Numerical Results for American-Style Asian OptionsOptions

Experimental results for nUnifSplA, nUnifSplA2, and nUnifCvgA will be presented below. All the experimental results reported here are based on an Athlon Thunder-bird 1.33GHz PC with 1GB DRAM. Recall that nUnifSplA and nUnifSplA2 provide upper bounds, whereas nUnifCvgA gives lower bounds (the claims will be proved in the next section). Fig. 4.11 plots the pricing errors of nUnifCvgA:nUnifSplA and nUnifCvgA:nUnifSplA2 as functions of n. The pricing errors are measured by nUnifSplA− nUnifCvgA and nUnifSplA2 − nUnifCvgA. Not surprisingly, both in-crease with n. But nUnifSplA2 has a much smaller pricing error than nUnifSplA.

Because the pricing errors of nUnifCvgA:nUnifSplA2 are extremely small, the algo-rithm practically gives the desired option value.

300 350 400 450 500 n 0.0005

0.001 0.0015 0.002

nUnifSplA-nUnifCvgA

nUnifSplA2-nUnifCvgA Error

Figure 4.11: Pricing errors of nUnifCvgA:nUnifSplA and nUnifCvgA:nUnifSplA2.

The data are: S₀ = X = 100, σ = 30%, r = 20% per annum, and T = 1. All use an average of k = 500 buckets per node.

To further investigate the performance of nUnifCvgA:nUnifSplA2 vs. nUnifCvgA:nUnifSplA, we perform a comprehensive test in Table 4.5. Although both algorithms perform well

with small pricing errors, nUnifCvgA:nUnifSplA2 is more competitive. Perhaps the most important lesson to draw from that table is that the pricing error rises as σ in-creases. This phenomenon is most apparent in the case of nUnifCvgA:nUnifSplA. For the algorithm with a tighter range bound, nUnifCvgA:nUnifSplA2, the absolute pric-ing errors never exceed 0.000454. The advantage of the two-phase nUnifCvgA:nUnifSplA2 again demonstrates the benefit of taking advantage of limited prefix sum ranges. In-deed, the data in Table 4.5 suggest that a k of only 300 is sufficient in most cases to price the option accurately.

We next investigate the convergence behavior of nUnifCvgA:nUnifSplA2. The results tabulated in Table 4.6 are based on k = 8n; hence the running time is O(n³).

To our knowledge, no papers in the literature on American-style Asian options men-tion numerical results for volatilities larger than 50%. Interestingly, Table 4.6 shows our O(n³) algorithm produces very tight range bounds for σ as high as 100%. Even where the algorithm fails to converge when σ = 100% and T = 5 year, the relative errors are less than 0.14% up to n = 400.

As mentioned above, nUnifCvgA:nUnifSplA2 fails to converge only when σ and T are both large (σ = 100% and T = 5 year in Table 4.6). This is in contrast to the case of European-style Asian options, where the related nUnifCvg:nUnifSpl’s convergence is not affected by a large σ as shown in Table 4.4. The discrepancy can be attributed to the fact that prefix sum ranges in the case of European-style Asian options are limited from above by (n + 1)X but no such limits are available in the case of American-style Asian options. Indeed, without this limit, the

full-σ X r nUnifCvgA nUnifSplA2 nUnifSplA nUnifSplA nUnifSplA2

−nUnifCvgA −nUnifCvgA 0.1 95 0.05 8.088364 8.088422 8.088522 0.000158 0.000058 0.1 95 0.15 11.267781 11.267846 11.267954 0.000173 0.000065 0.1 105 0.05 1.344226 1.344292 1.344403 0.000177 0.000066 0.1 105 0.15 3.623832 3.623887 3.623980 0.000148 0.000055 0.3 95 0.05 12.358376 12.358517 12.359182 0.000806 0.000141 0.3 95 0.15 14.428086 14.428229 14.428934 0.000848 0.000143 0.3 105 0.05 6.311839 6.311984 6.312741 0.000902 0.000145 0.3 105 0.15 8.208416 8.208553 8.209280 0.000864 0.000137 0.5 95 0.05 17.341037 17.341237 17.344196 0.003159 0.000200 0.5 95 0.15 18.922948 18.923150 18.926233 0.003285 0.000202 0.5 105 0.05 11.623434 11.623636 11.627077 0.003643 0.000202 0.5 105 0.15 13.214077 13.214273 13.217725 0.003648 0.000196 0.7 95 0.05 22.536275 22.536540 22.552333 0.016058 0.000265 0.7 95 0.15 23.775811 23.776080 23.792101 0.016290 0.000269 0.7 105 0.05 17.065704 17.065979 17.084335 0.018631 0.000275 0.7 105 0.15 18.382506 18.382779 18.401274 0.018768 0.000273 0.9 95 0.05 27.841546 27.841955 27.952798 0.111252 0.000409 0.9 95 0.15 28.797383 28.797804 28.908081 0.110698 0.000421 0.9 105 0.05 22.587415 22.587869 22.719667 0.132252 0.000454 0.9 105 0.15 23.650191 23.650639 23.779582 0.129391 0.000448

Table 4.5: Comprehensive Tests for nUnifSplA, nUnifSplA2, and nUnifCvgA.

