Pricing American-Style Options by Monte Carlo Simulation ...

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Table IV<strong>American</strong>-Asian Call OptionPanel A. Polynomial basisN b Asymptotic OLS Quantile Tikhonov MPP MMPP CARTNumber of Paths = 1000 5.42±0.00 5.38±0.10 5.44±0.11 5.44± 0.08 5.39±0.11 5.32±0.11 5.39±0.101 5.52±0.00 5.27±0.10** 5.59±0.11 5.60± 0.10 5.39±0.09 5.36±0.122 5.52±0.00 5.75±0.11** 5.57±0.11 5.47± 0.12 5.56±0.10 5.57±0.093 5.52±0.00 5.52±0.09 5.65±0.13 5.29± 0.10** 5.19±0.08*** 5.33±0.114 5.52±0.00 5.52±0.13 5.59±0.12 5.52± 0.13 5.56±0.10 5.40±0.115 5.52±0.00 5.41±0.10 5.40±0.13 5.39± 0.12 5.36±0.11 5.31±0.1**6 5.52±0.00 5.56±0.11 5.34±0.11 5.49± 0.08 5.31±0.07** 5.42±0.127 5.52±0.00 5.56±0.11 5.36±0.10 5.31± 0.08** 5.34±0.08** 5.41±0.128 5.52±0.00 5.38±0.13 5.53±0.12 5.42± 0.11 5.30±0.10** 5.26±0.09**9 5.52±0.00 5.78±0.11** 5.54±0.11 5.54± 0.11 5.49±0.08 5.35±0.08**10 5.52±0.00 5.66±0.08 5.54±0.11 5.72± 0.14 5.33±0.09** 5.39±0.09Number of Paths = 10000 5.42±0.00 5.42±0.03 5.33±0.03** 5.41± 0.04 5.42±0.04 5.41±0.04 5.43±0.03**1 5.52±0.00 5.47±0.04 5.53±0.04 5.38± 0.03*** 5.39±0.03*** 5.53±0.032 5.52±0.00 5.54±0.04 5.49±0.03 5.41± 0.04** 5.48±0.04 5.42±0.04**3 5.52±0.00 5.51±0.03 5.49±0.03 5.43± 0.03** 5.39±0.03*** 5.45±0.054 5.52±0.00 5.56±0.04 5.53±0.04 5.42± 0.04** 5.42±0.04** 5.44±0.045 5.52±0.00 5.55±0.03 5.53±0.03 5.56± 0.04 5.38±0.04*** 5.46±0.036 5.52±0.00 5.59±0.03** 5.47±0.04 5.43± 0.03** 5.39±0.03*** 5.44±0.047 5.52±0.00 5.57±0.03 5.49±0.03 5.49± 0.04 5.39±0.04*** 5.38±0.03***8 5.52±0.00 5.52±0.03 5.47±0.02** 5.46± 0.04 5.43±0.03** 5.41±0.03***9 5.52±0.00 5.50±0.04 5.48±0.03 5.51± 0.04 5.45±0.04 5.41±0.03***10 5.52±0.00 5.54±0.03 5.53±0.03 5.49± 0.04 5.45±0.03** 5.47±0.04Number of Paths = 50000 5.42±0.00 5.43±0.02 5.43±0.01 5.39± 0.02 5.42±0.01 5.43±0.02 5.44±0.02***1 5.52±0.00 5.51±0.02 5.52±0.01 5.42± 0.01*** 5.45±0.01*** 5.53±0.012 5.52±0.00 5.57±0.01*** 5.52±0.01 5.40± 0.01*** 5.39±0.01*** 5.42±0.01***3 5.52±0.00 5.52±0.02 5.52±0.01 5.39± 0.02*** 5.43±0.01*** 5.41±0.02***4 5.52±0.00 5.52±0.02 5.51±0.02 5.39± 0.01*** 5.41±0.02*** 5.45±0.01***5 5.52±0.00 5.50±0.01 5.51±0.02 5.50± 0.02 5.41±0.01*** 5.46±0.02**6 5.52±0.00 5.50±0.02 5.51±0.02 5.49± 0.02 5.42±0.01*** 5.49±0.027 5.52±0.00 5.53±0.01 5.50±0.02 5.48± 0.02 5.41±0.01*** 5.47±0.01***8 5.52±0.00 5.51±0.01 5.54±0.02 5.45± 0.02*** 5.40±0.02*** 5.41±0.02***9 5.52±0.00 5.52±0.02 5.51±0.02 5.45± 0.01*** 5.39±0.01*** 5.42±0.01***10 5.52±0.00 5.53±0.01 5.52±0.02 5.48± 0.02 5.40±0.01*** 5.43±0.01***Number of Paths = 100000 5.42±0.00 5.41±0.01 5.41±0.01 5.39± 0.01** 5.42±0.01 5.43±0.01 5.48±0.01***1 5.52±0.00 5.53±0.01 5.52±0.01 5.40± 0.01*** 5.47±0.01*** 