Using Carr and Linetsky ((2006), Proposition 5.1), then Q (Su ≤ Eu| Sl= El) = Q
possesses a central chi-square distribution with 2 (1 + v) degrees of freedom and a non-centrality parameter equal to its starting value. Consequently,
(E-2) Q (Su ≤ Eu| Sl = El) = 1− Q which is equivalent to equation (64) for v ≥ 0.
If v < 0, then Carr and Linetsky ((2006), Proposition 5.3) can be used with µ = −v, and equation (E-1) is rewritten as
Q (Su ≤ Eu| Sl = El) = E(−v)k(l,l;El)
where the expectation is taken with respect to the law of a Bessel process of index (−v) and started at k (l, l; El). Since the process R
(−v) τ (l,u)
2
τ (l,u) follows a non-central chi-square law with 2 (1 − v) degrees of freedom and non-centrality parameter k(l,l;Eτ (l,u)l)2, then equation (E-3) yields which agrees with equation (64) for v < 0.
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Table 1: Prices of American Put Options under the Merton (1973) Model, with St0 = $100 and T − t0 = 0.5 years American put
Option Early exercise boundary specification
parameters Strike Europ. Exact Const. Exp. ExpC. 4d Pol. 5d Pol. CJM HSY2000 HSY10 EXP3
80 0.215 0.219 0.218 0.219 0.219 0.219 0.219 0.219 0.219 0.219 0.220
r = 7% 90 1.345 1.386 1.376 1.385 1.386 1.386 1.386 1.386 1.386 1.386 1.387
q = 3% 100 4.578 4.783 4.750 4.778 4.781 4.782 4.782 4.781 4.782 4.782 4.784
σ = 20% 110 10.421 11.098 11.049 11.092 11.097 11.097 11.097 11.095 11.097 11.098 11.099 120 18.302 20.000 20.000 20.000 19.999 19.999 19.999 19.996 20.000 20.002 20.000
80 2.651 2.689 2.676 2.687 2.688 2.688 2.688 2.688 2.689 2.689 2.690
r = 7% 90 5.622 5.722 5.694 5.719 5.721 5.721 5.721 5.720 5.722 5.722 5.724
q = 3% 100 10.021 10.239 10.190 10.233 10.237 10.237 10.237 10.236 10.239 10.238 10.240 σ = 40% 110 15.768 16.181 16.110 16.173 16.180 16.179 16.179 16.177 16.181 16.181 16.183 120 22.650 23.360 23.271 23.350 23.358 23.358 23.358 23.355 23.359 23.359 23.362
80 1.006 1.037 1.029 1.036 1.037 1.037 1.037 1.037 1.037 1.037 1.038
r = 7% 90 3.004 3.123 3.098 3.120 3.122 3.122 3.122 3.122 3.123 3.123 3.125
q = 0% 100 6.694 7.035 6.985 7.029 7.034 7.034 7.034 7.032 7.035 7.035 7.037
σ = 30% 110 12.166 12.955 12.882 12.946 12.953 12.953 12.953 12.951 12.955 12.953 12.957 120 19.155 20.717 20.650 20.710 20.716 20.716 20.716 20.713 20.717 20.718 20.719
80 1.664 1.664 1.664 1.664 1.664 1.664 1.664 1.664 1.664 1.664 1.664
r = 3% 90 4.495 4.495 4.495 4.495 4.495 4.495 4.495 4.495 4.495 4.495 4.495
q = 7% 100 9.251 9.250 9.251 9.251 9.251 9.251 9.251 9.251 9.251 9.251 9.251
σ = 30% 110 15.798 15.798 15.798 15.798 15.798 15.798 15.798 15.798 15.798 15.798 15.798 120 23.706 23.706 23.706 23.706 23.706 23.706 23.706 23.706 23.706 23.706 23.706 Mean Absolute Percentage Error 0.407% 0.054% 0.020% 0.017% 0.015% 0.026% 0.003% 0.005% 0.023%
CPU (seconds) 451.32 0.01 0.03 2.12 8.87 10.07 1.91 3,215.35 0.17 0.08
Table 1 values American put options under the Merton (1973) model and for different specifications of the exercise boundary. The third column contains European put prices, while the exact American put values (fourth column) are based on the binomial model with 15,000 time steps. The fifth, sixth and seventh columns report the American put prices associated with the constant, the exponential, and the exponential-constant boundary specifications, as given by equations (44), (45), and (46). The eighth and ninth columns are both based on a polynomial boundary–see equation (47)–
with four and five degrees of freedom, respectively. The American put prices contained in the tenth column are obtained from the exercise boundary of equation (48). The next two columns implement the full (with 2,000 time steps) and the 10-point recursive methods of Huang, Subrahmanyam, and Yu (1996). The last column presents the American put prices generated by the three-point multipiece exponential method of Ju (1998).
