This section introduces the results of interpolating the price of an American option in the table we built in the prior section by the method of polynomial and cubic spline interpolation. Table (4.1) and (4.2) shows the general cases in interpolating an American option based on polynomial and cubic spline interpolation, respectively.
Table (4.3) and (4.5) shows the polynomial wiggle problem arising from interpolating a nonpolynomial function with a polynomial function. Table (4.4) and (4.6) shows that the results of cubic spline interpolation are still acceptable no matter how fine we partition. Table (4.7) and (4.8) shows the results of interpolating the price of an American option straddling the exercise boundary. Table (4.9) and (4.10) shows the results after we partition the grid finer so as to lower the error causing by interpolating the price of an American option straddling the exercise boundary.
Numerical Results 24
Table 4.1: Polynomial interpolation
K S T σ2 r Binomial Look-Up Rel. Err. Abs. Err.
5000 4800 0.5 20 2 363.322 364.464 -0.003144 1.142460 5000 4850 0.5 20 2 334.678 334.144 0.001595 0.533864 5000 4900 0.5 20 2 308.873 308.82 0.000171 0.052936 5000 4950 0.5 20 2 284.213 284.342 -0.000454 0.129088 5000 5000 0.5 20 2 259.715 259.715 0.000002 0.000458 5000 5050 0.5 20 2 238.83 238.945 -0.000481 0.114774 5000 5100 0.5 20 2 218.277 218.231 0.000209 0.045603 5000 5150 0.5 20 2 198.202 197.901 0.001517 0.300750 5000 5200 0.5 20 2 181.331 182.398 -0.005885 1.067120 5000 5000 0.5 20 1 269.989 270.011 -0.000083 0.0224928 5000 5000 0.5 20 2 259.715 259.715 0.000002 0.000457825 5000 5000 0.5 20 3 250.098 250.096 0.000009 0.00223362 5000 5000 0.5 20 4 241.03 241.03 -0.000001 0.000139438 5000 5000 0.5 20 5 232.46 232.46 0.000002 0.000479013 5000 5000 0.5 20 6 224.337 224.337 -0.000001 0.000211508 5000 5000 0.5 20 7 216.627 216.627 0.000001 0.000261474 5000 5000 0.5 20 8 209.284 209.284 0.000002 0.000359369 5000 5000 0.5 20 9 202.31 202.291 0.000094 0.0190052 5000 5000 0.5 15 2 190.211 190.212 -0.000006 0.001107 5000 5000 0.5 20 2 259.715 259.715 0.000002 0.000458 5000 5000 0.5 25 2 329.283 329.283 -0.000001 0.000351 5000 5000 0.5 30 2 398.822 398.822 0.000000 0.000013 5000 5000 0.5 35 2 468.277 468.277 0.000001 0.000420 Means square error(n1Σ(pi− ˆpi))=-0.000280389
Root mean square error(
q1
nΣ(pi− ˆpi)2)=0.001472845 Mean absolute relative error(n1P
|pi− ˆpi|)=0.149349637 Maximum relative error(max(pi− ˆppi)
i )=0.00588493 Maximum absolute error(max|pi− ˆpi|)=1.142460 St−1 = 5000 and S is tomorrow’s stock price m = 21, n = 16, and o = 16
Numerical Results 25
Table 4.2: Cubic spline interpolation
K S T σ2 r Binomial Look-Up Rel. Err. Abs. Err.
