Pricing American Options Using a Nonparametric Entropy Approach

Table 3

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

Method↓	Asset price 𝑆𝑡	Crank-Nicolson; FD Formula 𝑃𝑡	Growth rate; 𝜇=6%	Difference (%); 𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑𝑣𝑎𝑙𝑢𝑒−𝑃𝑡𝑃𝑡	Growth rate; 𝜇=100%	Difference (%); 𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑𝑣𝑎𝑙𝑢𝑒−𝑃𝑡𝑃𝑡
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 (800×800 Grid) (𝑟=0.06). 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.