5.53±0.012 5.52±0.00 5.54±0.01 5.51±0.01 5.39± 0.01*** 5.42±0.01*** 5.43±0.01***3 5.52±0.00 5.53±0.01 5.51±0.01 5.38± 0.01*** 5.43±0.01*** 5.44±0.01***4 5.52±0.00 5.51±0.01 5.52±0.01 5.42± 0.01*** 5.43±0.01*** 5.51±0.015 5.52±0.00 5.52±0.01 5.50±0.01 5.48± 0.01*** 5.43±0.01*** 5.50±0.016 5.52±0.00 5.55±0.01** 5.54±0.01 5.49± 0.01** 5.38±0.01*** 5.49±0.01**7 5.52±0.00 5.52±0.01 5.52±0.01 5.47± 0.01*** 5.40±0.01*** 5.45±0.01***8 5.52±0.00 5.53±0.01 5.53±0.01 5.49± 0.01** 5.40±0.01*** 5.44±0.01***9 5.52±0.00 5.51±0.01 5.52±0.01 5.47± 0.01*** 5.43±0.01*** 5.46±0.01***10 5.52±0.00 5.52±0.01 5.51±0.01 5.49± 0.01** 5.42±0.01*** 5.43±0.01***
Panel B. Polynomial and European price basisN b Asymptotic OLS Quantile Tikhonov MPP MMPP CARTNumber of Paths = 1000 5.78±0.00 5.84±0.07 5.81±0.10 5.89± 0.09 5.74±0.12 5.73±0.12 5.50±0.11**1 5.78±0.00 5.73±0.12 5.93±0.09 5.84± 0.12 5.77±0.10 5.68±0.112 5.78±0.00 5.71±0.10 5.78±0.12 5.80± 0.09 5.66±0.11 5.72±0.103 5.78±0.00 5.73±0.09 5.75±0.11 5.81± 0.08 5.58±0.10 5.96±0.104 5.78±0.00 5.88±0.14 5.68±0.14 5.79± 0.08 5.74±0.09 5.89±0.125 5.78±0.00 5.84±0.08 5.97±0.13 5.75± 0.08 5.84±0.12 5.91±0.156 5.78±0.00 5.78±0.10 5.63±0.13 5.94± 0.14 5.76±0.08 5.88±0.167 5.79±0.00 5.63±0.10 5.80±0.10 5.76± 0.11 5.72±0.10 5.56±0.09**8 5.78±0.00 5.86±0.09 5.91±0.11 5.82± 0.10 5.81±0.09 5.76±0.119 5.79±0.00 5.88±0.11 5.76±0.13 5.60± 0.10 5.70±0.12 5.76±0.1310 5.78±0.00 5.83±0.09 5.90±0.09 5.75± 0.08 5.75±0.09 5.69±0.11Number of Paths = 10000 5.78±0.00 5.76±0.03 5.78±0.03 5.81± 0.02 5.81±0.03 5.76±0.03 5.48±0.03***1 5.78±0.00 5.82±0.04 5.77±0.03 5.74± 0.03 5.80±0.04 5.77±0.032 5.78±0.00 5.78±0.04 5.69±0.04** 5.74± 0.03 5.81±0.03 5.76±0.033 5.78±0.00 5.79±0.03 5.80±0.04 5.72± 0.03 5.75±0.03 5.78±0.024 5.78±0.00 5.82±0.03 5.73±0.03 5.80± 0.03 5.82±0.04 5.74±0.045 5.78±0.00 5.78±0.04 5.77±0.03 5.79± 0.03 5.76±0.04 5.77±0.036 5.78±0.00 5.81±0.04 5.73±0.03 5.77± 0.04 5.82±0.04 5.76±0.037 5.79±0.00 5.70±0.02*** 5.82±0.04 5.78± 0.03 5.72±0.03** 5.81±0.038 5.78±0.00 5.79±0.03 5.81±0.03 5.81± 0.04 5.79±0.03 5.77±0.039 5.79±0.00 5.80±0.03 5.83±0.03 5.77± 0.04 5.76±0.04 5.78±0.0310 5.78±0.00 5.81±0.03 5.86±0.04 5.81± 0.03 5.77±0.04 5.74±0.03Number of Paths = 50000 5.78±0.00 5.78±0.02 5.79±0.02 5.80± 0.02 5.78±0.01 5.78±0.02 5.44±0.02***1 5.78±0.00 5.80±0.02 5.78±0.02 5.79± 0.02 5.79±0.02 5.81±0.022 5.78±0.00 5.78±0.01 5.76±0.01 5.78± 0.01 5.76±0.02 5.77±0.023 5.78±0.00 5.77±0.02 5.79±0.02 5.80± 0.02 5.77±0.01 5.78±0.014 5.78±0.00 5.78±0.02 5.76±0.01 5.77± 0.01 5.78±0.02 5.80±0.015 5.78±0.00 5.76±0.01 5.76±0.01 5.80± 0.01 5.80±0.02 5.81±0.01**6 5.78±0.00 5.77±0.01 5.75±0.01** 5.78± 0.02 5.79±0.02 5.75±0.027 5.79±0.00 5.80±0.02 5.78±0.01 5.78± 0.02 