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This is a practice question for an exam:

  1. 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?

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    $\begingroup$ Note that: $S(T_0^+) = (1-\delta)S(T_0^-)$, where $T_0^-$ refers to the value before the dividend payment and $T_0^+$ after the payment. Hence, regarding your last paragraph, your dividends at $T_0^+$ will have value $\delta S(T_0^-)$ whereas your short position at $T_0^+$ will have value $-\delta S(T_0^+)=-\delta (1-\delta) S(T_0^-)$, so that strategy does not work. $\endgroup$ – Daneel Olivaw Jan 8 '18 at 17:21
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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 $X$
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