Why Option Prices Must Be Revalued Across Spot Prices

Summary

The document explains why an option’s value is often calculated at multiple underlying spot prices rather than inferred from delta alone. Under the Black–Scholes framework, option value is nonlinear in spot, so delta, as the first derivative of value with respect to spot, provides only a local, first-order estimate of price change. That estimate becomes less accurate as spot moves farther from the starting point.

Gamma measures how delta changes as the underlying price changes, allowing delta and gamma together to form a better second-order approximation. This remains an approximation because gamma itself varies with spot. Revaluing the option at different spot levels therefore shows how the price and Greeks behave across scenarios. The discussion is conceptual and does not provide numerical examples or a broader model comparison; its explanation is specific to the nonlinear option-price relationship and approximation limits described in the excerpt.

Key ideas

• Delta estimates the option price’s local sensitivity to changes in spot.
• Because option value is nonlinear in spot, delta alone cannot accurately reprice large moves.
• Gamma measures how delta changes as spot changes and improves a second-order estimate.
• Gamma also varies with spot, so delta and gamma remain approximations across wider moves.

Tags

# Option value with different spot prices


# Option value with different spot prices












I found this post online which is plotting different results for option value and greeks depending on spot price.

Why would someone want to do calculate the value of the option with different spot prices? Is not delta what you would use to see the how the option changes with respect to spot price.

## Answer by Sanjay (score 3, accepted)

https://quant.stackexchange.com/a/44331

"Why would someone want to do calculate the value of the option with different spot prices?"

Because Delta is the derivative of the option price function wrt spot. Since the option price function is not linear in spot according to the BS model you cannot get the option prices for different spot just by Delta. What you are asking about is simply using a first order approximation.

Here is a (primitive) graph demonstrating the problem:

As @Bikenfly mentions a keyword here is Gamma. Using Delta and Gamma will then be a second order approximation

## Answer by Bikenfly (score 2)

https://quant.stackexchange.com/a/44330

Understand gamma - delta is an approximation of the price change, but as you extrapolate out, the approximation gets worse. Gamma shows how much delta changes based upon a change in the underlying spot price. So delta and gamma together will give a better approximation. However, gamma is also an approximation of the delta change and also changes as the spot price changes.

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.