Table 2: Averaged prices of American call options for a range of asset prices .

Method

Asset price

Black-Scholes formula

Growth rate

Difference (%)

Growth rate

Difference (%)

RMEL

36

5.041

5.013

−0.555

5.029

−0.238

38

6.164

6.123

−0.665

6.151

−0.211

40

7.389

7.357

−0.433

7.376

−0.176

42

8.708

8.653

−0.632

8.679

−0.333

44

10.112

10.065

−0.465

10.092

−0.198

Liu10

36

5.041

5.076

0.694

5.074

0.655

38

6.164

6.186

0.357

6.201

0.600

40

7.389

7.426

0.501

7.458

0.934

42

8.708

8.730

0.253

8.723

0.172

44

10.112

10.169

0.564

10.158

0.455

Note: the numbers in the first two columns represent, respectively, asset prices and the corresponding true Black-Scholes prices (as the underlying asset pays no dividend). Columns 3 and 5 report the price estimates with the growth rates of 6% and 100% for the two methods, and each reported value represents an estimate 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 and the “true” Black-Scholes prices, respectively. The difference is calculated by dividing the estimated price minus the Black-Scholes price by the Black-Scholes price. For both RMEL and Liu10, each reported price estimate is the average of the prices over three independent simulations. In each simulation, 100,000 risk-neutral price paths are generated and each path is divided into exercise opportunities.