Spot and Forward Delta Under Put–Call Parity

Summary

This note examines how spot delta and forward delta relate through put–call parity. Differentiating the parity relationship with respect to spot gives a call-minus-put delta of one, while differentiating with respect to the forward gives a discounted difference. The author asks whether a synthetic long hedged with a forward is cash-delta neutral and whether a cited FX convention reflects a change of measure or simply a different underlying for the delta.

The document presents the question and the relevant parity equations, but it does not resolve the distinction. It is useful as a prompt to distinguish the variable with respect to which an option’s delta is calculated from the pricing measure used in valuation. The discussion also highlights that FX market conventions may report a discounted forward delta as spot delta. It provides no numerical example or derivation of the convention, so the reader should treat its proposed interpretation as an open question rather than a settled explanation.

Key ideas

• Put–call parity implies different call-minus-put delta relationships when differentiated with respect to spot and forward.
• A forward hedge and a spot hedge correspond to different delta definitions and should not be conflated.
• The note asks whether an FX delta convention reflects discounting or a change of measure but does not answer the question.
• The equations provide a starting point for checking hedge units and market conventions.

Tags

# Clarification about spot and forward delta given PCP


# Clarification about spot and forward delta given PCP












I would like to clarify the difference between the spot delta and forward delta.

This question is related to this post: Use of cash delta vs forward delta and the mirror image rule.

From the put call parity formula, we have $c-p = e^{-rt}(F-K)$. Hence taking $\partial S$ on each side, we get $\partial_S c - \partial_S p = 1$ (as $\partial_S F = e^{rt}$), and taking $\partial F$ on each side, we get $\partial_F c - \partial_F p = e^{-rt}$.

Would this mean that if we trade a long synthetic against the forward we are NOT cash delta-neutral, and instead need to trade it against the spot? What I am guessing is that in the original post, the author (Taleb) is implicitly multiplying each side of PCP by $e^{rt}$, which brings us into the forward measure space. Is this line of thinking correct?

I think what could be happening is that I'm confusing forward-measure delta with delta wrt forward. Because in this post FX Spot Delta market standard calculation (Trader View) Akdemy says "Ignoring all this, Spot delta is really just forward delta, 𝑁(𝑑1) , discounted $exp^{-ccy1*\tau}*FwdDelta$". This is the inverse of my understanding, $\Delta_s = N(d_1)$ and $\Delta_F = e^{-rt}N(d_1)$. Is Akdemy operating in forward meaasure space?

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.