Black–Scholes Hedging with Continuous Dividends
Summary
The document derives the Black–Scholes partial differential equation for a stock paying a continuous dividend yield. It forms a portfolio that is long the derivative and short its delta in shares, then applies Itô's lemma. The stock position earns both price changes and dividend income, so the dividend cash flow must be included when calculating the portfolio's change. Choosing the derivative's delta removes the stochastic term, and equating the remaining riskless return to the risk-free rate yields the dividend-adjusted equation.
The answer clarifies that the hedge holds shares, not units of a dividend-reinvested total-return process. If the stock variable is instead transformed to include reinvested dividends, the hedge ratio must be transformed as well; the result is the same PDE under the corresponding change of variable. The derivation assumes the standard frictionless continuous-time pricing framework and does not discuss discrete dividends or trading costs.
Key ideas
- A continuous dividend yield contributes cash income to a position in the stock.
- Delta hedging removes the stochastic component of the derivative-and-stock portfolio.
- The resulting Black–Scholes equation uses the stock's risk-neutral price drift net of its dividend yield.
- A hedge in a total-return stock variable requires a correspondingly transformed hedge ratio.
- Either stock variable gives the same pricing result when the hedge and derivatives are transformed consistently.
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Full text
# Black Scholes in the case of dividends
# Black Scholes in the case of dividends
Let's take the case where the underlying stock has the continuous dividend yield $\delta$. Then, in the risk-neutral world, $\frac{dS}{S}=(r-\delta)dt+\sigma dW^Q$. Suppose we want to price a derivative on the underlying stock. The standard way to go about it is to create a risk-less portfolio by using a combination of the derivative and the stock, applying Itô to calculate the small change in said portfolio, and equating its growth rate to the risk-free rate.
$\begin{align*}\pi=C-\Delta S\implies d\pi=dC-\Delta dS-(\delta\Delta) Sdt\\=C_tdt+C_{S}dS+\frac{C_{SS}}{2}\sigma^2S^2dt-\delta\Delta Sdt-\Delta dS\end{align*}$
If we choose $\Delta=C_S$, then all the stochastic terms would go away, and we would get-
$d\pi=C_tdt+\frac{C_{SS}}{2}\sigma^2S^2dt-\delta\Delta Sdt=\pi rdt$. Cancelling out $dt$ on both sides, and re-arranging we get the Black-Scholes equation in the case of dividends-
$C_tdt+\frac{C_{SS}}{2}\sigma^2S^2+(r-\delta)C_{S}S-rC=0$.
My question is regarding the expansion of $dS$ in the first step. If we hold $-\Delta$ of the stock, then shouldn't we be using the total return process of the stock, i.e. $S'=Se^{\delta t}$? Using Itô, $\frac{dS'}{S'}=rdt+\sigma dW^Q$ and $d\pi=dC-\Delta dS'$ and $dS'\neq dS+\delta Sdt$.
Where am I going wrong here?
## Answer by byouness (score 1, accepted)
https://quant.stackexchange.com/a/39678
Holding a quantity $\Delta$ of the stock over an infinitesimal period $dt$ gives you:
- $\Delta \times dS$ : return of the stock over $dt$
- $\Delta S \times \delta \times dt$: continuous dividends collected over $dt$
The hedge is to hold a quantity $\Delta$ of stocks, not $\Delta$ of the total return process. If you want to consider the total return process, the hedge ratio will be different, it will be equal to $\Delta e^{-\delta t}$:
$\frac{\partial C}{\partial S^{'}} = \frac{\partial C}{\partial S} \times \frac{\partial S}{\partial S^{'}} = \Delta \times e^{-\delta t}$
Using either $S$ or $Se^{\delta t}$ will give the same result, same PDE, same hedge ratio, up to a change of variable $S \leftrightarrow S'$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.