Matematik simulation and monte carlo with applications in finance and mcmc phần 8 pot

x ðÀ1Þ e Àxþ1=xðÞ=2 on support ð0; 1Þ using a proposal density proportional to x ðÀ1Þ e Àx=2 for selected values of  and . The envelope has been optimized by setting  ¼ 2 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ  2 = 2 p À 1   (see Section 4.8 of the text). 2 6 6 6 6 6 6 6 6 6 6 6 6 6 4 > lambda:=5; beta:=10; gam:=2*lambda^2*(sqrt(1+beta^2/lambda^2)-1)/beta;  :¼ 5  :¼ 10 gam :¼ 5 ffiffiffi 5 p À 5 2 6 6 6 6 6 6 6 4 > ha:=proc(x) global alpha,beta; x^(lambda-1)*exp(-beta*(x+1/x)/2); end proc; ha :¼ procðxÞ global ; ; x ^ ð À1ÞÃexpðÀ1=2ÃÃðx þ1=xÞÞ end proc 2 6 6 6 4 > ra:=proc(x) global alpha,gam; x^(lambda-1)*exp(-gam^*x/2); end proc; ra :¼procðxÞ global , gam; x ^ ð À1ÞÃexpðÀ1=2ÃgamÃxÞ end proc 2 6 6 6 4 > acceptance_prob:¼evalf(envelopeaccep(ha, ra, 0, infinity, 1)); ‘envelope is’, 0.00206972426167299210 x 4 e Àð5 ffiffi 5 p À5Þx=2 acceptance_prob:¼0.6528327067 2 6 6 6 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3.2 Computation of acceptance probabilities for arbitrary ratio of uniforms generators The procedure ‘ratioaccep’ is used to compute the acceptance probability for a ratio of uniforms generator for a p.d.f. proportional to h(x). The parameters are: h ¼ a procedure supplied by the user, ½x 1 ;x 2 ¼the connected support of h, xinit ¼ a suggested start point for the numerical maximization of h(x) over ½x 1 ;x 2 , 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 230 Appendices xinit1 ¼ a suggested start point for the numerical maximization of x 2 hðxÞ over ½x 1 ; 0, xinit2 ¼ a suggested start point for the numerical maximization of x 2 hðxÞ over ½0;x 2 . > restart;with(Optimization): > ratioaccep:=proc(h::procedure,x1,x2,xinit,xinit1,xinit2) local u,u1,v1,v11,v2,v22,acc; u:=NLPSolve(h(x) , x=x1 x2,initialpoint= { x=xinit } ,maximize);u1:=sqrt(op(1,u)); if x1<0 then v1:=NLPSolve(x*x*h(x), x=x1 0,initialpoint= { x=xinit1 } ,maximize); v11:=-sqrt(op(1,v1)); else v11:=0 end if; if x2>0 then v2:=NLPSolve(x*x*h(x), x=0 x2,initialpoint= { x=xinit2 } ,maximize); v22:=sqrt(op(1,v2)); else v22:=0; end if; print("u+"=u1); print("v-"=v11); print("v+"=v22); acc:=i nt(h(x),x=x1 x2)/u1/(v22-v11)/2; acc; end proc: 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 Use ‘ratioaccep’ to compute the acceptance probabilities and uþ;þ;À for a density proportional to 1= 1 þ x 2 =3ðÞ 2 on support ðÀ1; 1Þ. " > h1:=proc(x); 1/(1+x*x/3)^2; end proc; h1:=procðxÞ1=ð1 þ xÃxÃ1=3Þ ^ 2 end proc 2 6 6 6 4 > acceptance_prob:=ratioaccep(h1,Àinfinity,infinity,0,À2,2); Warning, no iterations performed. ‘u+’=1. ‘vÀ’=À0.8660254038 ‘v+’=0.8660254038 acceptance_prob:=0.1443375672 ffiffiffi 3 p 2 6 6 6 6 6 6 6 6 6 6 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 Appendices 231 The warning aris es because the initial point chosen for the maximization of h over the entire support happens to be the maximizing point. " Now repeat for a density proportional to 1 4 À x 2 ÀÁ 9 on support À 1 2 ; 1 2 ÀÁ . h > h2:=proc(x); (1/4-x*x)^9; end proc; h2:=procðxÞð1=4 À xÃxÞ ^ 9 end proc 2 6 6 6 4 > acceptance_prob:=ratioaccep(h2,-1/2,1/2,0,-1/4,1/4); ‘u+’=0.001953125000 ‘vÀ’=À0.0001922167969 ‘v+’=0.0001922167969 acceptance_prob:=0.7208585900 2 6 6 6 6 6 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 232 Appendices Appendix 4: