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Forward Delta, Spot Delta, and Delta-Neutral Option Hedges

Article Quant Q&A · Author: Trajan
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Summary

The document distinguishes forward delta from cash, or spot, delta by the underlying exposure each measures. Forward delta describes an option’s sensitivity to the present value of a forward contract on the same underlying and maturity; it is commonly used as a convention in foreign exchange markets. It corresponds to the number of forward contracts used to hedge the option’s delta. Cash delta measures sensitivity to the spot price and corresponds to the number of underlying shares in a spot hedge.

The answer explains a delta-neutral put example: a put with delta of negative thirty percent can be offset by holding 0.3 forward contracts, giving a net forward delta of zero. It also invokes call-put parity: a call minus a put has forward delta of one. The Black–Scholes derivation uses spot delta because its risk-free portfolio is formed from the option, underlying shares, and cash. The account is conceptual and does not derive the formulas or cover market-specific delta conventions in detail.

Key ideas

  • Forward delta measures sensitivity to the present value of a matching-maturity forward and is used in some FX conventions.
  • Spot delta measures sensitivity to the underlying’s spot price and corresponds to a share hedge.
  • A put with negative 0.3 forward delta is neutralized by holding 0.3 forward contracts.
  • Call-put parity implies that a call minus a put has forward delta equal to one.
  • The Black–Scholes setup uses spot delta for a hedge built from the option, shares, and cash.

Tags

OptionsDerivatives pricingForexRisk management
Full text
# Use of cash delta vs forward delta and the mirror image rule


# Use of cash delta vs forward delta and the mirror image rule












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?

## Answer by Antoine Conze (score 4, accepted)

https://quant.stackexchange.com/a/39719

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.

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.