Dividend-Adjusted Bounds on Black–Scholes Call Delta

Summary

The note considers why a target call delta may fail to produce a strike in a dividend-paying Black–Scholes setting. With continuous dividend yield, call delta is bounded above by the discount factor for dividends, so a target above that bound cannot be represented by any value of the normal cumulative distribution function. The put delta has a corresponding formula, and call-put delta parity must use the dividend-adjusted factor rather than the no-dividend relation.

The supplied answer points out that the stated target leads to an implied normal probability above one whether approached through the call or put equations. That confirms the target is infeasible under the given parameters; using the put equation does not create a valid strike. The discussion is limited to the stated delta formulas and does not address alternative delta conventions or other option-pricing models.

Key ideas

• With continuous dividends, call delta is capped by the dividend discount factor.
• The cumulative normal probability cannot exceed one, so targets above that cap have no Black–Scholes solution.
• Call-put delta parity includes the dividend discount factor.
• An infeasible call delta does not become feasible by expressing it through put delta.

Tags

# Get strikes from delta works with put but no with call function


# Get strikes from delta works with put but no with call function












From call-put parity one can derive that $\Delta_C - \Delta_P = 1$. We also know that $\Delta_C = e^{-qt}N(d_1)$. If $e^{-qt} = 0.85$ then there is no value for $d_1$ for a $\Delta_C = 0.9$ as $ 0 <= N(d_1) <= 1 $. Therefore there is no way to get a strike with $\Delta_C = 0.9$.

But from the first equation, we have that if $\Delta_C = 0.9$ then $\Delta_P = -0.1$. Also from the delta of a Put equation we have $-0.1 = -0.85N(-d_1) $ -> $0.1176 = N(-d_1)$.

This second equation does yield a result for $d_1$ and therefore a strike.

If put-call parity holds, why can't I get a result (and also same result) with $\Delta_C = 0.9$ instead of $\Delta_P = -0.1$?

## Answer by Hans-Peter Schrei (score 1, accepted)

https://quant.stackexchange.com/a/74942

If you have dividends, then $\Delta_C - \Delta_P = e^{-qt}$. As you note, you have $\Delta_C = e^{-qt}N(d_1)$ and $\Delta_P = e^{-qt}(N(d_1)-1)$.

For both $\Delta_C$ and $\Delta_P$ you then get $N(d_1)\approx 1.0588$.

[1] https://bookdown.org/maxime_debellefroid/MyBook/the-greeks.html, Section 5.2.5 Delta under Black-Scholes

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