37 lines
2.2 KiB
Plaintext
37 lines
2.2 KiB
Plaintext
468 Part IV: Additional Considerations
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TABLE 28-6.
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Computation of expected profit.
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Chance of Being Profit at Expected
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XYZ Price at at That Price That Price Profit:
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Expiration (A) (B) (A) x (8)
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Below 30 20% -$200 -$ 40
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31 10% - 100 - 10
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32 10% 0 0
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33 10% + 100 + 10
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34 10% + 200 + 20
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Above 35 40% + 300 + 120
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Total expected profit $100
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As is readily observable, the selection of what percentages to assign to the pos
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sible outcomes in the stock price is a crucial choice. In the example above, if one
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altered his assumption slightly so that XYZ had a 30% chance of being below 30 and
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a 30% chance of being above 35 at expiration, the expected return would drop con
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siderably, to 25%. Thus, it is important to have a reasonably accurate and consistent
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method of assigning these percentages. Furthermore, the example above was too sim
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plistic, in that it did not allow for the stock to close at any fractional prices, such as
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32½. A correct expected return computation must take into account all possible out
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comes for the stock.
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Fortunately, there is a straightforward method of computing the expected per
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centage chance of a given stock being at a certain price at a certain point in time. This
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computation involves using the distribution of stock prices. As mentioned earlier, the
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Black-Scholes model assumes a lognormal distribution for stock prices, although
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many modelers today use nonstandard (empirical or heuristic) distributions. No mat
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ter what the distribution, the area under the distribution curve between any two
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points gives the probability of being between those two points.
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Figure 28-1 is a graph of a typical lognormal distribution. The peak always lies
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at the "mean," or average, of the distribution. For stock price distributions, under the
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random walk assumption, the "mean" is generally considered to be the current stock
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price. The graph allows one to visualize the probability of being at any given price.
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Note that there is a fairly great chance that the stock will be relatively unchanged;
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there is no chance that the stock will be below zero; and there is a bullish bias to the
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graph - the stock could rise infinitely, although the chances of it doing so are
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extremely small. |