% Monte Carlo for a European put

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Monte Carlo for a European put   randn('state'
,100)   Problem and method
parameters S 4 E 5 sigma
0.3 r 0.04 T 1 Dt 1e-3 N T/Dt M
100000
  V zeros(M,1) for i
1M Sfinal Sexp((r-0.5sigma2)Tsigmasqr
t(T)randn) V(i) exp(-rT)max(E-Sfinal,0)
end aM mean(V) bM std(V) conf aM -
1.96bM/sqrt(M), aM 1.96bM/sqrt(M)
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randn('state',100)   Problem
and method parameters S 4 E
5 sigma 0.3 r 0.04 T 1 Dt 1e-3 N
T/Dt M 100000
  V zeros(M,1)
Vanti zeros(M,1) for i 1M Xrandn
Sfinal Sexp((r-0.5sigma2)Tsigmasqrt(T)X)
Sfinalanti Sexp((r-0.5sigma2)T-sigmasq
rt(T)X) V(i) exp(-rT)max(E-Sfinal,0)
Vanti(i) exp(-rT)max(E-Sfinalanti,0) end V
(VVanti)./2 aM mean(V) bM std(V) conf
aM - 1.96bM/sqrt(M), aM 1.96bM/sqrt(M)
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randn('state',100)   Problem and method
parameters S 4 E 4 sigma 0.25
r 0.03 T 1 Dt 1e-2 N T/Dt M
1e4 Geom Asian exact mean
sigsqT sigma2T(N1)(2N1)/(6N
N) muT 0.5sigsqT (r - 0.5sigma2)T(N1)/(
2N) d1 (log(S/E) (muT 0.5sigsqT))/(sqrt(s
igsqT)) d2 d1 - sqrt(sigsqT) N1
0.5(1erf(d1/sqrt(2))) N2 0.5(1erf(d2/sqrt(2
))) geo exp(-rT)( Sexp(muT)N1 - EN2
) Sp
ath Scumprod(exp((r-0.5sigma2)Dtsigmasqrt(
Dt)randn(M,N)),2) Standard Monte
Carlo arithave mean(Spath,2) Parith
exp(-rT)max(arithave-E,0) payoffs Pmean
mean(Parith) Pstd std(Parith) confmc
Pmean-1.96Pstd/sqrt(M), Pmean1.96Pstd/sqrt(M)
  Control Variate geoave exp((1/N)sum(log(Sp
ath),2)) Pgeo exp(-rT)max(geoave-E,0)
geo payoffs Z Parith geo - Pgeo
control variate version Zmean
mean(Z) Zstd std(Z) confcv
Zmean-1.96Zstd/sqrt(M), Zmean1.96Zstd/sqrt(M)

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