Dividend Treatment in Black-Scholes Delta-Hedged Portfolios

Summary

The document asks why a Black-Scholes derivation for a dividend-paying underlying includes dividends in the change of a delta-hedged portfolio but omits them from the portfolio's risk-free return equation. The included answer separates these steps: dividend income matters when calculating the hedge portfolio's value change, while the no-arbitrage argument applies the risk-free growth rate to the completed, locally riskless portfolio.

This distinction explains why recognizing dividends in the hedge calculation does not mean adding a dividend yield to the bank-account return. The note gives a conceptual explanation rather than a full derivation, and its equations use a simplified time-step argument. It does not discuss assumptions such as continuous dividend yield, discrete payment timing, transaction costs, or the limits of continuous rebalancing, so those details must be handled separately in a complete model.

Key ideas

• Dividend income affects the change in value of a delta-hedged position.
• The no-arbitrage step relates the locally riskless portfolio's return to the risk-free rate.
• The dividend term in hedge accounting is distinct from the bank-account return.
• The explanation is schematic and does not specify dividend timing or transaction costs.

Tags

# Why the Inconsistency in the Derivation of BS for Dividend-Paying Underlying?


# Why the Inconsistency in the Derivation of BS for Dividend-Paying Underlying?












The basic idea is that we get two expressions for $\Delta \Pi = ...$ and equate them.

The thing that does not make sense is that in one we take into account the dividend

$$\Delta \Pi = \frac{d}{dS}V \cdot \Delta S -1 \cdot \Delta V \boxed{+ (D \cdot \Delta t) \cdot \frac{d}{dS}V \cdot S}$$

where as in the other we totally ignore it

$$\Delta \Pi = \Pi \cdot k$$ $$\Delta \Pi = \bigg(\frac{d}{dS}V \cdot S -1 \cdot V \bigg) \cdot \bigg(r \cdot \Delta t \bigg)$$

How come?

## Answer by A.L. Verminburger (score 1)

https://quant.stackexchange.com/a/49978

In the first $\Delta \Pi$ expression we are trying to eliminate risk from the portfolio. It just so happens that to do the delta hedging in this case we need to take into account the dividend.

The second $\Delta \Pi$ expression comes from the no-arbitrage principle, which is the same (as is its equation $\Delta \Pi = r \cdot \Pi \cdot \Delta t$) be it dividend or no divident. It is based on wishful, albeit logical, thinking that if we manage to create a synthetic asset (a delta-hedged portfolio) that is risk (volatility) free, then it must be equivalent to an interest-bearing bank account (because, well, it too is risk .., I mean volatility, free). And there are certainy no dividends in interest-bearing bank accounts.

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.