Estimating Greek Standard Errors with Monte Carlo Bump-and-Revalue

Summary

The document raises a statistical question about uncertainty in option Greeks estimated with Monte Carlo simulation. It starts from a simulated option value with a standard error, then applies central finite differences to estimate Delta from prices at upward and downward spot bumps. It also gives a second difference for Gamma using the bumped and unbumped values.

The central issue is how to propagate the simulation errors through these formulas. The answer is not included, so the document provides no prescription for combining standard errors or accounting for dependence between estimates. That dependence matters when simulations use shared random draws across bumps, as is common with common random numbers; treating the price estimates as independent would imply a different error calculation. The post is therefore useful as a clearly framed estimation problem, but it does not establish a solution, provide results, or discuss finite-difference bias and bump-size trade-offs.

Key ideas

• Monte Carlo option values have sampling error that carries into finite-difference Greek estimates.
• Central differences estimate Delta using prices at symmetric spot bumps.
• A second difference estimates Gamma from bumped and unbumped option values.
• The uncertainty calculation depends on covariance between the simulated price estimates.
• The source poses the error-propagation question but does not answer it.

Tags

# Combine standard error in finite difference with Monte Carlo


# Combine standard error in finite difference with Monte Carlo












I'm using Montecarlo to estimate the value of an option,

$$\overline V(S_T, r, \sigma, T;N)=\mathbb{E} \left[V(S_T, r, \sigma, T)\right]$$ which comes with a standard error $SE$.

I'm using "bump-and-reval" (finite differences) to compute the greeks. So for example for Delta:

$$ \Delta \approx \frac{\overline V(S_T+\Delta S, r, \sigma, T;N)-\overline V(S_T-\Delta S, r, \sigma, T;N)}{2\Delta S} $$

my question is, what's the right way to combine the standard errors to get the right one for $\Delta$?

What about for a second order greek, like Gamma:

$$ \Gamma \approx \frac{\overline V(S_T+\Delta S, r, \sigma, T;N)-2\overline V(S_T, r, \sigma, T;N)+\overline V(S_T-\Delta S, r, \sigma, T;N)}{(\Delta S)^2} $$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.