Random variate generators (standard distributions) The following variate generation procedures appear to be faster than the proprietary Maple functions and are therefore recommended. Note that the proprietary Maple beta and negative binomial generators do not accept noninteger shape parameters, whereas the corresponding procedures below do so. In all cases faster generators than the ones listed below could be devised. However, these ones are selected for their brevity and reasonable efficiency. No attempt has been made to save parameter values between successive calls of the same generator. Doing so would improve the efficiency when the parameter(s) of the distribution remains unchanged between successive calls. 4.1 Standard normal generator This is a polar implementation of the Box–Mu ¨ ller generator, as described in Section 4.1. Note the warning to set i and X 2 to global variables and to set i to ‘false’ on the first call. > STDNORM:=proc( ) local ij,U1,U2,S,B,X1;global i,X2; # # PROCEDURE GENERATES A STANDARD RANDOM NORMAL # DEVIATE, USI NG THE POLAR BOX MULLER METHOD. # SET i to ’false’ ON FIRST CALL. Note that i and 2 6 6 6 6 6 6 6 6 6 6 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 Simulation and Monte Carlo: With applications in finance and MCMC J. S. Dagpunar Ó 2007 John Wiley & Sons, Ltd # X2 are global variables and have to be declared as # such in any procedure that calls STDNORM. if type(i,boolean)then if (i) then i:=not(i);X2; else for ij from 1 to infinity do U1:=evalf(2.0*rand()/10^12)-1.0; U2:=evalf(2.0*rand()/10 ^12)-1.0; S:=U1*U1+U2*U2; if(S>=1.0 or S<=0.0) then next end if; B:=sqrt(-2.0*ln(S)/S); X1:=B*U1; X2:=B*U2; break ; end do; i:=not(i);X1; end if; else ERROR("i should be boolean") end if; end proc: 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 4.2 Beta generator This is an implementation of Cheng’s log-logis tic generator as described in Section 4.5.1. It generates a beta(a,b) variate. > beta:=proc(a,b) local z,r1,r2,w,rho,m,la; if a>0 and b>0 then rho:=a/b;m:=min(a,b);if m<=1 then la:=m else la:=sq rt((2*a*b-a-b)/(a+b-2)) end if; do; r1:=rand()/10^12; r2:=rand()/10^12; z:=(1/r1-1)^(1/la); if evalf(4*r1*r1* r2-evalf(z^(a-la)*((1+rho)/(1+rho*z))^ (a+b)))<0 then break end if; end do; evalf(rho*z/(1+rho*z)); else ERROR( "a and b should be positive") end if; end proc: 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 234 Appendices 4.3 Student’s t distribution This procedure generates a Student’s t variate with n degrees of freedom using the method described in Section 4.7. If n=1 then a Cauchy variate is delivered. > tdistn:=proc(n) local r1,r2,c,x; if type(n,posint)then; r1:=evalf(rand()/10^12); if n>1 then r2:=evalf(rand() /10^12); c:=cos(6.2831853071795864770*r2); sqrt(n/(1/(1-r1^(2/(n-1)))-c^2))*c; else tan(1.570796326794896 6192*(r1+r1-1)); end if; else ERROR("n should be positive integer") end if; end proc: 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 4.4 Generalized inverse Gaussian This procedure generates a variate from a p.d.f. that is proportional to x À1 e Àðxþ1=xÞ=2 on support ð0; 1Þ where 0 < and 0 <. The method is described in Section 4.8. > geninvgaussian:=proc(lambda,beta) local gam,a1,a2,a3,a4,r,j,x; if lambda>0 then if beta>0 then gam:=2*lambda^ 2*(sqrt(1+(beta/lambda)^2)-1)/beta; a1:=0.5*(beta-gam); a2:=0.5*beta; a3:=sqrt(beta*(beta-gam)); a4:=2.0/ gam; for j from 1 to infinity do; r:=evalf (rand()/10^12); x:=random [ gamma [ lambda,a4 ]] (1); if -ln(r)>x*a1+a2/x-a3 then break end if; end do; else ERROR("beta must be positive") end if; else ERROR("lambda m ust be positive") end if; end proc: 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 Appendices 235 4.5 Poisson generator This procedure uses unstored inversion of the c.d.f. as described in Section 4.9. It generates a Poisson ðÞ varia te. > ipois1:=proc(mu) local y,x,r; if mu>=0 then y:=evalf(exp(-mu)); x:=0; r:=evalf(rand()/10^12); while r>y do; x:=x +1; r:=r-y; y:=mu*y/x; end do; x; else ERROR("mu should be non-negative") end if; end proc: > 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 4.6 Binomial This procedure uses unstored inversion of the c.d.f. as described in Section 4.10. It generates a binomial ðn; pÞ variate. > ibinom:=proc(n,p) local ps,q,r,y,x,qs; if type(n,posint) then if p>=0 and p<=1 then if p>0.5 then ps:=1-p else ps:=p ; end if; q:=1-ps; qs:=ps/q; y:=evalf(q^n); r:=evalf(rand()/10^12); x:=0; while r>y do; x:=x+1; r:=r-y; y:=evalf(y *(n-x+1)*qs/x); end do; if p>0.5 then n-x else x end if; else ERROR("p should belong to [ 0,1 ] ") end if; else ERROR("n should be a positive integer") end if; end proc: 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 236 Appendices 4.7 Negative binomial This procedure generates a variate from a p.m.f. where fðxÞ¼ Àðx þ kÞp x q k ÀðkÞ Àðx þ 1Þ ðx ¼ 0; 1; 2; Þ k is a positive real. It uses unstored inversion as mentioned in Section 4.11. It is fast providing that kp=q is not too large. In that case the Poisson gamma method described in Section 4.11 is better. > negbinom:=proc(k,p) local q,r,y,x; if k>0 then if p>=0 and p<=1 then q:=1-p; y:=evalf(q^k); r:=evalf(rand()/10^12 ); x:=0; while r>y do; x:=x+1; r:=r-y; y:=evalf(y*p*(x+k-1)/x); end do; x; else ERROR("p should belong to [ 0,1 ] ") end if; else ERROR("k should be positive") end if; end proc: 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 Appendices 237 [...]... [describe[mean](u),describe[standarddeviation[1]](u)^2/m,u]; 6 6 print("theta1_hat"=describe[mean](u)); 6 6 print("standard 6 6 error"=evalf(describe[standarddeviation[1]](u)/sqrt(m))); 6 4 u; end proc: h Compute the estimate and estimated standard error for a sample size of 1000 2 6 > seed:=randomize(341): 6 print("seed"=seed); 6 4 res1:=theta1_2(1000): Simulation and Monte Carlo: With applications in finance and MCMC Ó... 0.9259670 987 ‘standard error’ ¼ 0.02797125535 Now replace r by 1 À r and theta1_hat by theta2_hat in the print statement of procedure ‘theta1_2’ and run the simulation again with the same seed 2 6 > seed:=randomize(341): 6 print("seed"=seed); 6 6 res2:=theta1_2(1000): 6 6 ‘seed’ ¼ 341 6 4 ‘theta2_hat’ ¼ 1.00915 386 5 ‘standard error’ ¼ 0.02773074613 2 The procedure ‘theta_combined’ does the two jobs above in. .. writing and hedging a call option, 6 6 where the hedging is performed nn þ 1 times The asset earns interest at rate 6 6 rf The risk-free interest rate is r and the (unknown) expected growth rate is  6 6 Print statements are currently suppressed using ‘#’ Removing ‘#’ will show 6 6 6 the price ðxÞ of the asset, the D, the change in D since the last hedge, and the 6 6 cumulative borrowings to finance. .. 