# Monte Carlo Method for American Call Option (No Dividends)

I tried to pricing the American Call option using "Longstaff-Schwartz" least squares method. However, I found the American call option is always lower than the Monte Carlo European call option (they should be equal to each other).

1. I collected one stock's daily returns over past 10 years.
2. Plot the frequency distribution of all daily returns.
3. Found the "t Location-Scale Distribution" is the best fitted distribution, where $$\mu=1.0118\times10^{-4}$$, $$\sigma = 0.0076$$ and $$\nu=2.5977$$, the probability density function is given by $$\begin{equation} p(x)=\frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\sigma\sqrt{\nu\pi}\Gamma\left(\frac{\nu}{2}\right)}\left[\frac{\nu+\left(\frac{x-\mu}{\sigma}\right)^2}{\nu}\right]^{-\frac{\nu+1}{2}}. \end{equation}$$
4. The set of Monte Carlo daily returns is given by $$x_{MC}=\{-20\% : 0.001\%:20\%\}$$.
5. Select 2-million numbers randomly from set $$x_{MC}$$ according to the "t Location-Scale Distribution".
6. Initial stock price $$S_0 = 16.86$$, generate Mont-Carlo price paths for $$60$$ days.
7. The strike price $$K=S_0=16.86$$, risk-free rate $$r=3.95\%$$.

One of results is: American Call option $$V_0^{A_c} = 0.7895$$ and European Call option $$V_0^{E_c} = 0.7907$$.

• For "Longstaff-Schwartz" least squares method, I also tried different orders of "basis functions", but it doesn't change the situation. – Stephen Ge Mar 7 '19 at 2:35
• One suggestion: Check to see how many paths are such that early exercise is occurring. – dm63 Mar 7 '19 at 3:40
• Regarding point 3, what distribution is this $p(x)$ fitting to? – Hans Mar 7 '19 at 8:21
• @Hans p(x) is the pdf of t-location scale distribution – Stephen Ge Mar 7 '19 at 8:39
• You are fitting this t-distribution to something (a distribution of some sort). What is this target distribution you are fitting to? – Hans Mar 7 '19 at 9:35