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CISC-Matchmaking-2018-Mar-6

 CISC-Matchmaking-2018-Mar-6

Presentation given at the Lunchtime Matchmaking Seminar at Illinois Tech

Fred J. Hickernell

March 06, 2018
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  1. Efficient Monte Carlo Simulations
    Fred J. Hickernell
    Department of Applied Mathematics
    Center for Interdisciplinary Scientific Computation
    Illinois Institute of Technology
    [email protected] mypages.iit.edu/~hickernell
    Supported by NSF-DMS-1522687 and DMS-1638521 (SAMSI)
    CISC Lunchtime Matchmaking Seminar, March 6, 2018

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  2. Background Save Computer Time Save User Time Summary References
    Problems Amenable to Monte Carlo Methods
    answer =

















    posterior mean
    option price
    probability of an event
    .
    .
    .

















    =
    Rd
    f(x) (x) dx
    Glasserman, P. Monte Carlo Methods in Financial Engineering. (Springer-Verlag, New York, 2004),
    Kalos, M. H. & Whitlock, P. A. Monte Carlo Methods, Volume I: Basics. (John Wiley & Sons, New York, 1986),
    Robert, C. P. & Casella, G. Monte Carlo Statistical Methods. Second (Springer-Verlag, New York, 2010). 2/8

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  3. Background Save Computer Time Save User Time Summary References
    Problems Amenable to Monte Carlo Methods
    answer =

















    posterior mean
    option price
    probability of an event
    .
    .
    .

















    =
    Rd
    f(x) (x) dx
    approx =
    1
    n
    n
    i=1
    f(xi), the xi ∼ are



    independent and identically distributed (IID)
    low discrepancy
    via a Markov chain



    Glasserman, P. Monte Carlo Methods in Financial Engineering. (Springer-Verlag, New York, 2004),
    Kalos, M. H. & Whitlock, P. A. Monte Carlo Methods, Volume I: Basics. (John Wiley & Sons, New York, 1986),
    Robert, C. P. & Casella, G. Monte Carlo Statistical Methods. Second (Springer-Verlag, New York, 2010). 2/8

    View Slide

  4. Background Save Computer Time Save User Time Summary References
    Problems Amenable to Monte Carlo Methods
    answer =

















    posterior mean
    option price
    probability of an event
    .
    .
    .

















    =
    Rd
    f(x) (x) dx
    approx =
    1
    n
    n
    i=1
    f(xi), the xi ∼ are



    independent and identically distributed (IID)
    low discrepancy
    via a Markov chain



    Want to save time for
    computer
    user
    Glasserman, P. Monte Carlo Methods in Financial Engineering. (Springer-Verlag, New York, 2004),
    Kalos, M. H. & Whitlock, P. A. Monte Carlo Methods, Volume I: Basics. (John Wiley & Sons, New York, 1986),
    Robert, C. P. & Casella, G. Monte Carlo Statistical Methods. Second (Springer-Verlag, New York, 2010). 2/8

    View Slide

  5. Background Save Computer Time Save User Time Summary References
    Problems Amenable to Monte Carlo Methods
    answer =

















    posterior mean
    option price
    probability of an event
    .
    .
    .

















    =
    Rd
    f(x) (x) dx
    approx =
    1
    n
    n
    i=1
    f(xi), the xi ∼ are



    independent and identically distributed (IID)
    low discrepancy
    via a Markov chain



    Want to save time for
    computer
    user
    Seeking collaborators with applications
    Glasserman, P. Monte Carlo Methods in Financial Engineering. (Springer-Verlag, New York, 2004),
    Kalos, M. H. & Whitlock, P. A. Monte Carlo Methods, Volume I: Basics. (John Wiley & Sons, New York, 1986),
    Robert, C. P. & Casella, G. Monte Carlo Statistical Methods. Second (Springer-Verlag, New York, 2010). 2/8

