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Moreover, assuming that the bond interest rate $r$ is constant, the above-mentioned equation can be simplified to:
\begin{equation}
P(S,t,T,K,r,\sigma)=C(S,t,T,K,r,\sigma)+K \cdot e^{-r(T-t)}-S(t)
\end{equation}
\subsection{Valuing European options}
\label{Valuing European options}
Assume that the stock price dynamics is described by the following stochastic differential equation:
%
\begin{equation} dS(t)=\mu \cdot S(t)dt + S(t)dW(t)
\end{equation}
% which is equivalent to the stock price following a geometric Brownian motion:
%
\begin{equation}
S(t) = S(0) \cdot \exp\left[(\mu - \frac{\sigma^{2}}{2})t + \sigma W(t)\right]
\end{equation}
% where $\mu$ is the instantaneous return of the stock, $\sigma$ is the immediate standard deviation (or volatility) of the underlying asset, while $dW(t)$ represents a small, uncertain movement specified by a normal distribution. Strictly speaking, the stock movement can be decomposed into two parts: the expected drift or expected returns and the uncertain …show more content…

\section{Vanilla options}
The following section illustrates the empirical results of the above-mentioned options pricing techniques for one sample, consisting of 20 American put options, referred to as small sample. Moreover, the comparison of the number of polynomial families, as well as the contrast between regression methods are conducted.
\section{Small sample results}
The analysis starts with the valuation of 20 American put options with the same set of parameters usually considered in the literature, implemented by Longstaff and Schwartz as well. The following Table 4.1 presents the results of pricing estimation of 20 options resulting from the combination of the following parameters:
\begin{gather*}
S(0) = \{ 36, 38, 40, 42, 44 \}\\
\sigma = \{ 0.2, 0.4 \}\\
T = \{ 1, 2 \}
\end{gather*}
The strike price for all options was considered to be $K = 40$, the risk-free interest rate $r = 6\%$ and the dividend yield $\delta = 0$. The simulation was performed with 100,000 paths, with 50 time steps per year and three weighted Laguerre

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