Approximating Option Delta Differences with Gamma

Summary

The document asks whether the difference in delta between two options with otherwise identical Black model inputs can be represented by the delta of another option. Its answer gives a first-order approximation instead: treat the option price as a function of its forward, so delta is its first derivative with respect to that forward. A Taylor expansion then relates the delta difference across two forward levels to the forward change multiplied by gamma.

For a finite forward gap, the approximation depends on where gamma is evaluated. The response recommends the midpoint between the two forwards as a theoretically appropriate point and says averaging the endpoint gammas is also reasonable. This is a local approximation, not an exact identity; accuracy depends on the size of the forward move and the curvature of the pricing function. The question does not specify the option strike, which the answer notes is held fixed along with other inputs.

Key ideas

• Option delta is the derivative of option value with respect to the forward when other inputs are fixed.
• A Taylor expansion approximates the delta difference as the forward difference multiplied by gamma.
• Gamma evaluated at the midpoint between forwards is suggested for the approximation.
• The relationship is first order and may be less accurate for larger forward changes.

Tags

# Difference in delta of options with same parameters


# Difference in delta of options with same parameters












Let's consider we have two hypothetical options with the same parameters under the Black Model (implied Vol, time to maturity etc.), the only difference being the value of the forward.

I am interested in the difference of delta between these two options. Instead of computing the delta of the two options and taking the difference, is there any equivalent where I could compute the delta of another hypothetical option with same parameters than above but with a forward price equal to the difference of the forwards of my two hypothetical options and a specific strike?

## Answer by Andrea (score 1)

https://quant.stackexchange.com/a/80542

If you are happy with a first order approximation, the difference between the delta of the options with different forwards is the gamma.

Let's call $c(x)$ the Black Scholes price when the forward is $x$, while all other values are fixed (including the strike, which you did not mention).

Then you have $\delta(x) = \frac{\partial}{\partial x}c(x)$.

You are interested in the difference $\delta(x_1) - \delta(x_2)$.

Using the Taylor expansion, this is $\delta(x_1) - \delta(x_2) \sim (x_1-x_2) \frac{\partial^2}{\partial x^2}c(x) + o(x_1-x_2)^2$

The question at which point to compute the gamma is more theoretical than anything else. The best would be at the middle point $\frac{x_1+x_2}{2}$, but the average of the gammas is good too.

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