5000 4800 0.5 20 2 363.322 363.348 -0.000072 0.0261708 5000 4850 0.5 20 2 334.678 334.496 0.000542 0.181521 5000 4900 0.5 20 2 308.873 308.893 -0.000064 0.0197189 5000 4950 0.5 20 2 284.213 284.256 -0.000152 0.0430896 5000 5000 0.5 20 2 259.715 259.715 0.000002 0.000457825 5000 5050 0.5 20 2 238.83 238.86 -0.000125 0.0298431 5000 5100 0.5 20 2 218.277 218.303 -0.000118 0.0257643 5000 5150 0.5 20 2 198.202 198.239 -0.000189 0.037442 5000 5200 0.5 20 2 181.331 181.331 0.000001 0.000238648 5000 5000 0.5 20 1 269.989 269.979 0.000035 0.00950098 5000 5000 0.5 20 2 259.715 259.715 0.000002 0.000457825 5000 5000 0.5 20 3 250.098 250.097 0.000005 0.00129289 5000 5000 0.5 20 4 241.03 241.03 -0.000001 0.000139438 5000 5000 0.5 20 5 232.46 232.459 0.000002 0.000524934 5000 5000 0.5 20 6 224.337 224.337 -0.000001 0.000211508 5000 5000 0.5 20 7 216.627 216.626 0.000005 0.00101464 5000 5000 0.5 20 8 209.284 209.284 0.000002 0.000359369 5000 5000 0.5 20 9 202.31 202.307 0.000014 0.0028402 5000 5000 0.5 15 2 190.211 190.212 -0.000007 0.001417 5000 5000 0.5 20 2 259.715 259.715 0.000002 0.000458 5000 5000 0.5 25 2 329.283 329.283 -0.000001 0.000373 5000 5000 0.5 30 2 398.822 398.822 0.000000 0.000013 5000 5000 0.5 35 2 468.277 468.277 0.000000 0.000081 Means square error(n1Σ(pi− ˆpi))=-0.000005
Root mean square error(
q
1
nΣ(pi− ˆpi)2)=0.000130753 Mean absolute relative error(n1P
|pi− ˆpi|)=0.01664915 Maximum relative error(max(pi− ˆppi)
i )=0.000542 Maximum absolute error(max|pi− ˆpi|)=0.181521 St−1 = 5000 and S is tomorrow’s stock price m = 21, n = 16, and o = 16
Table (4.1) and (4.2) are the results of interpolating 21 points in the S dimension, 16 points in the σ2 dimension, and 16 points in r dimension based on polynomial and cubic spline interpolation. We can see that holding other parameters the same as tomorrow’s stock price is more away from today’s stock price, 5000, the error tends to become larger in both interpolation method, which is just coincidence. If the error is not acceptable, the most simple way we can do is to increase the node we partition in S dimension. We can also see that holding other parameters the same, to change tomorrow’s σ2 or r causes very small error. Likewise, we can increase the partition for smaller error.
Numerical Results 26
Table 4.3: Polynomial interpolation with wiggle problem in the r dimension K S T σ2 r Binomial Look-Up Rel. Err. Abs. Err.
5000 4800 0.5 20 2 363.322 2040.7 -4.61679 1677.38 5000 4850 0.5 20 2 334.678 194.062 0.420152 140.616 5000 4900 0.5 20 2 308.873 -1242.58 5.02295 1551.45 5000 4950 0.5 20 2 284.213 270.965 0.0466145 13.2485 5000 5000 0.5 20 2 259.715 -490.134 2.8872 749.849 5000 5050 0.5 20 2 238.83 -1085.29 5.5442 1324.12 5000 5100 0.5 20 2 218.277 1322.85 -5.0604 1104.57 5000 5150 0.5 20 2 198.202 -14.7026 1.07418 212.905 5000 5200 0.5 20 2 181.331 1942.79 -9.71403 1761.46 Means square error(n1Σ(pi− ˆpi))=-0.488435944
Root mean square error(
q
1
nΣ(pi− ˆpi)2)=4.794921438 Mean absolute relative error(n1P
|pi− ˆpi|)=948.3998333 Maximum relative error(max(pi− ˆppi)
i )=9.71403 Maximum absolute error(max|pi− ˆpi|)=1761.46 St−1 = 5000 and S is tomorrow’s stock price m = 21, n = 16, and o = 100
Table 4.4: Cubic spline interpolation without wiggle problem in the r di-mension
K S T σ2 r Binomial Look-Up Rel. Err. Abs. Err.
5000 4800 0.5 20 2 363.322 363.349 -0.000074 0.026723 5000 4850 0.5 20 2 334.678 334.496 0.000543 0.181572 5000 4900 0.5 20 2 308.873 308.893 -0.000064 0.0197623 5000 4950 0.5 20 2 284.213 284.256 -0.000152 0.0431654 5000 5000 0.5 20 2 259.715 259.714 0.000002 0.000619204 5000 5050 0.5 20 2 238.83 238.86 -0.000125 0.0298814 5000 5100 0.5 20 2 218.277 218.303 -0.000119 0.0258755 5000 5150 0.5 20 2 198.202 198.24 -0.000190 0.0375936 5000 5200 0.5 20 2 181.331 181.331 0.000002 0.000324378 Means square error(n1Σ(pi− ˆpi))=-0.000020
Root mean square error(
q
1
nΣ(pi− ˆpi)2)=2.088586E-04 Mean absolute relative error(n1P
|pi− ˆpi|)=0.040613 Maximum relative error(max(pi− ˆppi)
i )=0.000543 Maximum absolute error(max|pi− ˆpi|)=0.181572 St−1 = 5000 and S is tomorrow’s stock price m = 21, n = 16, and o = 100
Numerical Results 27
Table 4.5: Polynomial interpolation with wiggle problem in the σ2dimension K S T σ2 r Binomial Look-Up Rel. Err. Abs. Err.