The data are: S₀ = 100, n = 300, k = 500 and T = 1. Algorithms nUnifCvgA, nUnifSplA2, and nUnifSplA take an average of 4.60 seconds in generating their respective results.

range nUnifCvg:nUnifSpl also fails to converge for large σ or large T . (Because u = e^σ

√T /n, the prefix sum ranges depend on σ and T in an exponential manner.) This fact is confirmed in Table 4.7. We conjecture that, without some limits on prefix sums, deterministic algorithms will eventually fail when σ is large enough unless T is small or the number of buckets per node, k, is properly increased.

σ T n nUnifCvgA nUnifSplA2 nUnifSplA2 − nUnifCvgA

10% 0.25 50 1.937256 1.937271 0.000015

100 1.947621 1.947626 0.000005

200 1.953399 1.953401 0.000002

400 1.956484 1.956485 0.000001

50% 1.00 50 14.763087 14.763184 0.000097 100 14.912143 14.912180 0.000037 200 14.996588 14.996602 0.000014 400 15.042595 15.042600 0.000005 50% 5.00 50 33.444456 33.444608 0.000152 100 33.837743 33.837809 0.000066 200 34.062623 34.062648 0.000025 400 34.184574 34.184584 0.000010 100% 1.00 50 27.595989 27.596134 0.000145 100 27.963737 27.963799 0.000062 200 28.175147 28.175170 0.000023 400 28.290796 28.290804 0.000008 100% 5.00 50 58.262845 58.262854 0.000009 100 59.448244 59.448330 0.000086 200 60.130631 60.130817 0.000186 400 60.501092 60.582166 0.081074

Table 4.6: Convergence of nUnifCvgA:nUnifSplA2.

Both algorithms use k = 8n. The data are: S₀= X = 100 and r = 10% per annum.

σ T n nUnifCvg nUnifSpl nUnifSpl − nUnifCvg 10% 0.25 50 1.848515 1.848533 0.000018

100 1.850035 1.850044 0.000009 200 1.850809 1.850813 0.000004 400 1.851199 1.851201 0.000002 50% 1.00 50 13.185396 13.185639 0.000243 100 13.195530 13.195701 0.000171 200 13.200738 13.200898 0.000160 400 13.203354 13.203612 0.000258 50% 5.00 50 28.387935 28.389159 0.001224 100 28.395811 28.398385 0.002574 200 28.397866 28.413588 0.015722 400 28.370135 28.920558 0.550423 100% 1.00 50 23.410075 23.411095 0.001020 100 23.434776 23.436654 0.001878 200 23.446473 23.453710 0.007237 400 23.442168 23.561833 0.119665 100% 5.00 50 42.747360 42.835029 0.087669 100 42.135964 44.997394 2.861430 200 38.257653 69.258656 31.001003 400 38.224317 184.271619 146.047302

Table 4.7: Convergence of the Full-Range nUnifCvg:nUnifSpl.

The setup is identical to Table 4.4 except that k = 8n. The full-range algorithm may fail to converge despite that more buckets are allocated than in Table 4.4

Conclusions

Asian options are a kind of strongly path-dependent derivatives. How to price such derivatives efficiently and accurately has been a long-standing research and practical problem. Asian options can be priced on the lattice. Unfortunately, only exponential-time algorithms are currently available if such options are to be priced on the lattice exactly. Although efficient approximation methods are available, most of them lack convergence guarantees or error controls. This dissertation addresses the Asian op-tion pricing problem with two different lattice methods to meet the efficiency and accuracy requirements. First, a new trinomial lattice for pricing Asian options is pro-posed, and the resulting exact pricing algorithm is proved to be the first one to break the exponential-time barrier. Second, range-bound algorithms are developed. These approximation algorithms are proved to converge to the true option value for pricing European-style Asian options. Extensive experiments also reveal that the extension of these algorithms work well numerically for American-style Asian options.

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Arbitrage, 20

Lattice and the PDE approach, 5, 29

Monte Carlo approach, 3, 26 Backward induction

Exercise boundary, 62, 63, 86 Financial engineering, 1

Maximum prefix sum, 35, 82 Minimum prefix sum, 35, 82 Multiresolution lattice, 41

Path-dependent option, 2 Practical lattice, 82 Prefix sum, 35, 82

Maximum prefix sum, 35, 82 Minimum prefix sum, 35, 82 Prefix sum range, 35, 82 Prefix sum range, 35, 82 Put option, 2, 14

Range bound algorithm, 8, 60 Convergence, 11, 61 Error analysis

European-style option, 71 Full range algorithm, 83 Prefix sum range reduction

American-style option, 86 European-style option, 65 Proof

American-style option, 88 European-style option, 75 Survey, 60

Replication, 19

Risk neutral probability, 20 Risk neutral valuation, 20 Round-down bucket, 72 Round-up bucket, 73 Speculator, 17

Terminal bucket, 88 Trinomial lattice, 45 UnifAvg, 72

UnifDown, 69 UnifUp, 69 Up bucket, 68