When Call Delta Is a Martingale Under the Stock Measure

Summary

The document asks whether a call option’s delta is a martingale when probabilities are measured using the stock as numeraire. It notes that in the Black–Scholes example, delta can be written as the stock-measure probability that the underlying finishes above the strike, matching the expected expiration delta under that measure.

The answer gives a broader condition: the result holds for a large class of models in which the distribution of the asset’s log return does not depend on its starting level. In that setting, delta can be represented as the stock-measure probability of finishing above the strike, and this quantity is a martingale under the stock measure. The response points to a mathematical finance text for a proof but does not provide the derivation. The claim is conditional on the stated class of models; it does not establish the property for every option-pricing model.

Key ideas

• In Black–Scholes, call delta can be expressed as a stock-measure probability of finishing above the strike.
• Under the stock measure, the expected expiration delta matches current delta in the described setting.
• The result extends to models where log-return distributions are independent of the asset’s starting level.
• The answer states a conditional result for a broad class of models, not a universal rule.
• A proof is referenced in a mathematical finance text but is not included in the document.

Tags

# Is the delta of a call option a martingale using the stock numeraire?


# Is the delta of a call option a martingale using the stock numeraire?












For example in the Black_scholes case the delta N(d1) does appear to be equal to the expectation (under the stock measure) of the delta at expiration, which is the expectation of I(S(T)>K).

Is there a fundamental reason to believe that the delta will always be a martingale under the stock numeraire?

## Answer by Mark Joshi (score 2, accepted)

https://quant.stackexchange.com/a/21880

for a large class of models that is ones where $\log S_T - \log S_0$ has distribution independent of level, it is possible to show that the delta is $$ \mathbb{P}_S(S_T>K) $$ and this is a martingale in the stock measure. (For a proof see More Mathematical Finance by me)

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.