Finite-Difference Monte Carlo Delta and a Reversed Numerator
Summary
The document examines how to estimate a European option’s delta by finite differences when option values are computed with risk-neutral Monte Carlo expectations. For a positive bump in the underlying price, the forward difference compares the bumped-price value with the original value, divided by the bump size. The questioner spots that a cited expression orders the two conditional expectations in the reverse direction, which would change the sign of the estimate.
The accepted response agrees that the displayed expression is inconsistent with the stated finite-difference approximation and that the questioner’s sign intuition is correct. No corrected derivation, simulation results, or discussion of alternative estimators is provided. The material is therefore useful as a narrow sign-convention check, but offers little guidance on Monte Carlo variance reduction, bump-size selection, or delta estimation more broadly.
Key ideas
- A forward finite difference estimates delta by subtracting the unbumped option value from the bumped value.
- The risk-neutral pricing formula expresses each option value as a discounted expected payoff.
- Reversing the two values in the difference reverses the sign of the estimated delta.
- The accepted response confirms the stated ordering in the source expression is inconsistent.
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Full text
# How to approximate a delta using monte carlo methods and finite differences via Higham's book?
# How to approximate a delta using monte carlo methods and finite differences via Higham's book?
I'm currently taking a Mathematical Finance module at University and one of the recommended texts is “An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation” by D.J. Higham. One of the chapters in this book is about using Monte Carlo Methods to value an option and approximating Greeks.
In this chapter, he defines $V(S,t)$ to be the time t value of a European option with payoff $F(S_{T})$, when the underlying stock's price is S, and we aim to approximate the partial derivative of V with respect to S (the delta) at time zero. Using finite differences, we can say that:
$$ \frac{\partial V}{\partial S} \approx \frac{V(S+h,t)-V(S,t)}{h} $$
Hence, we can use the risk neutral valuation formula:
$$ V(S_{0},0) = e^{-rT}\mathbb{E}_{\mathbb{Q}}(F(S_{T})) $$
However, Higham goes on to write that we can hence approximate the time zero delta by computing Monte Carlo estimates of the two expected values in:
$$ e^{-rT}\frac{\mathbb{E}_{\mathbb{Q}}(F(S_{T}) \mid S(0)=S_{0})-\mathbb{E}_{\mathbb{Q}}(F(S_{T}) \mid S(0)=S_{0}+h)}{h} $$
I really don't understand why this is the case - shouldn't the numerator be the other way round? I'd have passed it off as a printing error but all the examples in that chapter (e.g. writing out a Monte Carlo algorithm) following the above expression are consistent with it. Could someone explain to me why the numerator isn't the other way round i.e. multiplied by -1?
## Answer by Quantuple (score 4, accepted)
https://quant.stackexchange.com/a/63987
You are right, this does not make sense. Your intuition is the correct one.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.