Upon introduction of a dividend paid just before expiry or exercise, why do we have to reduce the share price by the amount of the dividend?

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Under the dividend discount model, when modelling the value of a company it is assumed to be equivalent to the present value of all future dividends.

Therefore, if a company pays away a dividend, it is reducing its value by the amount of cash that is being removed from the cash and cash equivalents line of the company's net asset value. The applicable discount factor is one (i.e. has no effect) because the cash gets transferred now. This decreases its ability to pay future dividends by that amount too.

Hence, as the dividend gets removed, the value of the company decreases by the exact amount of the dividend.

Mathematically the value before the dividend is paid is:

$$S_t = \sum_{i = 1}^{\infty} D_i \times DF(t_i)$$

Where D is the cash amount of a dividend and DF is the discount factor. This becomes:

$$S_t = \sum_{i = 1}^{\infty} D_i \times DF(t_i) - D_1\times DF(t_1)$$