Approximating an Option Price Change with Delta and Gamma

Summary

The document estimates how an option price changes when the underlying stock rises while volatility stays constant. It uses a second-order Taylor approximation: add delta times the underlying price change, then add half gamma times the squared change, to the current option value. Delta captures the local first-order sensitivity, while gamma adjusts for curvature in that sensitivity.

For the stated in-the-money option, the starting value is €11.50, delta is 0.58, gamma is 2, and the underlying rises by €0.50. Applying the approximation gives an estimated option value of €12.04. The calculation assumes the other inputs remain unchanged and is a local approximation, not a full repricing. Its accuracy can decline for larger underlying moves or when volatility, time, rates, or other pricing inputs change.

Key ideas

• Delta estimates the option price change for a small move in the underlying.
• Gamma adds a second-order adjustment for curvature in the option price response.
• The example holds volatility fixed and applies both sensitivities to a €0.50 underlying increase.
• The result is an approximation whose reliability is strongest for relatively small moves and stable inputs.

Tags

# Approximation of an option price


# Approximation of an option price












The value of an option in the money is 11.50 Euros. The parameters of the market are: -The price of the underlying stock: 81.4 Euros.

-The volatility ofthe underlying is : 34.65 %

The sensitivities are:

Delta = 58%

Gamma = 2

I would like to approximate the price of this option if the price of the underlying increase by 1/2 Euros (volatility not changed)

Thank you

## Answer by Gordon (score 4, accepted)

https://quant.stackexchange.com/a/23105

Since the volatility is not changing, we can assume that the only change is the underlying asset price $S$. Then \begin{align*} C(S+\Delta) &\approx C(S) + Delta \times\Delta +\frac{1}{2} Gamma \times \Delta^2 \\ &=11.50 + 0.58 \times 0.5 + \frac{1}{2}\times 2 \times (0.5)^2\\ &=12.04. \end{align*}

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