Why Longstaff–Schwartz Can Undervalue a Non-Dividend American Call
Summary
The document examines a Monte Carlo valuation in which a simulated American call on a non-dividend-paying stock is priced slightly below a European call with the same strike and maturity. The example fits a Student t location-scale distribution to historical daily returns, simulates stock paths, and applies the Longstaff–Schwartz least-squares method to estimate early-exercise value.
The explanation is that Longstaff–Schwartz approximates continuation value using a finite set of basis functions. Approximation error can lead to an underestimate, including for an American call whose theoretical value should match the European call when there are no dividends. The response also notes that too few exercise dates leave a Bermudan-style discretization effect, while reusing the same random paths for regression and valuation can introduce upward bias. The reported price gap illustrates the issue but does not establish its size generally; accuracy depends on simulation design, exercise frequency, and regression choices.
Key ideas
- Longstaff–Schwartz estimates continuation value through regression and can undervalue an option when that estimate is imperfect.
- A non-dividend-paying American call should have the same value as its European counterpart under the stated setup.
- More exercise dates reduce the gap between discrete Bermudan exercise and continuous exercise.
- Using identical random paths for fitting and pricing can add upward bias.
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Full text
# Monte Carlo Method for American Call Option (No Dividends)
# 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).
- I collected one stock's daily returns over past 10 years.
- Plot the frequency distribution of all daily returns.
- 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}
- The set of Monte Carlo daily returns is given by $x_{MC}=\{-20\% : 0.001\%:20\%\}$.
- Select 2-million numbers randomly from set $x_{MC}$ according to the "t Location-Scale Distribution".
- Initial stock price $S_0 = 16.86 $, generate Mont-Carlo price paths for $60$ days.
- 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 by Yian Pap (score 1)
https://quant.stackexchange.com/a/44504
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).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.