Adaptive Quasi-Monte Carlo Where Each Integrand Value Depends on All Data Sites: the American Option

Transcription

1 Adaptive Quasi-Monte Carlo Where Each Integrand Value Depends on All Data Sites: the American Option Lluís Antoni Jiménez Rugama Joint work with Fred J. Hickernell (IIT) and Da Li (IIT) Room 12, Bldg E1, Department of Applied Mathematics Illinois Institute of Technology, Chicago, 6616 IL Friday 19 th August, 216 MCQMC / 24

2 Outline American Options Quasi-Monte Carlo Cubatures The Challenge Improving Cubature Efficiency Conclusions and Future Work MCQMC / 24

3 Outline American Options The American option and the Longstaff and Schwartz (21) pricing method. Quasi-Monte Carlo Cubatures The Challenge Improving Cubature Efficiency Conclusions and Future Work MCQMC / 24

4 The American Put Option Important parameters: Strike price K Maturity T Exercise time t P r, T s Payoff maxpk Sptq, q For our examples we will use: K 4 T 1 year Weekly monitoring, r1{52, 2{52,..., 1s Geo. Brownian motion with S 36, r 6%, and σ 5% ljimene1@hawk.iit.edu MCQMC / 24

5 The Longstaff and Schwartz (21) Method 8 Exercise boundary 6 Price Weekly monitoring ljimene1@hawk.iit.edu MCQMC / 24

6 Regression at Week j 47 Value of holding Should exercise Should hold Holding expected value Payoff S i47 ljimene1@hawk.iit.edu MCQMC / 24

7 Outline American Options Quasi-Monte Carlo Cubatures Estimating integrals automatically. The Challenge Improving Cubature Efficiency Conclusions and Future Work MCQMC / 24

8 Low Discrepancy Sequences Consider an embedded sequence in base b, P tu Ă Ă P m tx i u bm i Ă Ă P 8 tx i u 8 i. Each P m forms a group: Digital nets 8ÿ x t rpx jl ` t jl q mod bsb l l 1 d pmod 1q. j 1 Rank-1 lattices x t px ` tq mod 1. ljimene1@hawk.iit.edu MCQMC / 24

9 Scrambled and Digitally Shifted Sobol Sequence 1 P ljimene1@hawk.iit.edu MCQMC / 24

10 Scrambled and Digitally Shifted Sobol Sequence 1 P ljimene1@hawk.iit.edu MCQMC / 24

11 Scrambled and Digitally Shifted Sobol Sequence 1 P ljimene1@hawk.iit.edu MCQMC / 24

12 Scrambled and Digitally Shifted Sobol Sequence 1 P ljimene1@hawk.iit.edu MCQMC / 24

13 Scrambled and Digitally Shifted Sobol Sequence 1 P ljimene1@hawk.iit.edu MCQMC / 24

14 Scrambled and Digitally Shifted Sobol Sequence 1 P ljimene1@hawk.iit.edu MCQMC / 24

15 Adaptive Algorithm for f P C ş r,1q fpxq dx 1 ř d P m fpxq xpp m ď Dual net/lat Fourier coef hkikj ÿ ď Cpr, mq ř b m r 1 κ tb m r 1 u Want fm,κ ď ε ˆfκ error bound ˆf κ, 2 12 κ ˆf κ, 2 7 κ < κ But we Do not want to assume that decay at a given rate. We cannot get from data. ljimene1@hawk.iit.edu MCQMC / 24

16 Adaptive Algorithm for f P C ş r,1q fpxq dx 1 ř d P m fpxq xpp m ď Dual net/lat Fourier coef hkikj ÿ ď Cpr, mq ř b m r 1 κ tb m r 1 u Want fm,κ ď ε 1 ˆfκ error bound ˆf κ, 2 12 κ ˆf κ, 2 7 κ < κ C #ÿ bounds ÿ ÿ bounds ÿ + ljimene1@hawk.iit.edu MCQMC / 24

17 Automatic Algorithm We choose an initial number of points b m. Then, Step 1 Generate ty i u bm 1 i tfpx i qu bm 1 i and compute its discrete Fourier/Walsh coefficients. Step 2 Estimate the error bound. If it is smaller than ε, STOP. Step 3 Otherwise, set m m ` 1. Step 3.1 Generate ty i u bm 1 i b m 1, and update the discrete Fourier/Walsh transform. Step 3.2 Update the error bound. If it is smaller than ε, STOP. Step 3.3 Increment m by one, and go to Step 3.1. ljimene1@hawk.iit.edu MCQMC / 24

18 Outline American Options Quasi-Monte Carlo Cubatures The Challenge Each payoff depends on all function values. Improving Cubature Efficiency Conclusions and Future Work MCQMC / 24

19 Updating the Function Values The American option payoff computed according to the Longstaff and Schwartz (21) method is not a regular function over r, 1s d : payoff i fpx i q becomes tpayoff i u bm 1 i f tx i u bm 1 i. ljimene1@hawk.iit.edu MCQMC / 24