Table 2: Prices of American Put Options under the CEV Model, with β = 3, St0 = $100and T − t0 = 0.5 years American put
Option Early exercise boundary specification
parameters Strike Europ. Exact Const. Exp. ExpC. 4d Pol. 5d Pol. CJM DT KY 80 0.159 0.162 0.162 0.162 0.162 0.162 0.162 0.162 0.162 0.162
r = 7% 90 1.255 1.297 1.287 1.296 1.296 1.296 1.296 1.294 1.297 1.297
q = 3% 100 4.579 4.792 4.759 4.787 4.789 4.789 4.790 4.786 4.792 4.791
δ = 0.02 110 10.542 11.215 11.164 11.208 11.213 11.213 11.213 11.209 11.214 11.214 120 18.452 20.025 20.011 20.021 20.023 20.024 20.024 20.024 20.024 20.027 80 2.293 2.331 2.319 2.329 2.330 2.330 2.330 2.329 2.331 2.331
r = 7% 90 5.385 5.491 5.463 5.487 5.489 5.489 5.489 5.486 5.491 5.491
q = 3% 100 10.030 10.262 10.212 10.255 10.260 10.260 10.260 10.256 10.261 10.261 δ = 0.04 110 16.043 16.474 16.401 16.464 16.471 16.471 16.471 16.465 16.472 16.472 120 23.132 23.843 23.751 23.832 23.840 23.840 23.841 23.833 23.839 23.840 80 0.822 0.852 0.844 0.851 0.851 0.851 0.851 0.850 0.851 0.851
r = 7% 90 2.843 2.969 2.944 2.965 2.967 2.967 2.967 2.964 2.968 2.969
q = 0% 100 6.698 7.060 7.008 7.053 7.058 7.058 7.058 7.053 7.060 7.060
δ = 0.03 110 12.371 13.175 13.099 13.165 13.173 13.172 13.173 13.166 13.174 13.172 120 19.493 20.992 20.915 20.982 20.989 20.989 20.990 20.983 20.989 20.993 80 1.419 1.419 1.419 1.419 1.419 1.419 1.419 1.419 1.419 1.419
r = 3% 90 4.311 4.311 4.311 4.311 4.311 4.311 4.311 4.311 4.311 4.311
q = 7% 100 9.254 9.254 9.254 9.254 9.254 9.254 9.254 9.254 9.254 9.254
δ = 0.03 110 15.980 15.980 15.980 15.980 15.980 15.980 15.980 15.980 15.980 15.980 120 23.978 23.978 23.978 23.978 23.978 23.978 23.978 23.978 23.978 23.978 Mean Absolute Percentage Error 0.384% 0.054% 0.023% 0.022% 0.021% 0.064% 0.006% 0.007%
CPU (seconds) 931.30 7.71 10.40 11.21 30.10 34.14 5.29 5,339.28 0.35
Table 2 values American put options under the CEV model and for different specifications of the exercise boundary. The third column contains European put prices, while the exact American put values (fourth column) are based on the Crank-Nicolson method with 15,000 time intervals and 10,000 space steps. The fifth, sixth and seventh columns report the American put prices associated with the constant, the exponential, and the exponential-constant boundary specifications, as given by equations (44), (45), and (46). The eighth and ninth columns are both based on a polynomial boundary–see equation (47)–with four and five degrees of freedom, respectively. The American put prices contained in the tenth column are obtained from the exercise boundary specification of equation (48). The last two columns implement the full (with 2,000 time steps) and the 10-point recursive methods of Huang, Subrahmanyam, and Yu (1996), as suggested by Detemple and Tian (2002) and Kim and Yu (1996), respectively.
