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<p>The ratio is a tool that enables us to summarize the overall exposure
of portfolios of options with various exercise prices and maturity
periods. An options ratio is the change in the option price for a 1\$
increase in the share price. A call option has a positive hedge ratio and
a put option has a negative hedge ratio.</p>
<p>Under the black scales option formula, the hedge ratio of a call
option is N (d1) and the hedge ratio for a put is N (d1)-1. Recall that N
(d) stands for the area under the standard normal curve up to d.
Therefore, the call option hedge ratio must be positive and the put
option hedge ratio is negative and of smaller absolute value than
1.0.</p>
<p>Implied Volatility</p>
<p>The black scales option valuation assumes that the volatility is
given. We can ask a different question. What is the volatility (or
standard deviation) for the observed option price to be consistent with
the black scales formula? This is implied volatility of the stock.
Implied volatility is the volatility that the option price implies. An
investor can compare the actual and implied volatility.</p>

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<p>If the actual volatility is higher than the implied volatility, the
investor may conclude that the options fair price is more than the
observed price. Hence, she may consider option as potentially a good
investment. You can use the excel spreadsheet to calculate the black
scales option price and implied volatility.</p>
<p>Dividend paying share option</p>
<p>The share prices go down by an amount reflecting the payment of
dividend. As a consequence, the value of a call option will decrease and
the value of a put option will increase. The share price is assumed to
have a risk less component and a risky component. The black scales model
includes the risky component of the share price. The present value of
dividends (from ex-dividend dates to present) can be treated as the risk-
less component of the share price. Thus, for valuing a call option, we
should adjust downwards the share price for the present value of the
dividend payments during the life of the option, and then use the black
scales model. We also need to adjust the volatility in case of a
dividend-paying share since in the black scales model it is the
volatility of the risky part of the share price. This is generally
ignored in practice.</p>

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