# Question: Suppose IBM pays a dividend D on their shares S at time τ. Show that S(τ+) = S(τ−) − D. Actually, to be precise, τ should be what is called the ex-dividend date. You should again argue your solution from the assumption of no arbitrage. S(τ+) means the value of S just after τ, and S(τ−) the value just before. – Free Chegg Question Answer

Suppose IBM pays a dividend D on their shares S at time τ. Show that S(τ+) = S(τ−) − D. Actually, to be precise, τ should be what is called the ex-dividend date. You should again argue your solution from the assumption of no arbitrage. S(τ+) means the value of S just after τ, and S(τ−) the value just before.

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`Answer:`

We know, the change of the stock price (S) after declaring a dividend (D) is calculated as:

$\delta S = \frac{D *(1-T_D)}{1-T_{CG}}$

………….(1)

$T_D$ = tax rate payable on dividend

$T_{CG}$= tax rate payable on capital gain

In case of no-arbitrage possibility there will be no tax payable, i.e., $T_D = 0$

and $T_{CG} = 0$

So from (1)

But $\delta S$

= price before dividend paid (value before ex-dividend)- price after dividend paid (value after ex-dividend) $= S_{(T-)} -S_{(T+)}$

So $D = S_{(T-)} - S_{(T+)}$

So $S_{(T+)} = S_{(T-)} -D$