252,0.2,0.05,97,100); 6 6 6 6 6 seed: =87 125635 6 6 6 6 6 4 ‘call_price’=7 .83 5472640, ‘std_error’=0.04129 181 009 6 6 6 ‘confidence interval (large m) is’, 7.754540692, 7.916404 588 6 6 6 6 2 6 6 6 6 6 6 6 6 Now reduce the remaining life (from 103 to 50 trading days) of the option 6 6 6 and make the option more in the money’ by reducing the strike price from 6 6 6 4 £97 to 85 Note how the standard error is now smaller,... distributions) Simulation and Monte Carlo: With applications in finance and MCMC Ó 2007 John Wiley & Sons, Ltd J S Dagpunar 234 Appendices 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 # # 4.2 X2 are global variables and have to be declared as such in any... 100 6 0:0 485 43 689 32 ln þ 2: 481 11124 À 47:5 188 887 6 erf 6 2 97 6   6 pffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffi pffiffiffi 6 6 þ0:0 286 1111111Þ 103 252 À 0:0007936507936 103 252 2 6 4 7 .84 025659 6.4 Monte Carlo estimation of the price of a European call option The procedure ‘BS’ below simulates m independent payoffs together with their antithetic counterparts; x0 is the current asset price at time t, and te is the... return in 1 year is 0.2), and where the option has 103 trading days till expiry (252 trading days in a year) The strike price is £97 The risk-free interest rate is 5 % Assume no dividends h > with( finance) : 2 6 > B:=blackscholes(100, 97, 0.05, 103/252, 0.2); 6 evalf(B); 6 6       6 pffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffipffiffiffi 100 6 þ 0:0 286 1111111 103 252 2 B :¼50 erf 0:02427 184 466 ln 6 97 6      6 6 1 100 6 0:0 485 43 689 32... ‘variates exceeds’, 18, ‘is’, 0.00176175 780 3 6 6 ‘standard error’ ¼ 0.000 084 14497712 6 4 ‘variance reduction ratio’ ¼ 49.67 684 543 ‘central limit approx’ ¼ 0.0023 388 67496 5.3 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 Stratified sampling 5.3.1 Estimating Z 0 1 Z 1 ½À ln ðr1 ފ2=3 þ ½À lnðr2 ފ2=3 5=4 dr1 dr2 0 using naive Monte Carlo The procedure... in one run, giving 6 4 an estimate and estimated standard error using primary and antithetic variates 2 6 > theta_combined:=proc(m) local j,r,x,y,u,r1,z; 6 for j from 1 to m do; 6 6 r:=evalf(rand()/10^12);r1:=1-r; 6 6 x:=-ln(r);z:=-ln(r1); 6 6 y[j]:=(x^0.9+z^0.9)/2; 6 6 end do; 6 6 u:=[seq(y[j],j=1 m)]; 6 6 print("mean"=describe[mean](u)); 6 6 print("standard 6 6 error"=evalf(describe[standarddeviation[1]](u)/sqrt(m)));... i and X2 are global variables and have to be declared as such in any procedure that calls this STDNORM if type(i,boolean)then if (i) then i:=not(i);X2; else for ij from 1 to infinity do U1:=evalf(2.0*rand()/10^12)-1.0; U2:=evalf(2.0*rand()/10^12)-1.0; S:=U1*U1+U2*U2; if(S>=1.0 or S . and 2 6 6 6 6 6 6 6 6 6 6 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 Simulation and Monte Carlo: With applications in finance and MCMC J. S. Dagpunar Ó 2007 John Wiley & Sons, Ltd # X2 are global variables and have to be declared as # such in any procedure. seed:=randomize(341): print("seed"=seed); res1:=theta1_2(1000): 2 6 6 6 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 Simulation and Monte Carlo: With applications in finance and MCMC J. S. Dagpunar Ó 2007 John Wiley & Sons, Ltd ‘seed’ ¼ 341 ‘thetal_hat’ ¼ 0.9259670 987 ‘standard error’ ¼. do; i:=not(i);X1; 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 Simulation and Monte Carlo: With applications in finance and MCMC J. S. Dagpunar Ó 2007 John Wiley & Sons, Ltd