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A stock price S follows a geometric Brownian motion and pays a constant dividend yield q. Suppose that you hold a shares of the stock

A stock price S follows a geometric Brownian motion and pays a constant dividend yield q. Suppose that you hold a shares of the stock and b dollars in cash for a discrete period of time t . At the end of the time period, you change your share holdings from a to a'. What will the change in your cash position be over this time period, including the effect of the change in share holdings? This is typically how we estimate the profit from a delta hedge [that is, holding c = e q T N ( d 1 ) shares and borrowing e r T K N ( d 2 ) to hedge a written call, so a = e q T N ( d 1 ) and b = c e r T K N ( d 2 ), where c is the call price). How would you estimate the profit from a hedge that is both delta and gamma neutral?

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