Pricing American Options Using a Nonparametric Entropy Approach

Table 2

Averaged prices of American call options for a range of asset prices (𝑆_𝑡=36,38,40,42,44;𝐾=40).

Method↓	Asset price 𝑆𝑡	Black-Scholes formula 𝐶𝑡	Growth rate 𝜇=6%	Difference (%) 𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑𝑣𝑎𝑙𝑢𝑒−𝐶𝑡𝐶𝑡	Growth rate 𝜇=100%	Difference (%) 𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑𝑣𝑎𝑙𝑢𝑒−𝐶𝑡𝐶𝑡
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 𝑃=73 exercise opportunities.