35 lines
2.5 KiB
Plaintext
35 lines
2.5 KiB
Plaintext
Chapter 28: Mathematical Applications 459
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Now, returning to the formula for theoretical option price, we can complete the
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calculation of the July 50 call's theoretical value, called value here for short:
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value = 45 x N(d1) - 50 x e-·1 x ·16438 x N(d2)
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= 45 X .25134 - 50 X .9837 X .21421
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= .7746
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Thus, the theoretical value of the July 50 call is just slightly over¼ of a point.
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Note that the delta of the call was calculated along the way as N(d1) and is equal to
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just over .25. That is, the July 50 call will change price about¼ as fast as the stock
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for a small price change by the stock.
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This example should answer many of the questions that readers of the first edi
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tion have posed. The reader interested in a more in-depth description of the model,
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possibly including the actual derivation, should refer to the article "Fact and Fantasy
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in the Use of Options." 1 One of the less obvious relationships in the model is that call
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option prices will increase (and put option prices will decrease) as the risk-free inter
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est rate increases. It may also be observed that the model correctly preserves rela
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tionships such as increased volatility, higher stock prices, or more time to expiration,
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which all imply higher option prices.
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CHARACTERISTICS Of THE MODEL
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Several aspects of this model are worth further discussion. First, the reader will
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notice that the model does not include dividends paid by the common stock. As has
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been demonstrated, dividends act as a negative effect on call prices. Thus, direct
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application of the model will tend to give inflated call prices, especially on stocks that
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pay relatively large dividends. There are ways of handling this. Fisher Black, one of
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the coauthors of the model, suggested the following method: Adjust the stock price
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to be used in the formula by subtracting, from the current stock price, the present
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worth of the dividends likely to be paid before maturity. Then calculate the option.
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price. Second, assume that the option expires just prior to the last ex-dividend date
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preceding actual option expiration. Again adjust the stock price and calculate the
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option price. Use the higher of the two option prices calculated as the theoretical
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price.
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Another, less exact, method is to apply a weighting factor to call prices. The
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weighting factor would be based on the dividend payment, with a heavier weight
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being applied to call options on high-yielding stock. It should be pointed out that, in
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1Fisher Black, Financial Analysts Journal, July-August 1975, pp. 36-70. |