PRÓRROGA DE PLAZOS

As Weideman (1999) pointed out, the major advantage of the Laguerre method over other Laplace transform inversion algorithms is that it provides one with function expansions of the Laplace transform, namely the Laguerre coefficients qn. They can be evaluated without the information on the t value. This means we may compute qn once for any given f

`

HsL, and use the values to evaluate fHtL for any t value. In this way, we can make computations very quick, even immediately since evaluation of the associated Laguerre functions lnHtL has no difficulty at all. However, this great advantage cannot be applied to the problem of pricing Asian options. This is because the Laplace transform of an Asian option is a function of the complex variable Λ, the normalized interest rate Ν and the normalized strike price q. The

normalized time to maturity h served as the t value in fHtL is, unfortunately, embedded in the Laplace transform. Thus, the values of qn changes with h. Hence, the Laguerre method cannot gain the benefits from function expansions in the application of pricing Asian options.

As Weideman (1999) pointed out, the major advantage of the Laguerre method over other Laplace transform inversion algorithms is that it provides one with function expansions of the Laplace transform, namely the Laguerre coefficients qn. They can be evaluated without the information on the t value. This

means we may compute qn once for any given f

`

HsL, and use the values to evaluate fHtL for any t value. In this way, we can make computations very quick, even immediately since evaluation of the associated Laguerre functions lnHtL has no difficulty at all. However, this great advantage cannot be applied to the problem of pricing Asian options. This is because the Laplace transform of an Asian option is a function of the complex variable Λ, the normalized interest rate Ν and the normalized strike price q. The

normalized time to maturity h served as the t value in fHtL is, unfortunately, embedded in the Laplace transform. Thus, the values of qn changes with h. Hence, the Laguerre method cannot gain the benefits from function expansions in the application of pricing Asian options.

The Laguerre method requires selection of a number of parameters. First of all, the Laplace transform of the Asian option price needs to be shifted so that no singularities are to the right of the contour. This can be done by setting the shifting parameter Α =1. Then, we can choose suitable values for other

parameters to have desired accuracy. The parameter m determines the truncation size. Increasing m gains more accuracy. The parameter Γ is used to control the discretization error in qn. We set Γ =25 so that

the maximum accuracy of the final result is about 23 significant digits. The parameter k is associated with Wynn’s Ε-algorithm. The choice of k=18 is recommended for the problem of pricing Asian

options. When k=0, fHtL is approximated by Ε₀m-1_{, namely S}

m-1 as if Wynn’s Ε-algorithm is not used.

The scaling parameters Σ and b are used together with the parameter k to improve the accuracy. For best

performance, we set Σ =0. The value of b is critical to the accuracy of the Laguerre method. Weideman

(1999) proposed two algorithms for computing the optimal parameters Σ and b. Here, we choose the

optimal b by simply checking the numerical results. Different case may have different optimal b. The working precision is very important in the computations. When an infinite expression is encountered during the calculation, it implies the current working precision is not sufficient. We should increase the working precision to solve this problem.

Table 4.28 shows the numerical results for the prices of Asian options using Laguerre method with working precision equal to 55 digits. The optimal b is chosen based on trials and it is a rough estimate. We notice that the optimal b increases as q becomes small. The Laguerre method yields good accuracy for normal cases and requires more truncation size for difficult cases. We carry out the same experiments in machine precision to check the effect of multi-precision. Numerical results in Table 4.29 show moderate size of m gives about 6-digit accuracy and larger m in fact reduces the accuracy. To obtain a accuracy of more than 5 digits for difficult cases, we need to use m=130 to have desired accuracy as

shown in Table 4.30.

