Monte Carlo Sampling Error in Long-Maturity Option Pricing
Summary
The document investigates why a Monte Carlo estimate for a long-dated call differs substantially from binomial and Black–Scholes prices, while a shorter-dated example is closer. The response points to the very high volatility and long horizon, which produce a wide distribution of outcomes and increase sampling variance. Increasing the number of simulated paths can reduce the estimation error, though the computation becomes more expensive; the answer also notes that time-step size matters when the option is path dependent.
The option is clarified to be American, but with no dividends an American call should not be exercised early and has the same value as its European counterpart. Black–Scholes therefore provides a direct benchmark under the assumed model. The discrepancy is not proof that Monte Carlo is inherently unsuitable: the response raises possible implementation or convergence problems and suggests checking prices as maturity increases. It gives no path count, confidence interval, or code-level diagnosis, so the exact cause in the original simulation remains uncertain.
Key ideas
- Long horizons and high volatility can increase Monte Carlo sampling error by widening the distribution of simulated outcomes.
- Increasing the number of paths can improve precision, at greater computational cost.
- Time-step selection matters for path-dependent options.
- A nondividend-paying American call with nonnegative rates should not be exercised early and matches the European call value.
- Compare simulation estimates with an analytical benchmark and inspect convergence across maturities.
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# Discrepancy between binomial model, Black-Scholes and Monte-Carlo Simulation # 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. ## Answer by Richi Wa (score 3) https://quant.stackexchange.com/a/23057 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 ## Answer by user18663 (score 1) https://quant.stackexchange.com/a/23054 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.
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