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Combining Monte Carlo Estimators

This guide explores the combination of multiple Monte Carlo estimators employing variance reduction techniques. It discusses selecting the best linear combination from a vector of unbiased estimators, ensuring that the total coefficients equal one. An example involves estimating call option prices. A MATLAB function, OPTIMAL, is provided, yielding optimal linear combinations and their respective variances. Results demonstrate substantial efficiency gains in simulation, with specific attention to coefficient impacts and the potential for dropping underperforming estimators.

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Combining Monte Carlo Estimators

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  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 totwenty 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.

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