40 lines
3.0 KiB
Plaintext
40 lines
3.0 KiB
Plaintext
806 Part VI: Measuring and 1iading Vo/atillty
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at any time during the life of the probability study, usually the life of an option. It
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turns out that there are a couple of ways to approach this problem. One is with a
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Monte Carlo analysis, whereby one lets a computer run a large number of random
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ly-generated scenarios (say, 100,000 or so) and counts the number of times the tar
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get price is hit. A Monte Carlo analysis is a completely valid way of estimating the
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probability of an event, but it is a somewhat complicated approach.
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In reality, there is a way to create a single formula that can estimate the "ever"
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probability, although it is not any easy task either. In the following discussion, I am
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borrowing liberally from correspondence with Dr. Stewart Mayhew, Professor of
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Mathematics at the University of Georgia. For proprietary reasons, the exact formu
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la is not given here, but the following description should be sufficient for a mathe
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matics or statistics major to encode it. If one is not interested in implementing the
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actual formula, the calculation can be obtained through programs sold by McMillan
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Analysis Corp. at www.optionstrategist.com.
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This discussion is quite technical, so readers not interested in the description of
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the mathematics can skip the next paragraph and instead move ahead to the next sec
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tion on Monte Carlo studies.
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These are the steps necessary in determining the formula for the "ever" proba
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bility of a stock hitting an upside target at any time during its life. First, make the
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assumption that stock prices behave randomly, and perform at the risk-free rate, r.
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Mathematicians call random behavior "Brownian motion." There are a number of
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formulae available in statistics books regarding Brownian motion. If one is to esti
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mate the probability of reaching a maximum (upside target) point, what is needed is
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the known formula for the cumulative density function (CDP) for a running maxi
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mum of a Brownian motion. In that formula, it is necessary to use the lognormal
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function to describe the upside target. Thus, instead of using the actual target price
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in the CDF formula, one substitutes ln( qlp ), where q is the target price and p is the
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current stock price.
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The "ever" probability calculator provides much more useful information to a
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trader of options. Not only does a naked option seller have a much more realistic esti
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mate of the probability that he's going to have to make an adjustment during the life
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of an option, but the option buyer can find the information useful as well. For exam
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ple, if one is buying an option at a price of 10, say, then he could use the "ever" prob
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ability calculator to estimate the chances of the stock trading 10 points above the
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striking price at any time during the life of the option. That is, what are the chances
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that the option is going to at least break even? The option buyer can, cf course, deter
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mine other things too, such as the probability that the option doubles in price ( or
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reaches some other return on investment, such as he might deem appropriate for his
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analysis). |