20 Former Automatic Algorithm We choose an initial number of points b m. Then, Step 1 Generate tpayoff i u bm 1 i tfpx i qu bm 1 i and compute its discrete Fourier/Walsh coefficients. Step 2 Estimate the error bound. If it is smaller than ε, STOP. Step 3 Otherwise, set m m ` 1. Step 3.1 Generate tpayoff i u bm 1 i b m 1, and update the discrete Fourier/Walsh transform. Step 3.2 Update the error bound. If it is smaller than ε, STOP. Step 3.3 Increment m by one, and go to Step 3.1. ljimene1@hawk.iit.edu MCQMC / 24

21 New Automatic Algorithm We choose an initial number of points b m. Then, Step 1 Generate tpayoff i u bm 1 i fptx i u bm 1 i q and compute its discrete Fourier/Walsh coefficients. Step 2 Estimate the error bound. If it is smaller than ε, STOP. Step 3 Otherwise, set m m ` 1. Step 3.1 Generate tpayoff i u bm 1 i fptx i u bm 1 i q, and recompute the discrete Fourier/Walsh coeff. Step 3.2 Update the error bound. If it is smaller than ε, STOP. Step 3.3 Increment m by one, and go to Step 3.1. ljimene1@hawk.iit.edu MCQMC / 24

22 Outline American Options Quasi-Monte Carlo Cubatures The Challenge Improving Cubature Efficiency Techniques that improve the computational cost. Conclusions and Future Work MCQMC / 24

23 Control Variates Introduction If Ipgq ş r,1q d gpxq dx is known, then ż ż fpxq dx fpxq ` βpipgq gpxqq dx, r,1q d r,1q d and, based on our cubatures, we choose β such that b m 1 ÿ κ tb m 1 u ˆf κ βĝ κ mñ8 ùñ faster than b m 1 ÿ κ tb m 1 u ˆf κ mñ8 ùñ. This choice is not necessarily β covpfpxq, gpxqq VarpgpXqq argmin b 8ÿ ˆf 2 κ bĝ κ, κ as noted by Hickernell et al. (25). ljimene1@hawk.iit.edu MCQMC / 24

24 American Option Walsh Coefficients: Control Variates 1 1 ˆfκ 1-2 ˆfκ error bound ˆf κ, 2 5 κ < error bound ˆf κ, 2 5 κ < Cholesky (time differencing) European put control variate For ε.5, 131, 72 points and 3.12 seconds. 32, 768 points and.75 seconds. ljimene1@hawk.iit.edu MCQMC / 24

25 American Option Walsh Coefficients: BM Construction 1 1 ˆfκ 1-2 ˆfκ error bound ˆf κ, 2 5 κ < error bound ˆf κ, 2 5 κ < Cholesky (time differencing) PCA For ε.5, 131, 72 points and 3.12 seconds. 16, 384 points and.44 seconds. ljimene1@hawk.iit.edu MCQMC / 24

26 American Option Walsh Coefficients: BM Construction 1 1 ˆfκ 1-2 ˆfκ error bound ˆf κ, 2 5 κ < error bound ˆf κ, 2 5 κ < Cholesky (time differencing) PCA For ε.5, 131, 72 points and 3.12 seconds. 16, 384 points and.44 seconds. 8, 192 points and.28 seconds. ljimene1@hawk.iit.edu MCQMC / 24

27 Outline American Options Quasi-Monte Carlo Cubatures The Challenge Improving Cubature Efficiency Conclusions and Future Work MCQMC / 24

28 Future Work Additional capabilities, Cost from Opn logpnqq to Opn logpnq 2 q, n b m. Allows for shifts and relative error tolerances. Future work, Test the algorithm with rank-1 lattices. Apply multilevel quasi-monte Carlo. Design adaptive cone conditions. Study other asset price models such as the Heston model. ljimene1@hawk.iit.edu MCQMC / 24

29 References References I Cools, R. and D. Nuyens (eds.) 216. Monte Carlo and quasi-monte Carlo methods 214, Springer-Verlag, Berlin. Glasserman, P. 24. Monte Carlo methods in financial engineering, Applications of Mathematics, vol. 53, Springer-Verlag, New York. Hickernell, F. J. and Ll. A. Jiménez Rugama Reliable adaptive cubature using digital sequences, Monte Carlo and quasi-monte Carlo methods 214. arxiv: [math.na]. Hickernell, F. J., C. Lemieux, and A. B. Owen. 25. Control variates for quasi-monte Carlo, Statist. Sci. 2, Longstaff, F. A. and E. S. Schwartz. 21. Valuing american options by simulation: A simple least-squares approach, Review of Financial Studies 14, Niederreiter, H Random number generation and quasi-monte Carlo methods, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia. ljimene1@hawk.iit.edu MCQMC / 24

30 References Regression Basis The basis proposed in Longstaff and Schwartz (21) and used for our examples in our slides is: e x 2S. e x 2S p1 x S q. e x 2S p1 2 x S ` x2 q. 2S 2 ljimene1@hawk.iit.edu MCQMC / 24

31 References f (x) f (x) Inside and Outside C x x fˆκ 1 fˆκ ljimene1@hawk.iit.edu 1 κ κ MCQMC / 24