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This is the formula for a Cashless Exercise of Warrants:

X(A-B)/A = Y

Where:

Y = the number of shares received

X = the number of shares purchasable

A = price of the stock

B = exercise price

The dollar value of the warrants is the amount of shares received times the price of the stock.

With this formula, at what point do we see diminishing returns in the dollar value of the warrants with respect to the amount of shares received.

For example when we are giving the option to purchase 1mm shares, the maximum amount of shares we will get is 1mm when the price of the stock approaches infinity. However when the price of the stock = the exercise price, we will receive 0 shares. If the strike price we have is 1 and the stock is trading at 1 and the stock is trading at 1.01 we will get roughly 9,900 shares, and then at 1.02 we will receive 19,607 shares resulting in an increase of from 0 to 1.02 we will receive 19,607 shares ​resulting in an increase of from 0 to 10,000 and 20,000 respectively.However the relationship is not linear; as we get more and more shares the increase will not jump up 20,000 respectively. However the relationship is not linear; as we get more and more shares the increase will not jump up 10,000 per cent increase (it can go larger depending on the variable values we choose, but at some point it will plateau at a max and start diminishing, while still remaining positive). I know this has to do with calculus and the first and potentially the second derivative, but I am having trouble solving this one my own or finding a suitable solution to this​.​ I am curious to know at what point does the returns start diminishing in terms of the dollar amount of the warrants per increase in stock price.

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