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Example of Risk-Neutral Valuation Work

A stock whose value is now $44.75 is growing on average by 15 percent per annum. Its volatility is 22 percent. The interest rate is 4 percent. You need to value a call option along with a strike of $45, expiring in two months’ time. So, what can you do?

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First of all, the 15 percent average growth is completely irrelevant. The stock’s growth and therefore its real direction do not influence the value of derivatives. What you can do is simulate many, various future paths of a stock with an average growth of 4 percent per annum, as those are the risk-free interest rate, and a 22 percent volatility, to determine where this may be in two months’ time. Then compute the call payoff for each of these paths. Present value every of these back to today, and estimates the average over all paths. That is your option value. (For this easy illustration of the call option there is a formula for its value, therefore you don’t require to do all these simulations. And in which formula you’ll notice an r for the risk-free interest rate, and also no mention of the real drift rate.)

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