Longstaff–Schwartz Monte Carlo Pricing and Exercise-Date Bias
Summary
The document examines an apparent inconsistency when using an R implementation of the Longstaff–Schwartz method to price an at-the-money put. A simulation with monthly exercise dates returns a value slightly below the stated Black–Scholes European put value, even though an American option should be at least as valuable as its European counterpart. The response explains that with a zero risk-free rate, no dividends, and equal spot and strike, the American and European put values coincide under the stated parity argument.
The key limitation is the finite number of exercise dates in the simulation: the option is Bermudan in this setup, and the estimate can fall below the theoretical European value. Adding exercise dates should increase the Bermudan value toward the continuous-exercise benchmark. The document offers a conceptual explanation and an example, but no systematic convergence study or detailed analysis of Monte Carlo estimation error.
Key ideas
- With zero interest and no dividends, the response argues that American and European put values coincide in the stated at-the-money case.
- A finite exercise schedule makes the simulated contract Bermudan rather than continuously exercisable.
- The response attributes the low estimate to the limited exercise dates and expects more dates to move the value toward the theoretical benchmark.
- The example does not quantify simulation error or demonstrate convergence across exercise schedules.
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Full text
# Longstaff Schwartz Algrorithm in R
# Longstaff Schwartz Algrorithm in R
I recently discovered the `LSMonteCarlo` library in R which basically determines the price of American options via Longstaff Schwartz method.
I tried the `AmerPutLSM` which per description first simulates paths of a geometric Brownian Motion and then uses the Longstaff Schwartz method.
The first few applications did non convince so I tried to price an European Put Option with this function:
```
library(LSMonteCarlo)
s0 <- 100
strike <- 100
sigma <- 0.03
set.seed(123)
AmerPutLSM(Spot = s0, sigma = sigma, n = 10000,
m = 12, Strike = strike, r = 0, mT = 1, dr = 0)
American Put Option
Price: 1.187689
```
So basically, I consider 10000 paths of a geometric Brownian motion with $\sigma = 0.03$ and an ATM put option (strike = $S_0$ = 100). The maturity of the option is set to 1 (`mT = 1`) and the number of time steps in the simulation is also set to 12 (`m = 12`). This means that the option can be exercised at the end of each month (Bermudan type). For simplicity I assumed no interest rate (`r = 0`) and zero dividends (`dr = 0`). This function tells me that the price of this option is about 1.188
But, if we compare this with the Black-Scholes Put Price of a European Put Option we get that
$$ V_{\text{European}} = \text{Strike} \cdot \Phi\Bigl(\frac \sigma 2 \Bigr) - S_0 \cdot \Phi\Bigl(-\frac \sigma 2 \Bigr) = 1.196782. $$
```
strike * pnorm(sigma/2) - s0 * pnorm(-sigma/2)
[1] 1.196782
```
This makes no sense since the value of the European Option should always be less then its American (or Bermudan) counterpart.
Does anyone have an explanation for this?
Thank you.
## Answer by Valometrics.com (score 1)
https://quant.stackexchange.com/a/51460
It is due the number of timestamps in your case. Actually, as the ZC rate is zero, the price of European and american options should be the same.
EDIT PROOF:
You know that for american options (see proof in pages 4,5 HERE): $S_T-K\leqslant C-P \leqslant S_T-Ke^{-rT}$
When the risk free rate is zero, you get that the call put parity remains valid $S_T-K= C-P$ As the european call price is the same as american call price without dividends, you can conclude that the the put prices are also the same. IN ADDITION, when the strike and spot are equal, you have: $C_A=C_E=P_A=P_E$
On the other hand, more you increase the number of timestamps, more the american option price should be higher (more stopping times) and converges to BS put price when the number of timestamps goes to $+\infty$. It explains the reason behind the fact that LS montecarlo price is lower than european put price.
I've done the test using DMS software and I've a smaller value than BS price: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.