐โ (2๐)
๐/๐! as ๐ โ +โ. The solutions for
the first few orders are
๐1= 2๐ โ
1
โ๐
+ ๐ถ1[
๐โ๐2
โ๐
โ ๐ erfc (๐)] ,
๐2= 2๐2โ
2๐
โ๐
+
๐2+ 1
2
โ ๐ถ2[
2๐๐โ๐2
โ๐
โ (2๐2+ 1) erfc (๐)]
โ ๐ถ1๐2[
๐๐โ๐2
โ๐
โ ๐2 erfc (๐)] ,
๐3=
โ8 + 12๐1โ 12๐
2โ 3๐2
2โ 24๐2
12โ๐
+
๐
3
(4๐2+ 3๐2+ 3)
โ ๐ถ3[(1 + ๐
2)
๐โ๐2
โ๐
โ (๐3+
3๐
2
) erfc (๐)]
โ 2๐ถ2๐2[
๐โ๐2
โ๐
โ ๐ erfc (๐)]
โ ๐ถ1[(๐1+
3
4
๐2
2)
๐โ๐2
โ๐
โ ๐ (๐1+ ๐2
2) erfc (๐)] ,
๐4=
๐ (โ8๐2+ 12๐
1โ 12๐
2โ 8 โ 3๐
3
2)
6โ๐
+
2๐4
3
+ (๐2+ 1) ๐
2+
(๐2+ 1)2
4
โ
๐1(๐2+ 1)
2
โ ๐ถ4[
[
(5๐ + 2๐3) ๐โ๐2
3โ๐
โ (
4
3
๐4+ 4๐2+ 1) erfc (๐)]
]
โ ๐ถ3๐2[
[
(๐ + ๐3) ๐โ๐2
โ๐
โ (
3
2
๐4+ ๐2) erfc (๐)]
]
โ ๐ถ2[
[
((4๐1โ 5๐2
2) ๐ + (4๐
1โ 2๐2
2) ๐3) ๐โ๐2
3โ๐
โ (
2
3
(2๐1โ ๐2
2) ๐4+ 2 (๐
1โ ๐2
2๐2)) erfc (๐) ]
]
โ ๐ถ1๐2[((
3
4
๐2
2+ 1) ๐ +
๐2
2๐3
3
)
๐โ๐2
โ๐
โ ((๐2
2+ ๐1) ๐2+
๐2
2๐4
3
) erfc (๐) ] ,
(24)
which differ from their counterparts (14) for the call only inthe signs of various terms.
6 ISRN Applied Mathematics
To apply the conditions (9) at the free boundary, weproceed as for the call, and at leading order, we obtain thepair of equations for ๐ฅ
1and ๐ถ
1,
๐ถ1[
๐โ๐ฅ2
1/4
โ๐
โ
๐ฅ1
2
erfc(๐ฅ12
)] + ๐ฅ1โ
1
โ๐
= 0,
๐ถ1
2
erfc(๐ฅ12
) โ 1 = 0,
(25)
so that ๐ฅ1obeys
erfc(๐ฅ12
) = 2๐โ๐ฅ2
1/4, (26)
with ๐ถ1given by
๐ถ1=
2
erfc (๐ฅ1/2)
= ๐โ๐ฅ2
1/4, (27)
which can be solved numerically to give
๐ฅ1= โ1.030396155,
๐ถ1= 1.303990345.
(28)
The coefficient ๐ถ1is the same as for the call but the sign of ๐ฅ
1
is changed.At the next order, we have
๐ถ2
๐ถ1
[2 โ
๐ฅ1
โ๐
+ ๐ฅ2
1] +
๐2
2
[1 โ
๐ฅ1
โ๐
+ ๐ฅ2
1]
+
1
2
[1 โ
2๐ฅ1
โ๐
+ ๐ฅ2
1] = 0,
๐ถ2
๐ถ1
[4๐ฅ1โ
1
โ๐
] + ๐2[2๐ฅ1โ
1
โ๐
] + [2๐ฅ1+
๐ฅ2โ 2
โ๐
] = 0,
(29)
with a solution
๐ฅ2=
โ2๐๐ฅ1(1 + ๐
2) + โ๐ (2 โ ๐ฅ
2
1(๐2+ 2)) โ ๐ฅ
1(๐2+ 2)
โ๐ (2 + ๐ฅ2
1) โ ๐ฅ1
= 0.5516261066๐2+ 0.6496056829,
๐ถ2=
๐ถ1
2
[โ
โ๐ (1 + ๐2) (1 + ๐ฅ
2
1) + ๐ฅ1(2 + ๐
2)
โ๐ (2 + ๐ฅ2
1) โ ๐ฅ1
]
= โ 0.4730258268๐2โ 0.5770676475,
(30)
with ๐ฅ2the same as for the call but the sign of ๐ถ
2reversed.
