3.5 Numerical Results
3.5.2 The American-Style Case: AGMAL
Nextwechecktheconvergence ofthepricing. Figure3.2showsthepriceconverges
quickly, up totwodecimalplaces whenL3. Thepatternof convergence oscillates.
Speci cally,the odd-Lpointsand theeven-L oneseachconverge monotonically. This
immediately suggests Richardson's extrapolation: 2V
i V
i 2
, where i 3, and it
indeedleadstotighteroptionprices. Thepatternofovervaluationandundervaluation
in Figure3.2alsocarries over toTable 3.1.
3.5.2 The American-Style Case: AGMAL
We use both the CRR model and the LSM algorithm to price the 3-day and 5-day
AGMALs. Assume S
0
= UB=50,r =2%, q =4%, T =1, T
s
=1=12, and n 0
=50.
Let L=8 for the 3-day cases, and L=3 for the 5-day cases. The pricing results by
LSMarebasedon100,000paths: 50,000plus50,000antithetic. Theresultsareshown
in Table 3.2 and Figure 3.3. We make the following observations. First, compared
with thetree algorithm,the pricescalculated by the LSMmethodaresystematically
8.18 8.19 8.20 8.21 8.22
1 2 3 4 5 6 7 8 9 10
L
EG EG*
MC
Figure 3.2: Convergence of EGMAL. The parameters are S
0
= UB = 50, =
40%,r =2%, q=4%,T =1,T
s
=1=12(n=22),and a=3. EG*andMCrepresent
the price obtained by Richardson's extrapolation and the Monte Carlo simulation
(MC),respectively. Here,MCgivesavlaueof8.1942withasamplestandarddeviation
of 0.0029. A band with a width of 2 times the standard deviation above and below
MC isplotted for reference.
a =3 a=5
LB CRR LSM Std. CRR LSM Std.
0.3 6.3228 6.2870 (0.0256) 6.2280 6.1555 (0.0247)
45 0.4 8.3571 8.3265 (0.0360) 8.2558 8.1852 (0.0352)
0.5 10.3149 10.2864 (0.0476) 10.2117 10.2090 (0.0474)
0.3 6.4256 6.3706 (0.0253) 6.3099 6.2832 (0.0255)
40 0.4 8.5921 8.5353 (0.0359) 8.4505 8.4115 (0.0355)
0.5 10.6800 10.6045 (0.0459) 10.5220 10.5057 (0.0472)
0.3 6.4277 6.4245 (0.0255) 6.3111 6.2814 (0.0256)
35 0.4 8.6118 8.5807 (0.0357) 8.4646 8.3980 (0.0356)
0.5 10.7436 10.7014 (0.0465) 10.5708 10.5530 (0.0476)
Table 3.2: Pricing AGMAL. The parameters are S
0
=UB=50, r=2%, q =4%,
T =1, T
s
=1=12 (n=22), L=8for the a=3 cases, L=3 forthe a =5 cases, and
n 0
=50. \Std." is the sample standard deviationof Monte Carlo simulations(MC).
undervalued. The reason can be traced to Proposition 1. Second, the fact that the
pricescalculatedbythetreealgorithmarewithintwicethesamplestandarddeviation
given by the LSM methodshows that the tree algorithm is likely to converge to the
correctvalue. Third,theearly-exercisepremiumsarerelativelystable. Intheunusual
case where r = 2% and q = 4%, they are just slightly over 0.165. In fact, as show
in Figure 3.4, they are roughly xed when L 3. In that Figure, we move ahead
of ourselves by plottingtheearly-exercise premiumof the arithmetic-moving-average
lookback option aswell.
In order to examine the option values of the AGMAL between Scenario one and
Scenariotwobyourlatticealgorithm,the mostimportantfactors qandT
s
arevaried.
Table 3.3 shows that the prices are more sensitive to T
s
than q. However, there is
littledi erence whenT
s
is at most two months whatever the value of q. And even if
T
s
isthreemonthslong,the di erenceisstillinsigni cant. Theseresultshowthatthe
probability of early exercise for the AGMAL before T
s
with a relatively short reset
period istoosmall tobenoticed.
8.32 8.34 8.36 8.38 8.40 8.42
1 2 3 4 5 6 7 8 9 10
L
AG AG*
LSM
Figure 3.3: Convergence of AGMAL. The parameters are S
0
= UB = 50, =
40%, r = 2%, q = 4%, T = 1, T
s
= 1=12 (n = 22), a = 3 and n 0
= 50. AG* and
LSMrepresentthe priceobtainedbyRichardson's extrapolationand the least-square
simulationmethod,respectively. LSMgivesavalueof 8.3265withasamplestandard
deviation of 0.0360. A band with a width of 2 times the standard deviation above
and belowLSM isplotted forreference.
0.160 0.165 0.170
1 2 3 4 5 6 7 8 9 10
L
Geometric Arithmetic
Figure 3.4: Stability of the Early Exercise Premium. The parameters are
S
0
= UB= 50, = 40%, r =2%, q =4%, T = 1, T
s
= 1=12 (n =22), a =3, and
n 0
= 50. Both the geometric- and arithmetic-moving-average lookback options are
plotted.