5000 4800 0.5 20 2 363.322 1727.57 -3.75492 1364.24 5000 4850 0.5 20 2 334.678 468.981 -0.40129 134.303 5000 4900 0.5 20 2 308.873 418.244 -0.354099 109.371 5000 4950 0.5 20 2 284.213 156.844 0.448145 127.369 5000 5000 0.5 20 2 259.715 268.458 -0.0336627 8.7427 5000 5050 0.5 20 2 238.83 308.644 -0.292315 69.8135 5000 5100 0.5 20 2 218.277 120.177 0.44943 98.1002 5000 5150 0.5 20 2 198.202 -1317.77 7.64861 1515.97 5000 5200 0.5 20 2 181.331 -6207.71 35.2341 6389.04 Means square error(n1Σ(pi− ˆpi))=4.327110922
Root mean square error(
q1
nΣ(pi− ˆpi)2)=12.0868077 Mean absolute relative error(n1P
|pi− ˆpi|)=1090.772156 Maximum relative error(max(pi− ˆpipi))=35.2341
Maximum absolute error(max|pi− ˆpi|)=6389.04 St−1 = 5000 and S is tomorrow’s stock price m = 21, n = 260, and o = 16
Numerical Results 28
Table 4.6: Cubic spline interpolation without wiggle problem in the σ2 di-mension
K S T σ2 r Binomial Look-Up Rel. Err. Abs. Err.
5000 4800 0.5 20 2 363.322 363.348 -0.000073 0.026415 5000 4850 0.5 20 2 334.678 334.495 0.000546 0.182720 5000 4900 0.5 20 2 308.873 308.893 -0.000064 0.019845 5000 4950 0.5 20 2 284.213 284.256 -0.000151 0.043022 5000 5000 0.5 20 2 259.715 259.715 0.000002 0.000458 5000 5050 0.5 20 2 238.83 238.86 -0.000125 0.029844 5000 5100 0.5 20 2 218.277 218.303 -0.000118 0.025763 5000 5150 0.5 20 2 198.202 198.239 -0.000189 0.037427 5000 5200 0.5 20 2 181.331 181.331 0.000001 0.000253 Means square error(n1Σ(pi− ˆpi))=-0.000019
Root mean square error(
q
1
nΣ(pi− ˆpi)2)=0.000209658 Mean absolute relative error(n1P
|pi− ˆpi|)=0.040638655 Maximum relative error(max(pi− ˆppi)
i )=0.000545959 Maximum absolute error(max|pi− ˆpi|)=0.182720 St−1 = 5000 and S is tomorrow’s stock price m = 21, n = 260, and o = 16
However, If we try to increase the partition in the σ2 or the r dimension, for example, we increase the points in r dimension in table (4.3) and (4.4) to 100 and σ2 dimension in table (4.5) and (4.6)to 260 , respectively. In table (4.3) and (4.5) we can see the polynomial wiggle problem occurs and the results are completely unacceptable, while in table (4.4) and (4.6) we can see that interpolating more points in the closed interval by cubic spline still makes the error acceptable, though it does not improve accuracy.
Therefore, it is very important to optimize the number of points partitioned to prevent polynomial wiggle problem for polynomial interpolation. Cubic spline interpolation is more preferable in this respect because we can partition these three dimensions as finer as we want. The reason why table (4.4) and (4.6) does not improve the accuracy compared with table (4.2) lies in that the majority of the error comes form the S dimension.
Numerical Results 29
Table 4.7: Polynomial interpolation straddling the exercise boundary K S T σ2 r Binomial Look-Up Rel. Err. Abs. Err.
5000 4178 0.25 20 2 822 822.019 -0.000024 0.019499 5000 4180 0.25 20 2 820 820.027 -0.000033 0.026761 5000 4182 0.25 20 2 818 818.035 -0.000043 0.035476 5000 4182.5 0.25 20 2 817.5 817.538 -0.000046 0.037898 5000 4183 0.25 20 2 817.0022 817.04 -0.000047 0.038222 5000 4184 0.25 20 2 816.011 816.046 -4.26E-05 0.0347828 5000 4186 0.25 20 2 814.03 814.058 -3.42E-05 0.0278103 Means square error(n1Σ(pi− ˆpi))=-0.000039
Root mean square error(
q
1
nΣ(pi− ˆpi)2)=0.000039 Mean absolute relative error(n1P
|pi− ˆpi|)=0.031492671 Maximum relative error(max(pi− ˆppi)
i )=0.000047 Maximum absolute error(max|pi− ˆpi|)=0.220449 St−1 = 4000 and S is tomorrow’s stock price m = 21, n = 16, and o = 16
Table 4.8: Cubic spline interpolation straddling the exercise boundary K S T σ2 r Binomial Look-Up Rel. Err. Abs. Err.
