# Option Delta

### From Riski

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.

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## 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 be the cumulative normal distribution function. Let *d*_{1} be given by

.

### European Vanilla Call Option Delta

The option delta of European vanilla call option is given by

.

#### Proof

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

We take the partial derivative of *V*_{call} with respect to the price *S* of the option's underlying asset.

### European Vanilla Put Option Delta

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

.