T
s
1/12 2/12 3/12
q Scenario 1 Scenario 2 Scenario 1 Scenario 2 Scenario 1 Scenario 2
2% 6.778140 6.778140 7.069512 7.069512 7.216221 7.216223
4% 6.322439 6.322439 6.607496 6.607502 6.750990 6.751187
6% 5.923372 5.923372 6.203751 6.203848 6.344009 6.345403
Table 3.3: Early Exercise before T
s
for AGMAL. The parameters are S
0
=
UB=50, =30%, r=2%, LB=45,T =1, n=22for the T
s
=1=12cases, n =45
for the T
s
=2=12 cases, n=67for the T
s
=3=12 cases, a=3, and n 0
=50.
Pricing Arithmetic-Mo
ving-Average-Lookback
Options
In this chapter, we price the arithmetic-moving-average-lookbackoptions (AMALs).
AMALissimilartoGMALexceptthatthegeometricmovingaverageisreplacedwith
the arithmetic version:
m
a
min
a 1tn P
t
i=t a+1 S
i
a :
Due tothe nonnormality of the arithmetic sum of the stock prices when modeledby
geometricBrowningmotion,we arenot expectedtoderivethe analyticalsolutionfor
the AMAL. Instead, we use the CRRmodel.
4.1 Pricing AMAL on the CRR Model
The basic price tree remains the same as the geometric version. The di erence lies
inthe auxiliarystate variables forthe nodes. LetC(i;j;k;b) denote the optionprice
onnode N(i;j). The meaning of b is the same asthe geometricversion. But that of
k di ers. The size of the auxiliarystate variables depends on the set formed by the
possible strike prices. In the geometric version, the set is
fS
0 u
k=a
:k isan integer;k
LB
k k
UB g:
In contrast, in the arithmetic version, it is no longer so simple. We rst nd all
the possible strike prices between LB and UB and put each in array M(k), where
k = 0;1;:::;k
max
, softed from the smallest to the largest one such that M(0)= LB
and M(k
max
)= UB. (Notice that M(0) may not equal LB. This happens when LB
issmaller than allpossible strikeprices.) The possiblestrikeprices inthe arithmetic
version formsthe set
fM(k):k is aninteger;0kk
max g:
Sok is not the powerindex but the rank of the movingaverage in the array.
Unfortunately,this set grows with O(L a 1
)whichisexponentialina. In orderto
overcomethis complexity,we observe thatthe accuracyof thestrikepriceis rounded
to 2 decimalplaces in the market (x=2). Consequently, the size of the set is much
more limited. Forexample, when UB=50,LB =45, there are atmost 501 possible
strike prices: f45:00;45:01;:::;50:00g.
The backward-inductionformulafor the AMAL is similar tothat for the GMAL
except for the function k(l). The function k(l)now becomes
k(l) = 8
>
<
>
:
k; if M(k)MA(b(l))
f(MA(b(l))); if LBMA (b(l))<M(k)
0; if MA (b(l))<LB
; (4.1)
whereMA (b(l)))isthearithmeticmovingaveragewithrespecttothepathfromnode
N(i;j) tonode N(i+1;j+l).
For example, suppose UB = 50, LB = 45, MA (b(l)) = 47, and M(k) = 48.
By function (4.1), we know that when the state moves to node N(i+1;j +l), the
minimum moving average moves down to 47. Hence, we have to search the correct
rankk whichcorresponds toa minimummoving average of 47in the successor node
N(i+1;j+l). However, searching the array can be time consuming. We now turn
to a more eÆcient way to invert M(k) to get rank k. The idea is to hard-code the
correspondence once and for all. Of course a correspondence is simply an integral
function, which can be coded as atable. Letf(i) bethe said array (calledthe
rank-inversion table), where bM(0) c i bM(k
max
) c and 10 z
with z being the
smallest nonnegativeinteger such that bM(i) c6=bM(j) c for i6=j. The table can
be constructed by the followingalgorithm.
Algorithm 1 Construction of the Rank-Inversion Table.
1: for k =0to k
max do
2: f(bM(k) c):=k;
3: end for
For example suppose M(0) = 1, M(1) = 1:11, M(2) = 1:33, M(3) = 1:55, and
M(4)=1:7. Itisobviousthat =10andf()worksasdesired. IfgivenM(k)=1:33,
we can nd k =2via rank-inversion table f() in constanttime.
Location Value
b10M(k)c f(b10M(k)c)
10 0
11 1
12
-13 2
14
-15 3
16
-17 4
Theoretically,the algorithmfortheAMAL takesO((L+1) a+1
)spaceandO((L+
1) a+1
) time. Although it may seem that the algorithmfor the AMAL is faster than
that for the GMAL, the numericalresults show that the opposite istrue in practice.
The reason is that thereis a bigconstant factor for the AMAL algorithm.
4.2 Numerical Results
We use the same parameters asin the geometric version to price both AMALs. T
a-bles 4.1 and 4.2 show that the prices are similar to the geometric counterparts. In
particular,the convergence isquitefastasshown inFig. 4.1. ComparingAMAL and
GMAL, we observe that, the price of GMAL is greaterthan that of AMAL because
of the smaller mean of the geometric average. Second, the price di erence increases
with the moving-average period a and volatility . Still, the di erence is hard to
detect. Figure4.2 shows that the di erence is quite stable when L 3. As it takes
much less time to calculate the price of GMAL, it is a good approximation to the
value ofAMAL. Finally,Table4.3shows that theprobabilityof earlyexercisebefore
T
s
with ashort reset periodis toosmallto be noticed.