Mathematical and Statistical Methods for Actuarial Sciences and Finance

A Monte Carlo approach to value exchange options using a single stochastic factor 313

We can conclude that efficiency Effav =σˆ
n^2 /^2 n
σˆa^2 v/nand the introduction of antithetic
variates has the same effect on precision as doubling the sample size of Monte Carlo
path-simulations.
Using the Stratified Sample, we concentrate the sample in the region where the
functiongis positive and, where the function is more variable, we use larger sam-
ple sizes. First of all, we consider the piecewiseagki(u 1 )+( 1 −a)gki(u 2 )where
u 1 ∼U[0,a]andu 2 ∼U[a,1], as an individual sample. This is a weighted average
of two function values with weightsaand 1−aproportional to the length of the
corresponding intervals. Ifu 1 andu 2 are independent, we obtain a dramatic improve-
ment in variance reduction since it becomesa^2 var[gki(u 1 )]+( 1 −a)^2 var[gki(u 2 )].

For instance, the payoff of SEEOgis(PTi)=max[0,PTi−1] withV 0 =180 and
D 0 =180 has a positive value starting fromas= 0 .60, as shown in Figure 1(a),
while the CEEO will be exercised whenPt 1 ≥ 0 .9878 and the payoff will be positive
fromac= 0 .50, as illustrated in Figure 1(b). Assuminga= 0 .90, Tables 1, 2 and
3 show the variance using the Stratified Sample (ST) and the efficiency index. For
the same reason as before, we should to compare this result with the Monte Carlo
variance with the same number (1 000 000) of path simulations. The efficiency index

Effst=σˆ
n^2 /^2 n
σˆst^2 /nshows that the improvement is about 4. We can assert that it is possibile
to use one fourth the sample size by stratifying the sample into two regions: [0,a]
and [a,1].
Finally, we consider the general stratified sample subdividing the interval [0,1]
into convenient subintervals. Then, if we use the stratified method with two strata
[0. 80 , 0 .90],[0. 90 ,1], Tables 1, 2 and 3 show the variance and also the efficiency

gain Effgst= σˆ
n^2
∑k
i= 1 niσˆ^2 gst

. Moreover, for the first simulation of SEEO we have that

the optimal choice sample size isn=66 477,433 522, for the first simulation of
CEEO we obtain thatn=59 915,440084, while for the PAEO it results thatn=
59 492,440 507. It’s plain that the functionsgkiare more variables in the the interval

1

F

Cumulative Normal Distribution

P

0.60

1

0

0.90

Asset Price

(a) SEEO

0.9878 Asset Price P

Cumulative Normal Distribution

0

0.50

1

0.90

F

(b) CEEO

Fig. 1.Cumulative normal distribution of assetP