Combining Monte Carlo Estimators

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,t1optimal(U)
• generates optimal linear combination of five
  estimators and outputs
• average estimator and variance.
• t1cputime
• Y1(.53/2)(fn(.47.53U)fn(1-.53U))
  t1t1 cputime
• Y2.37.5(fn(.47.37U)fn(.84-.37U)).16.5(fn
  (.84.16U)fn(1-.16U))
• t1t1 cputime
• Y3.37fn(.47.37U).16fn(1-.16U)
  t1t1 cputime
• intg2(.53)3.532/2
  Y4intgfn(U)-GG(U) t1t1 cputime
• Y5importance('fn','importancedens','Ginverse',U)
  t1t1 cputime
• XY1' Y2' Y3' Y4' Y5'
• mean(X)
• Vcov(X) Zones(5,1) Cinv(V)
  bCZ/(Z'CZ)
• omean(Xb) this is mean of the optimal
  linear combinations
• t1t1 cputime
• v1/(Z'V1Z)
• t1diff(t1) these are the cputimes of the
  various estimators.

9
Results for option pricing

• o,v,boptimal(rand(1,100000))
• Estimators 0.4619 0.4617 0.4618
  0.4613 0.4619
• o 0.46151 best linear combination (true
  value0.46150)
• 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.