38 lines
2.5 KiB
Plaintext
38 lines
2.5 KiB
Plaintext
460 Part IV: Additional Considerations
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many of the applications that are going to be prescribed, it is not necessary to know
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the exact theoretical price of the call. Therefore, the dividend "correction" might not
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have to be applied for certain strategy decisions.
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The model is based on a lognormal distribution of stock prices. Even though the
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normal distribution is part of the model, the inclusion of the exponential functions
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makes the distribution lognormal. For those less familiar with statistics, a normal dis
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tribution has a bell-shaped curve. This is the most familiar mathematical distribution.
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The problem with using a normal distribution is that it allows for negative stock
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prices, an impossible occurrence. Therefore, the lognormal distribution is generally
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used for stock prices, because it implies that the stock price can have a range only
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between zero and infinity. Furthermore, the upward (bullish) bias of the lognormal
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distribution appears to be logically correct, since a stock can drop only 100% but can
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rise in price by more than 100%. Many option pricing models that antedate the
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Black-Scholes model have attempted to use empirical distributions. An empirical
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distribution has a different shape than either the normal or the lognormal distribu
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tion. Reasonable empirical distributions for stock prices do not differ tremendously
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from the lognormal distribution, although they often assume that a stock has a
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greater probability of remaining stable than does the lognormal distribution. Critics
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of the Black-Scholes model claim that, largely because it uses the lognormal distri
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bution, the model tends to overprice in-the-money calls and underprice out-of-the
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money calls. This criticism is true in some cases, but does not materially subtract
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from many applications of the model in strategy decisions. True, if one is going to buy
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or sell calls solely on the basis of their computed value, this would create a large prob
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lem. However, if strategy decisions are to be made based on other factors that out
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weigh the overpriced/underpriced criteria, small differentials will not matter.
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The computation of volatility is always a difficult problem for mathematical
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application. In the Black-Scholes model, volatility is defined as the annual standard
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deviation of the stock price. This is the regular statistical definition of standard devi
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ation:
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where
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P = average stock price of all P/s
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Pi = daily stock price
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n
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~ (Pi -P)2
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cr2 = _1=_1 __ _
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n-1
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v = a!P |