5.81±0.01 5.79±0.028 5.78±0.00 5.80±0.02 5.81±0.02 5.79± 0.02 5.77±0.02 5.77±0.029 5.79±0.00 5.77±0.01 5.77±0.01 5.76± 0.02 5.79±0.01 5.77±0.0110 5.78±0.00 5.77±0.01 5.74±0.01*** 5.77±0.02 5.78±0.01 5.77±0.02Number of Paths = 100000 5.78±0.00 5.80±0.01 5.75±0.01** 5.81± 0.01** 5.79±0.01 5.79±0.01 5.45±0.01***1 5.78±0.00 5.78±0.01 5.78±0.01 5.79± 0.01 5.78±0.01 5.77±0.012 5.78±0.00 5.77±0.01 5.76±0.01 5.77± 0.01 5.78±0.01 5.78±0.013 5.78±0.00 5.79±0.01 5.77±0.01 5.78± 0.01 5.78±0.01 5.78±0.014 5.78±0.00 5.78±0.01 5.76±0.01 5.79± 0.01 5.78±0.01 5.80±0.015 5.78±0.00 5.76±0.01 5.78±0.01 5.78± 0.01 5.79±0.01 5.78±0.016 5.78±0.00 5.78±0.01 5.78±0.01 5.81± 0.01** 5.78±0.01 5.78±0.017 5.79±0.00 5.77±0.01 5.77±0.01 5.78± 0.01 5.77±0.01 5.81±0.018 5.78±0.00 5.79±0.01 5.77±0.01 5.79± 0.01 5.79±0.01 5.79±0.019 5.79±0.00 5.79±0.01 5.78±0.01 5.77± 0.01 5.78±0.01 5.78±0.0110 5.78±0.00 5.78±0.01 5.78±0.01 5.77± 0.01 5.79±0.01 5.77±0.01Notes: The parameters are: σ = 0.2, r = 0.09, T = 120/365, t ′ = 91/365, S 0 = 100, K =100, and exercise points t ∈ {0, 105/365, 108/365, 111/365, 114/365, 117/365, 120/365}.N b denotes the highest degree of the polynomial basis functions. The standard errors arereported after the ± sign based on 20 replications. The asymptotic value is computed<strong>by</strong> the OLS regression method with 100,000 paths. The “**” sign means two to threestandard errors away from the asymptotic value. The “***” sign means more than threestandard errors away.
Page 3 and 4: information overload! - The single
Page 5 and 6: multi-collinearity problem. We find
Page 7 and 8: Step 4: for each path update the va
Page 9 and 10: time; i.e., y i = e −r(t j+1−t
Page 11 and 12: trade-off between size and quality.
Page 13 and 14: C.5. Modified Matching Projection P
Page 15 and 16: provide a numerical algorithm for e
Page 17 and 18: The actual stock price process is g
Page 19 and 20: Testcase 5: Max-Call Option on Mult
Page 21 and 22: and 100,000 paths are used. Given o
Page 23 and 24: the off-diagonal cubic terms. For e
Page 25 and 26: Table VII presents the case of a ma
Page 27 and 28: as well as European option prices.
Page 29 and 30: ReferencesBreiman, L., J.H. Friedma
Page 31 and 32: SS 0K(1)St N − 1(2)St N − 1(3)S
Page 33 and 34: Table IBasis Functions of Max-Call
Page 35 and 36: Panel B. Polynomial and European pr
Page 37: Panel B. Polynomial and European pr
Page 41 and 42: Panel B. Polynomial and European pr
Page 43 and 44: Panel C. K=100, polynomial basisN b
Page 45 and 46: Table VIIMax-Call Option on Ten Ass
Page 47 and 48: Table VIII: Overall Performance Com
Page 49: Table X: Performance Comparison of

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