Canonical Least-Squares Monte Carlo Valuation of American Options: Convergence and Empirical Pricing Analysis

Table 3

Price estimates for both American puts with various levels of number of paths 𝑁, number of regressors 𝑀 and type of regressors.

Type of regressor ↓	𝑀↓	𝑁→	5,000			10,000			20,000			30,000
		BT	LS	Bias	S.E.	LS	Bias	S.E.	LS	Bias	S.E.	LS	Bias	S.E.
Monomials	2	7.289	6.994	−0.295	0.308	7.020	−0.269	0.218	7.015	−0.274	0.292	7.023	−0.266	0.215
		59.038	58.088	−0.950	0.413	58.136	−0.902	0.414	59.683	0.645	0.381	59.690	0.652	0.392
	3	7.289	7.101	−0.188	0.192	7.105	−0.184	0.098	7.104	−0.185	0.054	7.208	−0.081	0.126
		59.038	58.417	−0.621	0.345	59.280	0.242	0.332	58.497	−0.541	0.317	58.497	−0.541	0.312
	4	7.289	7.113	−0.176	0.155	7.122	−0.167	0.083	7.130	−0.159	0.076	7.164	−0.125	0.068
		59.038	59.494	0.456	0.353	58.459	−0.579	0.341	58.517	−0.521	0.298	58.531	−0.507	0.273
	5	7.289	7.129	−0.160	0.105	7.192	−0.097	0.055	7.187	−0.102	0.068	7.191	−0.098	0.071
		59.038	59.504	0.466	0.347	58.584	−0.454	0.265	58.785	−0.253	0.145	58.761	−0.277	0.165
Weighted Laguerre	2	7.289	7.004	−0.285	0.208	7.020	−0.269	0.198	7.023	−0.266	0.195	7.033	−0.256	0.169
		59.038	58.208	−0.830	0.392	58.430	−0.608	0.314	59.589	0.551	0.331	59.597	0.559	0.381
	3	7.289	7.108	−0.181	0.091	7.146	−0.143	0.076	7.164	−0.125	0.048	7.178	−0.111	0.034
		59.038	58.419	−0.619	0.353	58.443	−0.595	0.327	58.486	−0.552	0.313	58.470	−0.568	0.314
	4	7.289	7.167	−0.122	0.145	7.171	−0.118	0.041	7.169	−0.120	0.053	7.174	−0.115	0.078
		59.038	58.540	−0.498	0.352	58.539	−0.499	0.323	59.450	0.416	0.285	59.447	0.409	0.301
	5	7.289	7.180	−0.109	0.009	7.201	−0.088	0.015	47.184	−0.105	0.028	7.193	−0.096	0.019
		59.038	59.524	0.486	0.358	58.570	−0.468	0.254	58.773	−0.265	0.157	58.802	−0.236	0.146
Shifted Legendre	2	7.289	7.087	−0.202	0.125	7.131	−0.158	0.071	7.168	−0.121	0.034	7.180	−0.109	0.032
		59.038	58.216	−0.822	0.300	58.275	−0.763	0.260	58.422	−0.616	0.206	58.318	−0.720	0.302
	3	7.289	7.160	−0.129	0.052	7.182	−0.107	0.045	7.227	−0.062	0.015	7.219	−0.070	0.007
		59.038	59.395	0.357	0.279	58.758	−0.280	0.132	58.810	−0.223	0.199	58.803	−0.235	0.287
	4	7.289	7.162	−0.127	0.150	7.184	−0.105	0.128	57.190	−0.099	0.051	7.223	−0.066	0.031
		59.038	58.69	−0.348	0.270	58.764	−0.274	0.166	58.822	−0.216	0.162	58.824	−0.214	0.118
	5	7.289	7.182	−0.107	0.110	7.222	−0.067	0.030	7.207	−0.082	0.030	7.232	−0.057	0.010
		59.038	58.722	−0.316	0.206	58.781	−0.257	0.165	58.829	−0.209	0.103	58.848	−0.190	0.114

Note. BT and LS represent the price from binomial tree model and the price from the least-squares algorithm used in CLM method, respectively. Bias is the difference between LS and BT and S.E. is the standard error. In each cell, the number above resulted from the first put option and the bottom is from the second put. The parameters for both puts are given in Table 1.