Table 4.28. Results of the Laguerre method with arbitrary precision

Laguerre : k=18;Γ =25;Α =1;Σ =0

Case Reference b

m=65; wp=55 m=80; wp=55 m=100; wp=55

Accu Price ED Accu Price CPU ED Accu Price CPU ED Accu Price CPU

1 38 0.1931737903 1500 16.5 10 0.19317 0.53 24.4 14 0.19317 0.73 26.7 18 0.19317 0.98 2 37 0.2464156905 1500 15.5 10 0.24642 0.51 24.6 13 0.24642 0.72 27.0 18 0.24642 0.95 3 37 0.3062203648 1500 11.4 10 0.30622 0.59 23.6 13 0.30622 0.78 26.6 17 0.30622 1.03 4 17 0.05598604154 12 000 16.7 3 0.055980 14.2 19.0 3 0.055981 18.4 15.2 3 0.055963 22.8 5 38 0.2183875466 5000 24.9 8 0.21839 1.28 16.4 12 0.21839 1.65 25.9 16 0.21839 2.17 6 39 0.1722687410 3200 21.1 10 0.17227 0.97 23.4 12 0.17227 1.25 22.5 17 0.17227 1.64 7 30 0.3500952190 700 18.0 11 0.35010 0.41 21.2 14 0.35010 0.56 22.7 17 0.35010 0.80 8 36 2.815862016 1600 24.6 10 2.8159 0.83 27.3 13 2.8159 1.01 23.9 16 2.8159 1.42 9 38 2.310878887 1600 25.2 9 2.3109 0.66 27.0 13 2.3109 0.81 25.3 19 2.3109 1.17 10 38 1.879023661 1600 24.9 9 1.8790 0.55 26.4 14 1.8790 0.73 26.0 17 1.8790 1.01 11 36 7.895795199 1600 24.8 8 7.8958 0.56 23.9 13 7.8958 0.73 23.4 16 7.8958 1.03 12 37 6.935422632 1600 24.2 9 6.9354 0.53 23.6 13 6.9354 0.67 25.0 16 6.9354 1.01 13 38 6.070987190 1600 25.5 9 6.0710 0.55 21.8 13 6.0710 0.69 25.7 16 6.0710 1.0 14 22 2.697871538 12 000 16.7 5 2.6979 14.6 19.0 4 2.6978 18.1 15.2 5 2.6979 23.1 15 20 1.134741432 12 000 16.7 4 1.1348 13.3 19.0 4 1.1348 16.8 15.2 5 1.1347 21.3 16 17 0.2853249387 12 000 16.7 4 0.28534 12.4 19.0 4 0.28533 15.6 15.2 6 0.28533 19.9 17 38 14.98395833 5700 26.0 6 14.984 1.44 21.1 10 14.984 1.72 20.2 15 14.984 2.32 18 37 8.828758224 4700 20.1 7 8.8288 1.26 19.3 11 8.8288 1.59 23.5 15 8.8288 2.06 19 38 4.696709132 4400 26.3 8 4.6967 1.15 26.8 13 4.6967 1.54 27.2 15 4.6967 2.03

wp: working precision; ED: effective number of significant digits; Accu: accuracy measured by the number of significant digits; CPU: computing time in seconds.

Laguerre: the Laguerre method.

Table 4.29. Results of the Laguerre method with machine precision

Laguerre : k=18;Γ =25;Α =1;Σ =0; wp=MachinePrecision

Case Reference b m=65 m=80 m=100

Accu Price Accu Price CPU Accu Price CPU Accu Price CPU

1 38 0.1931737903 1500 6 0.19317 0.22 6 0.19317 0.30 4 0.19319 0.47 2 37 0.2464156905 1500 7 0.24642 0.22 6 0.24642 0.30 4 0.24644 0.44 3 37 0.3062203648 1500 5 0.30622 0.23 6 0.30622 0.33 4 0.30619 0.47 4 17 0.05598604154 12 000 2 0.056109 15.1 2 0.056086 19.0 0 0.099675 23.9 5 38 0.2183875466 5000 4 0.21837 0.81 6 0.21839 1.05 5 0.21839 1.45 6 39 0.1722687410 3200 6 0.17227 0.47 6 0.17227 0.64 5 0.17227 0.87 7 30 0.3500952190 700 4 0.35011 0.22 5 0.35010 0.28 4 0.35009 0.45 8 36 2.815862016 1600 6 2.8159 0.34 4 2.8158 0.47 4 2.8160 0.62 9 38 2.310878887 1600 6 2.3109 0.28 6 2.3109 0.37 4 2.3108 0.55 10 38 1.879023661 1600 6 1.8790 0.25 6 1.8790 0.36 4 1.8791 0.53 11 36 7.895795199 1600 5 7.8958 0.28 6 7.8958 0.33 4 7.8957 0.50 12 37 6.935422632 1600 5 6.9354 0.22 5 6.9354 0.33 3 6.9355 0.48 13 38 6.070987190 1600 6 6.0710 0.23 6 6.0710 0.31 4 6.0709 0.47 14 22 2.697871538 12 000 3 2.6997 15.3 3 2.6973 18.8 1 3.4631 24.1 15 20 1.134741432 12 000 2 1.1363 14.0 4 1.1345 17.6 4 1.1349 22.0 16 17 0.2853249387 12 000 2 0.29150 13.1 2 0.28702 16.2 0 0.16815 20.5 17 38 14.98395833 5700 4 14.980 1.0 7 14.984 1.26 5 14.984 1.67 18 37 8.828758224 4700 4 8.8287 0.80 4 8.8287 0.98 4 8.8289 1.37 19 38 4.696709132 4400 5 4.6967 0.67 6 4.6967 0.89 6 4.6967 1.19

wp: working precision; ED: effective number of significant digits; Accu: accuracy measured by the number of significant digits; CPU: computing time in seconds.

Laguerre: the Laguerre method.

Table 4.30. Results of the Laguerre method on the difficult cases with enhanced parameter settings

Laguerre : k=18;Γ =25;Α =1;Σ =0

Case Reference b m

=130; wp=60

Accu Price ED Accu Price CPU

4 17 0.05598604154 12 000 19.2 5 0.055986 30.8 14 22 2.697871538 12 000 19.2 6 2.6979 31.1 15 20 1.134741432 12 000 19.2 6 1.1347 28.6 16 17 0.2853249387 12 000 19.2 7 0.28532 26.5

wp: working precision; ED: effective number of significant digits; Accu: accuracy measured by the number of significant digits; CPU: computing time in seconds.

Laguerre: the Laguerre method.