5000 4178 0.25 20 2 822 822.026 -0.000031 0.025683 5000 4180 0.25 20 2 820 820.034 -0.000041 0.033825 5000 4182 0.25 20 2 818 818.043 -0.000053 0.043135 5000 4182.5 0.25 20 2 817.5 817.546 -0.000056 0.045654 5000 4183 0.25 20 2 817.0022 817.048 -0.000056 0.046051 5000 4184 0.25 20 2 816.011 816.054 -0.000052 0.042686 5000 4186 0.25 20 2 814.03 814.066 -0.000044 0.035549 Means square error(n1Σ(pi− ˆpi))=-0.000048
Root mean square error(
q1
nΣ(pi− ˆpi)2)=0.000048 Mean absolute relative error(n1P
|pi− ˆpi|)=0.038940 Maximum relative error(max(pi− ˆppi)
i )=0.000056 Maximum absolute error(max|pi− ˆpi|)=0.046051 St−1 = 4000 and S is tomorrow’s stock price m = 21, n = 16, and o = 16
Numerical Results 30
Table 4.9: Polynomial interpolation with finer grid
K S T σ2 r Binomial Look-Up Rel. Err. Abs. Err.
5000 4178 0.25 20 2 822 822 0.000000 0.000036
5000 4180 0.25 20 2 820 820 0.000000 0.000118
5000 4182 0.25 20 2 818 818 0.000000 0.000000
5000 4182.5 0.25 20 2 817.5 817.501 -0.000001 0.001095 5000 4183 0.25 20 2 817.0022 817.003 -0.000001 0.001080 5000 4184 0.25 20 2 816.011 816.011 0.000000 0.000091 5000 4186 0.25 20 2 814.03 814.03 0.000000 0.000038 Means square error(n1Σ(pi− ˆpi))=-0.0000004
Root mean square error(
q
1
nΣ(pi− ˆpi)2)=0.000001 Mean absolute relative error(n1P
|pi− ˆpi|)=0.000351136 Maximum relative error(max(pi− ˆppi)
i )=0.000001 Maximum absolute error(max|pi− ˆpi|)=0.001095 St−1 = 4000 and S is tomorrow’s stock price m = 21, n = 16, and o = 16
Numerical Results 31
Table 4.10: Cubic spline interpolation with finer grid
K S T σ2 r Binomial Look-Up Rel. Err. Abs. Err.
5000 4178 0.25 20 2 822 822 0.000000 0.000007
5000 4180 0.25 20 2 820 820 0.000000 0.000090
5000 4182 0.25 20 2 818 818 0.000000 0.000000
5000 4182.5 0.25 20 2 817.5 817.501 -0.000001 0.001174 5000 4183 0.25 20 2 817.0022 817.003 -0.000001 0.001142 5000 4184 0.25 20 2 816.011 816.011 0.000000 0.000087 5000 4186 0.25 20 2 814.03 814.03 0.000000 0.000088 Means square error(n1Σ(pi− ˆpi))=-0.0000004
Root mean square error(
q
1
nΣ(pi− ˆpi)2)=0.000001 Mean absolute relative error(n1P
|pi− ˆpi|)=0.000370 Maximum relative error(max(pi− ˆppi)
i )=0.000001 Maximum absolute error(max|pi− ˆpi|)=0.001174 St−1 = 4000 and S is tomorrow’s stock price m = 21, n = 16, and o = 16
Interpolating in cubes that straddle the exercise boundary causes larger error as we can see in table (4.7) and (4.8). Our remedy for this problem is to keep track of the cubes straddling the exercise boundary and partition them finer. In table (4.9) and (4.10) we can see that the error becomes smaller. However, we should be careful that polynomial wiggle problem may happen because of finer grid for the polynomial interpolation.
Chapter 5 Conclusion
The results in this thesis presents the method of look-up table to price American op-tions. Polynomial interpolation and cubic spline interpolation are used to maintain its accuracy. One key factor in maintaining its accuracy depends on the number of partitions chosen. For the polynomial interpolation we should choose the number of partition properly so as to prevent the polynomial wiggle problem. Therefore, cubic spline interpolation is preferable because it becomes more accurate as the grid turns finer. The other key factor lies in that price function changes very rapidly at the exer-cise boundary. Therefore, keeping track of the cubes straddling the exerexer-cise boundary and partitioning the cubes finer are required to lower the error of interpolation.
32
Bibliography
[1] Lee W. Johnson and R. Dean Riess. Numerical Analysis. Addison-Wesley, 1982.
[2] Melvin J. and Maron. Numerical Analysis: A Pratical Approach. Macmillan, 1982.
[3] William H. Press., Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery Numerical Recipes in C. Cambridge, 1992.
[4] L.C.G.Rogers and D. Taylay. Numerical Methods in Finance. Prentice-Hall, 2000.
[5] Hull, John. Options, Futures, and Other Derivatives. 4th edition. Cambridge, 1997.
[6] Yuh-Dauh Lyuu “Financial Engineering and Computation.” Cambridge, 2002.
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