# What is Dual Delta?

I understand that it is the partial derivative of option price with respect to strike. What is it used for though? What does your dual delta signify?

## 2 Answers

Dual Delta is the derivative of option value with respect to the strike: $$\frac{\partial C}{\partial K}$$. The ordinary Delta is of course $$\frac{\partial C}{\partial S}$$.

In the BSM framework Dual Delta evaluates to $$\frac{\partial C}{\partial K}=-e^{-r T} N(d_2)$$, it is therefore closely related to the pseudo probability of exercise $$N(d_2)$$. In fact it is minus the price of an Arrow Debreu security that pays 1 USD at time T if the Call is in the money and 0 otherwise.

• probably derivative of Call by Strike K will be -e^(-rT) N(d_2). Cause when strike is increasing, the price of vanilla Call is decreasing. And I think yes, pseudo probability of exercise is N(d_2), it's correct
– Mike
Commented Jul 24, 2021 at 21:22
• You are correct, I left out a minus sign en.wikipedia.org/wiki/… Thanks for noticing this. Commented Jul 24, 2021 at 21:30

Dual Delta, dual Gamma and dual DdelV can be used to calculate the "local volatility" that is induced by a given volatility surface for example (the local volatility can be seen as the instantaneous volatility that the underlying would have at a given price and a given time).