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There has been no mention in this text of why this formula uses forward delta not cash delta. Why should have this been obvious to the reader?

How can a put be delta neutral at 30%, what does this mean?

Why does the Black-Scholes Merton use cash delta and not forward delta?

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  • $\begingroup$ Maybe it would help giving some context as to where this text comes from? $\endgroup$ – Quantuple Mar 15 '16 at 15:58
  • $\begingroup$ Dynamic hedging by taleb $\endgroup$ – Permian Mar 15 '16 at 19:48
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Forward delta is the option's sensitivity to the PV of a forward contract on the same underlying with same maturity as the option. It is a convention often used in FX markets (see for instance On a FX volatility smile, Is a-delta put volatility equal to (1-a)-delta call volatility?). It is the number of forward contracts required to delta hedge the option.

Cash delta (also called spot delta) is the option sensitivity to the underlying spot price. It is the number of shares required to delta hedge the option.

From the Call-Put parity a call minus a put has a forward delta of exactly 1, hence the 30%/70% example in the text you are referring to.

A put delta neutral at 30% means the sum of the put and 0.3 forward contracts has zero delta.

The original Black & Scholes derivation considers a risk free portfolio made of the option, shares of the underlying and cash. The number of shares that make the portfolio risk free is the spot delta.

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