Monte Carlo Method for American Call Option (No Dividends)

Question

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 $V_0^{E_c} = 0.7907$.

Answer 1

The LS algo only approximates the continuation value no matter which and how many basis functions you use (unless it's infinite). Therefore it will always undervalue any option and sure enough also an American Call with no dividends will be under-priced too. I also remember wondering about this when I first tried it, but like you I found it always gives a price lower than the European for an American call w/o dividends. That is if you do everything else right: use a lot of time steps, so that it's practically "continuous exercise" and not Bermudan (that would also take care of the time discretization error by the way), and don't introduce something else that would give it an upward bias (like say using the same random numbers for both the fitting and the pricing part).