For the higher orders, we find
๐ฅ3= โ 0.292207785๐
2
2+ 0.893808684๐
2
โ 0.3950836719๐1โ 0.610198033,
๐ฅ4= 0.203979819๐
3
2+ 0.9698633439๐
2
2
+ 1.452965717๐2โ 0.8345732085๐
1๐2
โ 0.9203954739๐1+ 0.6875373101,
๐ฅ5= โ 0.1638998310๐
4
2โ 1.037819212๐
3
2
โ 2.400660126๐2
2โ 2.386097546๐
2
+ 1.469136768๐1๐2
2+ 3.178694724๐
1๐2
+ 1.768309502๐1โ 0.4855736529๐
2
1
โ 0.8593396200,
(31)
with
๐ถ3= โ 0.2474229510๐
2
2+ 0.1188463797๐
2
โ 0.5304417776๐1โ 0.737535464,
๐ถ4= โ 0.00318404397๐
3
2+ 0.2473130714๐
2
2
โ 0.5280891544๐2+ 0.3593239715๐
1๐2
+ 0.4660110242๐1โ 0.2730318578,
๐ถ5= โ 0.04939807173๐
4
2+ 0.02503076966๐
3
2
โ 0.1638363804๐2
2+ 0.00645471784๐
2
โ 0.1970541793๐1๐2
2+ 0.08183232660๐
1๐2
โ 0.03734886087๐1โ 0.09802891395๐
2
1
โ 0.1461511036,
(32)
with (31) and (32) differing from their counterparts for the call((21), (22)) only in the sign of various terms. In (10), (24), (28),(30), and (32), we have an expression for the value of a shoutput close to expiry, with the location of the free boundarygiven by (11), (28), (30), and (31).
3. Discussion
In the previous section, we used the method of Tao [14โ22]to study the behavior of shout options close to expiry, thesebeing exotic options which allow the investor to receive aportion of the pay-off prior to expiry while still retaining theright to further upside participation, because of which thepay-off for early exercise is sweeter than for vanilla Americanoptions. Perhaps surprisingly, the behavior close to expiryis slightly different for shouts than for vanilla Americans, andwe can attribute a large part of this difference to the richnessof the pay-off for early exercise. For vanilla Americans [5โ13],
ISRN Applied Mathematics 7
the behavior of the free boundary has a strong dependenceon the relative values of the risk-free interest rate ๐ and thedividend yield ๐ท on the underlying stock. For the Americancall with 0 โค ๐ท < ๐ and the American put with๐ท > ๐ โฅ 0, thefree boundary started at ๐๐ธ/๐ท at expiry, with๐ธ as the exerciseprice of the option, and had the usual ๐1/2 behavior close toexpiry, meaning that, as ๐ก โ ๐, the free boundary behavedlike ๐ โผ ๐
0exp[๐(๐ โ ๐ก)1/2]; this was the ๐1/2 behavior which
Tao found in themajority of the physical problems he consid-ered. For the American call with๐ท > ๐ โฅ 0 and the Americanput with 0 โค ๐ท < ๐, the free boundary started at ๐ธ at expiryand behaved like ๐ โผ ๐
0exp[๐(โ(๐ โ ๐ก) ln(๐ โ ๐ก))1/2]; this
behavior is somewhat unusual in that Tao did not encounterthis sort of behavior in his studies. For shouts, we foundin Section 2 that, although the coefficients in the expansiondepended on ๐ and ๐ท, the qualitative behavior of the freeboundary did not: regardless of the values of ๐ and ๐ท, thefree boundary for a shout always starts from ๐ธ at expiry andalways has the usual ๐1/2 behavior close to expiry. Regardlessof the values of ๐ and ๐ท, the free boundary close to expiryfor shout options seems to be less steep than that for vanillaAmericans, and it would seem likely that this is because earlyexercise is more likely for a shout than a vanilla Americanon the same underlying with the same strike, simply becausethe rewards for early exercise are greater for a shout thanan American. For an American, early exercise involves atrade-off between receiving the pay-off earlier and receivingbenefits from any further upside, while with a shout earlyexercise results in receiving a portion of the pay-off earlierwhile still benefitting from further upsides. Because of this, itwould appear paradoxically that, although shout options aremore complex contracts than vanilla Americans, the analysisof shouts is actually a little simpler than that of Americans,primarily because logs are not present for the shouts.
Finally, we note that the behavior of shout puts and callsclose to expiry is very similar, suggesting that there is somesort of put-call symmetry for shout options, perhaps alongthe lines of that for vanilla Americans [28, 29], and it wouldbe interesting to find the exact forms of this symmetry forshouts and other American-style exotics.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
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