A Monte Carlo approach to value exchange options using a single stochastic factor 311wheregs(PT/ 2 )=(PT/ 2 − 1 )ifPT/ 2 ≥ P 2 ∗ andgs(PT)=max[PT− 1 ,0] if
PT/ 2 <P 2 ∗.
So, by the Monte Carlo simulation, we can approximate the PAEO as:
S 2 (V,D,T)$D 0
(∑
i∈Ag
i
s(Pˆi
T/ 2 )e−δdT/ (^2) +∑
i∈Bg
i
s(Pˆ
i
T)e
−δdT
n
)
, (29)
whereA={i= 1 ..ns.t.PˆTi/ 2 ≥P 2 ∗}andB={i= 1 ..ns.t.PˆTi/ 2 <P 2 ∗}.
5 Numerical examples and variance reduction techniques
In this section we report the results of numerical simulations of SEEO, CEEO and
PAEO and we propose a generalisation of the antithetic method and a newa-stratified
sampling in order to improve on the speed and the efficiency of simulations. To
compute the simulations we have assumed that the number of simulated pathsnis
equal to 500 000. The parameter values areσv= 0. 40 ,σd= 0. 30 ,ρvd= 0. 20 ,
δv= 0. 15 ,δd=0andT=2 years. Furthermore, to compute the CEEO we assume
thatt 1 =1 year and the exchange ratioq= 0 .10. Table 1 summarises the results of
SEEO simulations, while Table 2 shows the CEEO’s simulated values. Finally, Table
3 contains the numerical results of PAEO.
Using Equation (16), we can observe thatY=ln(PPt 0 )follows a normal distribution
with mean(−δ−σ
2
2 )tand varianceσ
(^2) t. So, the random variableYcan be generated
by the inverse of the normal cumulative distribution functionY =F−^1 (u;(−δ−
σ^2
2 )t,σ
(^2) t)whereuis a function of a uniform random variableU[0,1]. Using the
Matlab algorithm, we can generate thensimulated pricesPˆti,fori= 1 ...n,as:
Pt=P0exp(norminv(u,-dt-0.5sig^2t,sig*sqrt(t))),
whereu=rand( 1 ,n)are thenrandom uniform values between 0 and 1. As the
simulated pricePˆtidepends on random valueui, we write henceforth that the SEEO,
CEEO and PAEO payoffsgik,fork =s,cusing a single stochastic factor depend
Ta b le 1 .Simulation prices of Simple European Exchange Option (SEEO)
V 0 D 0 SEEO (true) SEEO (sim) σˆn^2 εn σˆa^2 v Effav σˆst^2 Effst σˆ^2 gst Effgst
180 180 19.8354 19.8221 0.1175 0.0011 0.0516 1.13 0.0136 4.32 1.02e-8 22.82
180 200 16.0095 16.0332 0.0808 8.98e-4 0.0366 1.10 0.0068 5.97 8.08e-9 19.98
180 220 12.9829 12.9685 0.0535 7.31e-4 0.0258 1.03 0.0035 7.56 5.89e-9 18.15
200 180 26.8315 26.8506 0.1635 0.0013 0.0704 1.16 0.0253 3.23 1.27e-8 25.54
200 200 22.0393 22.0726 0.1137 0.0011 0.0525 1.08 0.0135 4.19 1.03e-8 21.97
200 220 18.1697 18.1746 0.0820 9.05e-4 0.0379 1.08 0.0072 5.65 8.37e-9 19.58
220 180 34.7572 34.7201 0.2243 0.0015 0.0939 1.19 0.0417 2.68 1.54e-8 28.94
220 200 28.9873 28.9479 0.1573 0.0013 0.0695 1.13 0.0238 3.30 1.23e-8 25.45
220 220 24.2433 24.2096 0.1180 0.0011 0.0517 1.14 0.0135 4.35 1.03e-8 22.88