Table 3: Prices of American Call Options under the CEV Model, with β = 1, St0 = $100 and T − t0 = 1 year American call
Option Early exercise boundary specification
parameters Strike Europ. Exact Const. Exp. ExpC. 2d Pol. 3d Pol. CJM DT KY 80 23.370 23.370 23.370 23.370 23.370 23.370 23.370 23.370 23.370 23.373 r = 7% 90 15.735 15.735 15.735 15.735 15.735 15.735 15.735 15.735 15.735 15.732
q = 3% 100 9.635 9.635 9.635 9.635 9.635 9.635 9.635 9.635 9.635 9.626
δ = 2 110 5.315 5.315 5.315 5.315 5.315 5.315 5.315 5.315 5.315 5.306
120 2.630 2.630 2.630 2.630 2.630 2.630 2.630 2.630 2.630 2.625 80 28.249 28.254 28.253 28.253 28.253 28.253 28.253 28.254 28.254 28.243 r = 7% 90 22.204 22.205 22.205 22.205 22.205 22.205 22.205 22.205 22.205 22.183 q = 3% 100 17.083 17.083 17.083 17.083 17.083 17.083 17.083 17.083 17.083 17.053 δ = 4 110 12.870 12.870 12.870 12.870 12.870 12.870 12.870 12.870 12.870 12.835 120 9.499 9.499 9.499 9.499 9.499 9.499 9.499 9.499 9.499 9.465 80 28.022 28.022 28.022 28.022 28.022 28.022 28.022 28.022 28.022 28.023 r = 7% 90 21.061 21.061 21.061 21.061 21.061 21.061 21.061 21.061 21.061 21.051 q = 0% 100 15.221 15.221 15.221 15.221 15.221 15.221 15.221 15.221 15.221 15.201 δ = 3 110 10.567 10.567 10.567 10.567 10.567 10.567 10.567 10.567 10.567 10.542 120 7.047 7.047 7.047 7.047 7.047 7.047 7.047 7.047 7.047 7.024 80 20.301 21.882 21.794 21.871 21.801 21.879 21.879 21.878 21.881 21.871 r = 3% 90 14.257 15.187 15.100 15.175 15.105 15.184 15.184 15.184 15.186 15.164 q = 7% 100 9.552 10.084 10.015 10.074 10.018 10.081 10.081 10.082 10.082 10.058
δ = 3 110 6.106 6.401 6.353 6.394 6.354 6.398 6.398 6.400 6.400 6.380
120 3.728 3.886 3.855 3.881 3.856 3.884 3.884 3.885 3.885 3.875 Mean Absolute Percentage Error 0.161% 0.024% 0.154% 0.009% 0.009% 0.005% 0.005% 0.162%
CPU (seconds) 912.31 5.60 7.07 5.60 9.94 10.84 3.87 5,690.36 0.49
Table 3 values American call options under the CEV model and for different specifications of the exercise boundary. The third column contains European call prices, while the exact American call values (fourth column) are based on the Crank-Nicolson method with 15,000 time intervals and 10,000 space steps. The fifth, sixth and seventh columns report the American call prices associated with the constant, the exponential, and the exponential-constant boundary specifications, as given by equations (44), (45), and (46). The eighth and ninth columns are both based on a polynomial boundary–equation (47)–with two and three degrees of freedom, respectively. The American call prices contained in the tenth column are obtained from the exercise boundary specification of equation (48). The last two columns implement the full (with 2,000 time steps) and the 10-point recursive methods of Huang, Subrahmanyam, and Yu (1996), as suggested by Detemple and Tian (2002) and Kim and Yu (1996), respectively.
