# Least Square Monte Carlo - american Call Option

An American Call Option on an non dividend paying stock has the same value as a european one. I tired to compare the results given by the LSM with the results given by the B&S formular. It seems like LSM is really bad for valuing american call options. It always overestimate the value by 5-20%.

But why is the algorithm so bad for call options and so good for put options ? Are there some studies concerning this problem ?

If implemented properly, least-squares Monte Carlo as originally suggested by Longstaff-Schwartz should allow you to identify sub-optimal exercise dates and a lower bound of the true option price. There are many articles out there discussing this non trivial topic. @MarkJoshi can probably shed some more light, see this nice paper.

You claim that your LSM procedure overestimates the true option price: I guess that you did not use 2 independent set of paths i.e. one for calibrating the exercise boundary through regressions and a separate one for determining the exercise dates.

Regarding the precision of $5\%-20\%$, IMHO you must have a problem with your implementation. However, it is difficult to say anything more concerning your precision issue since you don't provide all the details: How many paths do you simulate and under what working modelling assumption, did you only use ITM paths during the regression pass, what type of basis functions did you choose and how many of them etc.

• I am using 100,000 paths, 50 exercise dates and the first three laguere polynomials. I also calculated some values for american put options with the LSM. I compared it with a 15000 steps binomial tree. The results are just fine. Always a sligt underestimation of 0.5 – 1%. This totally makes sense. However, for call options it always overestimate. For example a Call Option with strike price = 10$, stock price = 8$, risk free rate 0.05 and a volatility of 0.4, I get a price of 0.7569 (B&S = 0.6725). I did not use 2 independent set of paths. Should I have done that ? – FinanceStudent Sep 19 '16 at 13:56
• I would advise it yes. Also check and examine the paths for which you identify early exercise opportunities. You seem to have an abnormal number of them, this might give you an idea as to where to look next. What about the paths you use as part of the regression sweep? It helps to perform the regression in a standardised spot space + only consider in-the-money paths as suggested in the original L&S paper. – Quantuple Sep 19 '16 at 14:12
• @FinanceStudent, I rencetly came upon this SE question which may be of interest to you: quant.stackexchange.com/questions/10458/… – Quantuple Sep 21 '16 at 12:17
• Hi @Vim. Simulate a first set of $M_1$ paths. Work backwards applying the LS algorithm. At each call decision date, store the coefficients defining the continuation value at that date. This produces a biased estimator (limited number of simulations => high bias due to foresight; limited number of basis functions => low-bias due to suboptimality). – Quantuple Feb 25 at 8:27
• Now, simulate another set of $M_2$ independent paths. Work forward: at each call decision date compare the continuation value obtained using the coefficients stored as part of the previous simulation to the exercise value. The stopping time for a given path is then defined as the first time one can exercise. The PV for that path is then the exercise value at the identified exercise time. Taking the average yields to a low-biased estimate of the option price. – Quantuple Feb 25 at 8:28

Maybe I have something wrong with my calculation. But this is not my whole point. When calculating the value of an American put I totally understand why the LSM algorithms underestimates the value. Mainly because I only have an estimated conditional expectation function. But with Call option I don’t understand why the algorithms also have to underestimate the value. For me it makes total sense that the LSM approach overestimates the value. Again I only have an estimate of the conditional expectation function. So there can happen two things. Either the LSM algorithms determines that early exercise is favourable. Then the value has to be higher than the European value (overestimation). Or the LSM algorithm determines that early exercise is disadvantageous. Then the value must be exactly the same.

• The algorithm believes that an early opportunity to exercise the AmCall exists and will be profitable, but on average exeprience should show that these "opportunities" lose money (i.e you could have made more by holding). This is what is difficult to understand. – noob2 Sep 21 '16 at 11:52