Estimating Option Delta with Finite Differences

Summary

The document explains how to approximate an option’s delta numerically. Delta is the derivative of the option value with respect to the underlying asset price, describing how much the option value changes for a small change in that price.

When an analytic derivative is unavailable or a numerical estimate is desired, compare option values at two nearby underlying prices and divide their difference by the price change. The example describes a call price increase of 0.05 points as the underlying moves from 100 to 100.1, yielding a delta of 0.5 when the underlying change is interpreted as 0.1. The answer identifies this as a finite-difference approximation and notes that a related numerical approach is used for effective duration. The document does not specify how to choose the price increment or address the accuracy tradeoff between increment size and numerical error.

Key ideas

• Option delta is the derivative of option value with respect to the underlying price.
• A finite difference estimates delta by dividing the option value change by the underlying price change.
• The result depends on the size of the price increment used in the approximation.
• Effective duration uses a related numerical estimation approach.

Tags

# Taleb Modified Delta


# Taleb Modified Delta












How does one go about calculating the modified delta as proposed by Taleb in his book Dynamic hedging?

In his book he says its a change in the call price divided by a change in the underlying and provides the following example:

"If the call price picks up 0.05 points when the underlying asset moves from 100 to 100.1 then its delta will be 0.05/10 = 0.5."

How does he pick the price to change it at? Does he use the Black Scholes formula to calculate the the resulting option value when we change the price?

## Answer by Richi Wa (score 4)

https://quant.stackexchange.com/a/25036

What you (and he?) describe is the numerical derivative. The delta of an option is the infinitesimal change in value when the stock moves (infinitesimally) - thus $$ \Delta = \frac{dO}{dS}. $$ If you approximate this quantity be finite differences then you get $$ \Delta \approx \frac{O(S+\Delta S)-O(S)}{\Delta S}, $$ where $\Delta S$ is a change in the stock price (10 poins in your case). These numerical approaches are applied with duration too - there this is called effective duration.

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