Table 4: Accuracy of the Polynomial Specification for a Large Sample of Randomly Generated American Puts
Polynomial specifications KY
2nd degree 3rd degree 4th degree 5th degree 6th degree Percentage Errors
mean -0.0291% -0.0226% -0.0200% -0.0179% -0.0147% 0.0068%
maximum 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.3953%
minimum -0.1691% -0.1162% -0.1071% -0.1003% -0.0902% -0.0563%
99th percentile 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.1898%
1st percentile -0.1308% -0.1037% -0.0949% -0.0889% -0.0758% -0.0480%
Absolute Percentage Errors
mean 0.0291% 0.0226% 0.0200% 0.0179% 0.0147% 0.0158%
maximum 0.1691% 0.1162% 0.1071% 0.1003% 0.0902% 0.3953%
minimum 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000%
99th percentile 0.1308% 0.1037% 0.0949% 0.0889% 0.0758% 0.1898%
Table 4 reports the pricing errors associated with the valuation of 1,250 randomly generated American put options, under the CEV model and through different polynomial parameterizations of the exercise boundary, as given by equation (47). For comparison purposes, the last column contains the pricing errors associated with the 10-point recursive scheme of Huang, Subrahmanyam, and Yu (1996), as suggested by Kim and Yu (1996). The strike price is always set at $100 while the other option features were generated from uniform distributions and within the following intervals: instantaneous volatility between 10% and 60%; interest rate and dividend yield between 0% and 10%; underlying spot price between $70 an $130; beta between 0 and 4.0; and time-to-maturity ranging from 0 to 3.0 years. The pricing errors produced by the alternative boundary specifications were computed against the Crank-Nicolson method with 15,000 time intervals and 10,000 space steps.
Table 5: American Put Options under the JDCEV Model, with ¯β =−0.5, c = 0.5, b = 0.02, St0 = $100 and T − t0 = 5 years
Option European put American put
parameters Strike v0t0 vtD0 vt0 EEPt00 Vt00 VtD0 Vt0
80 1.095 9.772 10.866 1.005 2.100 11.602 13.702
r = 7% 90 1.842 10.993 12.835 2.098 3.940 13.053 16.992
q = 3% 100 2.895 12.215 15.110 4.129 7.024 14.503 21.527
a = 2 110 4.302 13.436 17.739 7.697 11.999 15.953 27.953
120 6.098 14.658 20.756 13.425 19.523 17.404 36.927
80 1.797 22.017 23.814 5.599 7.397 26.818 34.215
r = 7% 90 2.504 24.769 27.273 7.853 10.358 30.170 40.528
q = 3% 100 3.358 27.521 30.879 10.757 14.115 33.523 47.638
a = 4 110 4.363 30.273 34.636 14.430 18.793 36.875 55.668
120 5.525 33.025 38.550 18.973 24.498 40.227 64.726
80 1.400 14.570 15.970 2.590 3.990 17.688 21.678
r = 7% 90 2.054 16.391 18.445 4.247 6.301 19.899 26.200
q = 0% 100 2.884 18.213 21.096 6.752 9.636 22.110 31.746
a = 3 110 3.906 20.034 23.940 10.425 14.331 24.321 38.652
120 5.134 21.855 26.989 15.504 20.637 26.532 47.170
80 5.385 21.080 26.465 3.857 9.242 22.785 32.027
r = 3% 90 7.496 23.715 31.212 5.340 12.836 25.633 38.469
q = 7% 100 10.001 26.351 36.352 7.172 17.173 28.481 45.654
a = 3 110 12.891 28.986 41.877 9.392 22.283 31.329 53.612
120 16.148 31.621 47.768 12.045 28.193 34.177 62.370
Table 5 values American put options under a time-homogeneous JDCEV model, and for the parameter values considered in Table 3. The third column contains European put prices, conditional on no default, as given by equation (58). The fourth column reports the recovery component of the European put contract, which is computed from Carr and Linetsky ((2006), eq. 3.11), and the fifth column yields the sum of the previous two components. The early exercise premium (sixth column) is computed through Propositions 7 and 8, using a polynomial specification for the exercise boundary with four degrees of freedom. The seventh, eighth and ninth columns are given by equations (57), (56) and (54), respectively.