Table 3: Averaged prices of American puts for a range of asset prices .

Method

Asset price

Crank-Nicolson FD Formula

Growth rate

Difference (%)

Growth rate

Difference (%)

RMEL

36

7.109

7.094

−0.211

7.091

−0.253

38

6.154

6.139

−0.244

6.145

−0.146

40

5.318

5.305

−0.244

5.301

−0.320

42

4.588

4.575

−0.283

4.575

−0.283

44

3.953

3.941

−0.304

3.945

−0.202

Liu10

36

7.109

7.145

0.506

7.138

0.407

38

6.154

6.195

0.666

6.167

0.211

40

5.318

5.364

0.865

5.360

0.789

42

4.588

4.596

0.174

4.598

0.217

44

3.953

3.992

0.987

3.980

0.683

Note: the reported numbers in the first two columns represent, respectively, the asset prices and the corresponding “true” prices calculated using the Crank-Nicolson finite difference ( Grid) . Columns 3 and 5 report the price estimates with the growth rates of 6% and 100% for the two methods. Each reported value represents an estimate of option price for a particular combination of growth rate and asset price. The values reported in Columns 4 and 6 are the corresponding difference between the estimated price and the Crank-Nicolson finite difference prices, respectively. The difference is calculated by dividing the estimated price minus the “true” price of the Crank-Nicolson finite difference by the Crank-Nicolson finite difference price. For both RMEL and Liu10, each price estimate is the average of the values over three independent simulations. Each of the simulations generates 100,000 sample price paths and each path is divided into 73 potential exercise opportunities.