Replicating a Stock Payoff Across a Proportional Dividend
Summary
The document considers a claim that pays the stock price at a future date after the stock distributes a proportional cash dividend. It derives an arbitrage-free value by relating the stock price immediately before the dividend to its price immediately afterward. Under the stated no-arbitrage argument, the claim’s value at the initial date is the post-dividend fraction of the initial stock price; the answer says this does not require a Black–Scholes volatility assumption.
The replication holds that same fraction of a share initially. When the dividend arrives, the proceeds buy the remaining fraction, leaving one share to deliver the claim’s terminal stock-price payoff. This corrects the questioner’s proposed short-stock strategy. The result assumes the market permits the stated trades and dividend reinvestment, and treats the dividend as a deterministic proportional payment at a specified time. It does not discuss transaction costs, taxes, or other market frictions.
Key ideas
- The claim’s value follows from no-arbitrage relations across the dividend date.
- The pricing argument does not rely on a constant-volatility Black–Scholes model.
- Hold the post-dividend fraction of a share initially, then reinvest the dividend to reach one share.
- The replication depends on the assumed proportional dividend and frictionless trading setup.
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Full text
# Replicating portfolio for claim on stock with discrete dividend
# Replicating portfolio for claim on stock with discrete dividend
This is a practice question for an exam:
> Consider a market consisting of a bank account with a constant interest rate $r$ and a stock $S$. The stock pays a proportional dividend of size $\delta S(T_{0-})$ at time $T_0$. Consider a $T$-claim that pays $X = S(T)$ at time $T$, where $T > T_0$. a) What is the arbitrage-free price of $X$ at time $0$? b) Find a replicating strategy for $X$
For the first question, if we assume that volatility of the stock is constant (i.e Black-Scholes), then we have the relation
$$\Pi_{\delta}(t,s) = \Pi(t,(1-\delta)s)$$ where $\Pi_{\delta},\Pi$ are the pricing functions for a claim on the underlying with and without dividends respectively, and since the price of the claim $X=S(T)$ (in a non-dividend context) is simply $\Pi(t) = s$, where $S(t) = s$, we get $$\Pi_{\delta}(t,s) = s(1-\delta)$$ for all $t$.
The question for me now, is if it is reasonable to assume the Black Scholes model holds? I have no idea how one would approach it otherwise, is there a way to do it more generally?
For the second question I figure that we can buy the stock at $t=0$, and short $\delta$ units of the stock. At time $T_0$ we can use the dividend to settle our short position, and so the portfolio would pay out exactly $S(T)$ at time $T$. Does that make sense?
## Answer by Antoine Conze (score 4, accepted)
https://quant.stackexchange.com/a/37640
a) From the no arbitrage condition, and without ressorting to a specific model $$ PV[S(T)|S(T_0)] = S(T_0) $$ $$ S(T_0) = (1-\delta) S(T_0^-) $$ $$ PV[S(T_0^-)|S(0)] = S(0) $$ Therefore the PV of $X$ at time $0$ is $$ PV[S(T)|S(0)] = PV[S(T_0)|S(0)] = PV[(1-\delta) S(T_0^-)|S(0)] = (1-\delta) S(0) $$
b)
- on $t=0$ you buy $1-\delta$ units of the stock
- on $t=T_0$ you get a total dividend amount of $(1-\delta) \delta S(T_0^-)$ which you use to buy an additional $(1-\delta) \delta S(T_0^-)/S(T_0) = \delta$ units of stock, so that you now hold $1-\delta + \delta=1$ unit of stock
- on $t=T$ you sell your $1$ unit of stock to replicate the payoff $X$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.