1. Combining Monte Carlo Estimators If I have many MC estimators, with/without various variance reduction techniques, which should I choose?

2. Combining Estimators Suppose I have m unbiased estimators all of the same parameter Put these estimators in a vector Y Any linear combination of these estimators with coefficients that add to one is also an unbiased estimator of the parameter Which such linear combination is best?

3. Best linear combination of estimators.

4. Estimating Covariance

5. Theorem on Optimal Linear Combination of estimators

6. Example: combining the estimators of the call option price

7. Example: (cont)

8. MATLAB function OPTIMAL function [o,v,b,t1]=optimal(U) % generates optimal linear combination of five estimators and outputs % average estimator and variance. t1=cputime; Y1=(.53/2)*(fn(.47+.53*U)+fn(1-.53*U)); t1=[t1 cputime]; Y2=.37*.5*(fn(.47+.37*U)+fn(.84-.37*U))+.16*.5*(fn(.84+.16*U)+fn(1-.16*U)); t1=[t1 cputime]; Y3=.37*fn(.47+.37*U)+.16*fn(1-.16*U); t1=[t1 cputime]; intg=2*(.53)^3+.53^2/2; Y4=intg+fn(U)-GG(U); t1=[t1 cputime]; Y5=importance('fn','importancedens','Ginverse',U); t1=[t1 cputime]; X=[Y1' Y2' Y3' Y4' Y5']; mean(X) V=cov(X); Z=ones(5,1); C=inv(V); b=C*Z/(Z'*C*Z); o=mean(X*b); % this is mean of the optimal linear combinations t1=[t1 cputime]; v=1/(Z'*V1*Z); t1=diff(t1); % these are the cputimes of the various estimators.

9. Results for option pricing [o,v,b]=optimal(rand(1,100000)) Estimators = 0.4619 0.4617 0.4618 0.4613 0.4619 o = 0.46151 % best linear combination (true value=0.46150) v = 1.1183e-005 %variance per uniform input b’ = -0.5503 1.4487 0.1000 0.0491 -0.0475

10. Efficiency of Optimal Linear Combination Efficiency gain based on number of uniform random numbers 0.4467/0.00001118 or about 40,000. However, one uniform generates 5 estimators requiring 10 function evaluations. Efficiency based on function evaluations approx 4,000 A simulation using 500,000 uniform random numbers ; 13 seconds on Pentium IV (2.4 Ghz) equivalent to twenty billion simulations by crude Monte Carlo.

11. Interpreting the coefficients b. Dropping estimators. Variance of the mean of 100,000 is Standard error is around.00001 Some weights are negative, (e.g. on Y1) some more than 1 (on Y2), some approximately 0 (could they be dropped? For example if we drop

12. More examples Integrate the function (exp(u)-1)/(exp(1)-1), u is from 0 to 1 The efficiency gain is over 26000.