Skip to content
All library documents

Dual Delta, Exercise Probability, and Local Volatility

Article Quant Q&A · Author: jack klompus
Read the original

Summary

The document explains dual delta as the sensitivity of an option’s value to its strike, distinguishing it from ordinary delta, which measures sensitivity to the underlying price. In the Black–Scholes–Merton framework, call dual delta is negative and relates to the discounted risk-neutral probability of exercise. It can also be interpreted through an Arrow–Debreu security that pays one unit if the call finishes in the money.

Key ideas

  • Dual delta measures how option value changes when the strike changes.
  • Ordinary delta instead measures sensitivity to the underlying asset price.
  • For a Black–Scholes–Merton call, dual delta is the negative discounted in-the-money probability term.
  • Dual delta and related strike sensitivities can help infer local volatility from a volatility surface.

Tags

OptionsDerivatives pricingVolatilityStatistics
Full text
# What is Dual Delta?


# 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?

## Answer by nbbo2 (score 5)

https://quant.stackexchange.com/a/64142

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.

## Answer by Marses (score 2)

https://quant.stackexchange.com/a/64136

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).

See e.g. https://en.wikipedia.org/wiki/Local_volatility

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.