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.
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 d1 be given by
European Vanilla Call Option Delta
The option delta of European vanilla call option is given by
Consider the Black-Scholes closed form solution for a European Vanilla Call Option given by
We take the partial derivative of Vcall 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