    View Slide

  6. Background Save Computer Time Save User Time Summary References
    Error of Monte Carlo Methods
    error = answer − approx =
    Rd
    f(x) (x) dx −
    1
    n
    n
    i=1
    f(xi)
    = CNF(f, {xi}n
    i=1
    ) DSC({xi}n
    i=1
    ) VAR(f)
    VAR(f) 0 and measures the variation of f
    DSC({xi}n
    i=1
    ) → 0 as n → ∞ and measures the discrepancy sampling distribution
    deviates from the distribution defining the integral
    CNF(f, {xi}n
    i=1
    ) = O(1) and measures how confounded f is with the difference between
    the sampling and true distributions
    H., F. J. The Trio Identity for Quasi-Monte Carlo Error Analysis. in Monte Carlo and Quasi-Monte Carlo
    Methods: MCQMC, Stanford, USA, August 2016 (eds Glynn, P. & Owen, A.) to appear, arXiv:1702.01487
    (Springer-Verlag, Berlin, 2018), 13–37. 3/8

    View Slide

  7. Background Save Computer Time Save User Time Summary References
    Error of Monte Carlo Methods
    error = answer − approx =
    Rd
    f(x) (x) dx −
    1
    n
    n
    i=1
    f(xi)
    = CNF(f, {xi}n
    i=1
    ) DSC({xi}n
    i=1
    ) VAR(f)
    VAR(f) 0 and measures the variation of f.
    Can be decreased using importance sampling or control variates.
    H., F. J. The Trio Identity for Quasi-Monte Carlo Error Analysis. in Monte Carlo and Quasi-Monte Carlo
    Methods: MCQMC, Stanford, USA, August 2016 (eds Glynn, P. & Owen, A.) to appear, arXiv:1702.01487
    (Springer-Verlag, Berlin, 2018), 13–37. 3/8

    View Slide

  8. Background Save Computer Time Save User Time Summary References
    Error of Monte Carlo Methods
    error = answer − approx =
    Rd
    f(x) (x) dx −
    1
    n
    n
    i=1
    f(xi)
    = CNF(f, {xi}n
    i=1
    ) DSC({xi}n
    i=1
    ) VAR(f)
    DSC({xi}n
    i=1
    ) → 0 as n → ∞ and measures the discrepancy sampling distribution
    deviates from the distribution defining the integral.
    O(n−1/2) for IID xi.
    O(n−1+ ) or better for low discrepancy xi.
    O(n−r) for smooth f and unequal weights for the f(xi).
    H., F. J. The Trio Identity for Quasi-Monte Carlo Error Analysis. in Monte Carlo and Quasi-Monte Carlo
    Methods: MCQMC, Stanford, USA, August 2016 (eds Glynn, P. & Owen, A.) to appear, arXiv:1702.01487
    (Springer-Verlag, Berlin, 2018), 13–37. 3/8

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  9. Background Save Computer Time Save User Time Summary References
    Low Discrepancy Sampling
    error =
    Rd
    f(x) (x) dx −
    1
    n
    n
    i=1
    f(xi) = CNF(f, {xi}n
    i=1
    ) DSC({xi}n
    i=1
    ) VAR(f)
    Low discrepancy sampling places the xi more evenly than IID sampling
    IID points Sobol’ points Integration lattice points
    ···
    Dick, J. et al. High dimensional integration — the Quasi-Monte Carlo way. Acta Numer. 22, 133–288 (2013). 4/8

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  10. Background Save Computer Time Save User Time Summary References
    Automatically Determining the Sample Size
    error =
    Rd
    f(x) (x) dx −
    1
    n
    n
    i=1
    f(xi) = CNF(f, {xi}n
    i=1
    ) DSC({xi}n
    i=1
    ) VAR(f)
    How large should n be to ensure that |error| tolerance? We have answered this question
    For IID sampling
    For low discrepancy sampling
    Assuming f is a Gaussian stochastic process
    H., F. J. et al. Guaranteed Conservative Fixed Width Confidence Intervals Via Monte Carlo Sampling. in Monte
    Carlo and Quasi-Monte Carlo Methods 2012 (eds Dick, J. et al.) 65 (Springer-Verlag, Berlin, 2013), 105–128.
    H., F. J. & Jiménez Rugama, Ll. A. Reliable Adaptive Cubature Using Digital Sequences. in Monte Carlo and
    Quasi-Monte Carlo Methods: MCQMC, Leuven, Belgium, April 2014 (eds Cools, R. & Nuyens, D.) 163.
    arXiv:1410.8615 [math.NA] (Springer-Verlag, Berlin, 2016), 367–383, Jiménez Rugama, Ll. A. & H., F. J. Adaptive
    Multidimensional Integration Based on Rank-1 Lattices. in Monte Carlo and Quasi-Monte Carlo Methods: MCQMC,
    Leuven, Belgium, April 2014 (eds Cools, R. & Nuyens, D.) 163. arXiv:1411.1966 (Springer-Verlag, Berlin, 2016),
    407–422.
    Rathinavel, J. & H., F. J. Automatic Bayesian Cubature. in preparation. 2018+. 5/8

