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Moreover, assuming that the bond interest rate $r$ is constant, the above-mentioned equation can be simplified to:

P(S,t,T,K,r,\sigma)=C(S,t,T,K,r,\sigma)+K \cdot e^{-r(T-t)}-S(t)

\subsection{Valuing European options}
\label{Valuing European options}
Assume that the stock price dynamics is described by the following stochastic differential equation:
%
dS(t)=\mu \cdot S(t)dt + S(t)dW(t)

% which is equivalent to the stock price following a geometric Brownian motion:
%

S(t) = S(0) \cdot \exp\left[(\mu - \frac{\sigma^{2}}{2})t + \sigma W(t)\right]

% 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
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