Mathematical and Statistical Methods for Actuarial Sciences and Finance

312 G. Villani

Ta b le 2 .Simulation prices of Compound European Exchange Option (CEEO)

V 0 D 0 CEEO (true) CEEO (sim) σˆn^2 εn σˆa^2 v Effav σˆst^2 Effst σˆgst^2 Effgst

180 180 11.1542 11.1590 0.0284 2.38e−4 0.0123 1.15 0.0043 3.30 2.23e−9 25.44

180 200 8.0580 8.0830 0.0172 1.85e−4 0.0078 1.10 0.0019 4.64 1.61e−9 21.28

180 220 5.8277 5.8126 0.0103 1.43e−4 0.0048 1.06 0.0008 6.30 1.15e−9 17.89

200 180 16.6015 16.6696 0.0464 3.04e−4 0.0184 1.25 0.0089 2.60 2.99e−9 30.94

200 200 12.3935 12.4010 0.0283 2.37e−4 0.0124 1.14 0.0043 3.28 2.22e−9 25.40

200 220 9.2490 9.2226 0.0179 1.89e−4 0.0080 1.11 0.0020 4.42 1.67e−9 21.37

220 180 23.1658 23.1676 0.0684 3.69e−4 0.0263 1.30 0.0158 2.15 3.83e−9 35.71

220 200 17.7837 17.7350 0.0439 2.96e−4 0.0180 1.21 0.0083 2.65 2.91e−9 30.07

220 220 13.6329 13.6478 0.0285 2.38e−4 0.0122 1.17 0.0043 3.33 2.22e−9 25.66

Ta b le 3 .Simulation prices of Pseudo American Exchange Option (PAEO)

V 0 D 0 PAEO (true) PAEO (sim) σˆn^2 εn σˆa^2 v Effav σˆst^2 Effst σˆ^2 gst Effgst

180 180 23.5056 23.5152 0.0833 9.12e–4 0.0333 1.26 0.0142 2.93 3.29e–8 25.31

180 200 18.6054 18.6699 0.0581 7.62e–4 0.0250 1.16 0.0083 3.51 3.96e–8 14.65

180 220 14.8145 14.8205 0.0411 6.41e–4 0.0183 1.12 0.0051 4.00 3.72e–8 11.03

200 180 32.3724 32.3501 0.1172 0.0011 0.0436 1.34 0.0247 2.36 3.44e–8 24.86

200 200 26.1173 26.1588 0.0839 9.16e–4 0.0328 1.27 0.0142 2.95 3.27e–8 25.64

200 220 21.1563 21.1814 0.0600 7.74e–4 0.0253 1.18 0.0053 3.43 3.83e–8 15.63

220 180 42.5410 42.5176 0.1571 0.0013 0.0536 1.46 0.0319 2.46 3.97e–8 32.82

220 200 34.9165 34.9770 0.1134 0.0011 0.0422 1.34 0.0233 2.43 2.36e–8 27.90

220 220 28.7290 28.7840 0.0819 9.04e–4 0.0338 1.21 0.0142 2.87 3.35e–8 24.41

onui. We can observe that the simulated values are very close to true ones. In a

particular way, the Standard Errorεn=
σ∧
√n
nis a measure of simulation accurancy
and it is usually estimated as the realised standard deviation of the simulations√ σˆn=
∑n
i= 1

(

gik(ui)

) 2

n −

(∑n

i= 1 g

i

k(ui)

n

) 2

divided by the square root of simulations. Moreover,

to reduce the variance of results, we propose the Antithetic Variates (AV), the Stratified
Sample with two intervals (ST) and a general stratified sample (GST). The Antithetic
Variates consist in generatingnindependent pairwise averages^12 (gki(ui)+gik( 1 −ui))

withui∼U[0,1]. The functiongik( 1 −ui)decreases whenevergik(ui)increases, and
this produces a negative covariancecov[gik(ui),gik( 1 −ui)]<0 and so a variance
reduction. For instance, we can rewrite the Monte Carlo pricing given by Equation
(18) as:

E∼AV

Q

[gs(PT)]≈

1

n

(n

∑

i= 1

1

2

gsi(ui)+

1

2

gsi( 1 −ui))

)

. (30)

We can observe that the varianceσˆa^2 vis halved, but if we generaten=500 000
uniform variatesuandwealsousethevaluesof1−u, it results in a total of 1 000 000
function evaluations. Therefore, in order to determine the efficiency Effav, the variance
σˆn^2 should be compared with the same number (1 000 000) of function evaluations.