# 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

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*}