# Option Delta

In financial theory, an option's delta Δ measures an option's change in value due to a small change in the price of the option's underlying asset.

## Formulas

Let S be the price of the option's underlying asset. Let b be the cost of carry. Let r be the risk-free interest rate. Let t be the time until option expiry. Let $\Phi \left ( \cdot \right )$ be the cumulative normal distribution function. Let d1 be given by

$d_{1} = \frac{\ln \left ( \frac{S}{X} \right ) + \left ( b + \frac{\sigma^{2}}{2} \right ) t}{\sigma \sqrt{t}}$.

### European Vanilla Call Option Delta

The option delta of European vanilla call option is given by

$\Delta_{\text{call}} = \frac{\partial V_{\text{call}}}{\partial S} = e^{\left ( b - r \right ) t} \Phi \left ( d_{1} \right )$.

#### Proof

Consider the Black-Scholes closed form solution for a European Vanilla Call Option given by

$V_{\text{call}} = Se^{\left ( b - r \right ) t} \Phi \left ( d_{1} \right ) - Xe^{-rt} \Phi \left ( d_{2} \right )$

We take the partial derivative of Vcall with respect to the price S of the option's underlying asset. \begin{align} \Delta_{\text{call}} &= \frac{\partial V_{\text{call}}}{\partial S} \\ &= e^{\left ( b - r \right ) t} \Phi \left ( d_{1} \right ) + Se^{\left ( b - r \right ) t} \frac{\partial \Phi \left ( d_{1} \right )}{\partial d_{1}} \frac{\partial d_{1}}{\partial S} - Xe^{-rt} \frac{\partial \Phi \left ( d_{2} \right )}{\partial d_{2}} \frac{\partial d_{2}}{\partial d_{1}} \frac{\partial d_{1}}{\partial S} \\ &= e^{\left ( b - r \right ) t} \Phi \left ( d_{1} \right ) + \left [ Se^{\left ( b - r \right ) t} \phi \left ( d_{1} \right ) - Xe^{-rt} \phi \left ( d_{2} \right ) \right ] \frac{\partial d_{1}}{\partial S} \\ &= e^{\left ( b - r \right ) t} \Phi \left ( d_{1} \right ) \end{align}

### European Vanilla Put Option Delta

For a European vanilla put option, the option's delta may be expressed as

$\Delta_{\text{put}} = \frac{\partial V_{\text{put}}}{\partial S} = -e^{\left ( b - r \right ) t} \Phi \left ( - d_{1} \right )$.