European Option Model on Asset with Known Cash Payouts
When a stock issues dividend, cash is paid to the holder of the asset. The call holder does not receive any part of the payout. When the stock goes ex-dividend, its value will usually decreased by approximately the amount of the dividend distribution. Consider a stock that is expected to pay out one dividend of Q amount on N date before the option expiration date. On N date, the stock price will drop by Q amount in price.

To take this dividend effect into account, we can substract the stock price by the present value of the dividend paid. This is as if the stock did not pay out any dividend, but only began with a lower price. The new S price, S*, would be original stock price minus the present value of all future dividends.

The following demostrates the computation of option prices with the underlying asset that pays dividends. Suppose the underlying asset pay dividend 3 times before the option expired. The present value of these payouts is $8.95. We discount the stock price by this amount and get $91.05. This is the new stock price that we use to compute the option prices. The option price model used for this example is the same as the model used in the Black-Scholes European Option Pricing Model example.
The delta for call and put are computed as0.4208 and -0.5792, respectively.
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