# Discrepancy between binomial model, Black-Scholes and Monte-Carlo Simulation

I try to use Monte-Carlo Simulation to price a 10-year call option. Based on below parameter,

S = 1, X = 1, volatility = 80%, T = 10, risk-free rate = 0.22%

The option value based on Monte-Carlo Simluation (Longstaff and Schwartz regression) is 0.4634.

But using Binomial model, the value is 0.7943, while using Black-Scholes model, the value is 0.7965. Is there any reason of large discrepancy using Monte-Carlo Simulation model.

When I consider to value short-maturity option by consider similar parameter

S = 1, X = 1, volatility = 80%, T = 1, risk-free rate = 0.22%

Option value based on Monte-Carlo Simluation is 0.2938. Based on binomial model, the value is 0.3112 while the value based on Black-Scholes model is 0.3116.

What is the reason of large discrepancy when using Monte-Carlo Simulation to value long-maturity option? Thanks.

• Why LS regression? Is the option American? – Kiwiakos Feb 2 '16 at 9:04
• Yes, it is American option – Dennis Feb 2 '16 at 14:31
• Then what does Black-Scholes mean? Also, LS depends heavily on the basis used as far as I recall. Are you using polynomial in log-price? – Kiwiakos Feb 2 '16 at 14:36
• Yes, I use polynomial, but using constant, price, price^2 and price^3 as basis functions – Dennis Feb 2 '16 at 22:25
• If I use polynomial in log-price, using constant, log(price), (log(price))^2 and (log(price))^3 as basis functions, the discrepany is even larger – Dennis Feb 2 '16 at 22:51

As your code works for the short maturity case, I assume that it is correct. The volatility of $80 \%$ is simply huge. Thus the area covered by the paths is huge too. As you can read e.g. here the sampling error is proportional to the variance of the process, which is huge in your case.

As a brute force solution you can just enlarge the number of samples. If your option is path dependent then you could reduce the step size. In any case MC will take long.

The OP just added the fact that the call is American. As there are no dividends mentioned we can assume that the stock does not pay any. Therefore (see here) the American option will never be exercies eary. Thus is has the same value as the European option.

Finally you don't need MC at all.

If you still want to apply MC then you should take care for your time-stepsize.

EDIT: Thinking again about it: your American call is in fact European. You apply LS-algo to it and get a price that much off - is your code ok? Any MC pricer should have an if-statment where it says that if dividends=0 then price it analytically. However, you can use the code to check the implementation of LS in order to price the American options where it is needed correctly.

What about increasing the time-to-maturity 1,2,4,6,8 to 10 years. How do the prices behave?

You should see:

• 1 years: 0.3116
• 2 years: 0.42965
• 4 years: 0.57815
• 6 years: 0.67497
• 8 years: 0.74436
• 10 years: 0.79635
• I already run the simulation for 10,000 times in excel...If I increase the number of simulation, the result is still not closer to binomial model or black-scholes model. – Dennis Feb 2 '16 at 14:30
• See my edit. What is your stepsize? – Ric Feb 2 '16 at 14:38
• stepsize = 4/250 = 0.04 years – Dennis Feb 2 '16 at 22:19
• As you can see above the price of this Option is the same as the price of a European call... Thus you have the analytical value.. In each step – Ric Feb 3 '16 at 6:48
• I edited the answer. Please provide your prices for 1,2,4,6,8 years and compare to the numbers above. Does your error increase? – Ric Feb 3 '16 at 12:13

Increase the number of paths in your simulation for the getting the terminal prices, and at some point your monte carlo option price will finally converge to Black scholes option price as you are using a very longer maturity call option i.e. 10 year call option.

• I can only use Monte-Carlo simulation by using excel. The maximum number of trials should be below 40,000 times – Dennis Feb 3 '16 at 4:04