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Estimating Monte Carlo and Finite-Difference Pricing Accuracy

Article Quant Q&A · Author: Marquee
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Summary

The discussion asks how closely Monte Carlo (MC) and finite-difference methods (FDM) should match exact option prices and Greeks under specified path and grid counts. It describes a delta obtained by bumping an FDM price that differs modestly from the exact delta, and asks whether that precision is typical.

The reply says accuracy depends on volatility, moneyness, option type, and time to expiry. It contrasts cases needing many simulations, such as long-dated, far out-of-the-money options, with near-the-money, short-lived options that can require fewer. Repeated runs with different random seeds can show simulation variability, while MC standard error decreases in proportion to the inverse square root of the path count. The example is only a rough benchmark: it concerns an apparently near-the-money vanilla option, and it does not establish a general accuracy level for FDM grids or bumped Greeks.

Key ideas

  • Pricing and Greek accuracy depends on volatility, moneyness, option structure, and time to expiry.
  • Monte Carlo path requirements can vary substantially across option scenarios.
  • Repeated runs with different random seeds reveal sampling variability.
  • Monte Carlo standard error declines with the square root of the number of paths.
  • The reported delta difference is an example, not a universal FDM accuracy guarantee.

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OptionsStatisticsDerivatives pricing
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# How many decimals of accuracy can I expect from FDM and MC (both valuation and risk)


# How many decimals of accuracy can I expect from FDM and MC (both valuation and risk)












I have implemented some Monte Carlo and FDM code. I can then get greeks by bumping.

I am comparing to to exact formulas of price + greeks, and am wondering how many decimals of accuracy I can expect for reasonable inputs (say, 100.000 paths and 250 steps in MC, and 2000x2000 gridsize in FDM).

For example, take FDM and estimating the delta by bumping: I am personally getting 0.5281 in delta from exact formula, and 0.5298 from FDM by bumping, so about 2 decimals worth of accuracy. Is this normal?

## Answer by RandyF (score 1)

https://quant.stackexchange.com/a/42537

This depends on quite a few other inputs. If you're comparing deltas, what's the volatility of the underlying and the moneyness of the derivative (whether call, put, or something more exotic). I've done valuations that need 10 million simulations (5x out of the money options with 10 years until expiration) and others where 10,000 is perfectly sufficient (at or in the money with a short life).

That delta seems fair under common circumstances and 100k paths (since the option ,I presume something similar to a call option, is fairly near the money with a delta ~=0.5).

You can always run this a couple times with a different seed and see if the result changes (it should, but centered around the actual value). Additionally, as you increase or decrease the number of paths, the standard error will change by a factor of $\frac{1}{\sqrt{n}}$.

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.