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  11. Background Save Computer Time Save User Time Summary References
    Option Pricing
    µ = fair price =
    Rd
    e−rT max


    1
    d
    d
    j=1
    Sj − K, 0


    e−zTz/2
    (2π)d/2
    dz
    ≈ $13.12
    Sj = S0e(r−σ2/2)jT/d+σxj = stock price at time jT/d,
    x = Az, AAT = Σ = min(i, j)T/d
    d
    i,j=1
    , T = 1/4, d = 13 here
    Abs. Error Median Worst 10% Worst 10%
    Tolerance Method A Error Accuracy n Time (s)
    1E−2 IID diff 2E−3 100% 6.1E7 33.000
    1E−2 Scr. Sobol’ PCA 1E−3 100% 1.6E4 0.040
    1E−2 Scr. Sob. cont. var. PCA 2E−3 100% 4.1E3 0.019
    1E−2 Bayes. Latt. PCA 2E−3 99% 1.6E4 0.051
    6/8

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  12. Background Save Computer Time Save User Time Summary References
    Summary
    We can make Monte Carlo calculations faster through more clever sampling
    We can determine when to stop the calculation to meet your tolerance
    Theory is good, but we want more use cases to test or demonstrate its applicability.
    7/8

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  13. Thank you
    Please contact me at [email protected]
    These slides are available at
    speakerdeck.com/fjhickernell/CISC-Matchmaking-2018-Mar-6

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  14. Background Save Computer Time Save User Time Summary References
    Glasserman, P. Monte Carlo Methods in Financial Engineering. (Springer-Verlag, New
    York, 2004).
    Kalos, M. H. & Whitlock, P. A. Monte Carlo Methods, Volume I: Basics. (John Wiley &
    Sons, New York, 1986).
    Robert, C. P. & Casella, G. Monte Carlo Statistical Methods. Second (Springer-Verlag,
    New York, 2010).
    H., F. J. The Trio Identity for Quasi-Monte Carlo Error Analysis. in Monte Carlo and
    Quasi-Monte Carlo Methods: MCQMC, Stanford, USA, August 2016 (eds Glynn, P. &
    Owen, A.) to appear, arXiv:1702.01487 (Springer-Verlag, Berlin, 2018), 13–37.
    Dick, J., Kuo, F. & Sloan, I. H. High dimensional integration — the Quasi-Monte Carlo
    way. Acta Numer. 22, 133–288 (2013).
    H., F. J., Jiang, L., Liu, Y. & Owen, A. B. Guaranteed Conservative Fixed Width
    Confidence Intervals Via Monte Carlo Sampling. in Monte Carlo and Quasi-Monte Carlo
    8/8

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  15. Background Save Computer Time Save User Time Summary References
    Methods 2012 (eds Dick, J., Kuo, F. Y., Peters, G. W. & Sloan, I. H.) 65 (Springer-Verlag,
    Berlin, 2013), 105–128.
    H., F. J. & Jiménez Rugama, Ll. A. Reliable Adaptive Cubature Using Digital Sequences.
    in Monte Carlo and Quasi-Monte Carlo Methods: MCQMC, Leuven, Belgium, April 2014
    (eds Cools, R. & Nuyens, D.) 163. arXiv:1410.8615 [math.NA] (Springer-Verlag, Berlin,
    2016), 367–383.
    Jiménez Rugama, Ll. A. & H., F. J. Adaptive Multidimensional Integration Based on
    Rank-1 Lattices. in Monte Carlo and Quasi-Monte Carlo Methods: MCQMC, Leuven,
    Belgium, April 2014 (eds Cools, R. & Nuyens, D.) 163. arXiv:1411.1966 (Springer-Verlag,
    Berlin, 2016), 407–422.
    Rathinavel, J. & H., F. J. Automatic Bayesian Cubature. in preparation. 2018+.
    (eds Cools, R. & Nuyens, D.) Monte Carlo and Quasi-Monte Carlo Methods: MCQMC,
    Leuven, Belgium, April 2014. 163 (Springer-Verlag, Berlin, 2016).
    8/8

    View Slide