36 lines
1.7 KiB
Plaintext
36 lines
1.7 KiB
Plaintext
Chapter 28: Mathematical Applications
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Theoretical option price= pN(d 1) se-rtN(d2)
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p v2
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ln(8 )+ (r +2 )t
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where d1 = _ r.
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V-4 t
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d2 = d1 - v--ft
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The variables are:
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p = stock price
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s = striking price
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t = time remaining until expiration, expressed as a percent of a year
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r = current risk-free interest rate
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v = volatility measured by annual standard deviation
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ln = natural logarithm
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N(x) = cumulative normal density function
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457
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An important by-product of the model is the exact calculation of the delta - that
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is, the amount by which the option price can be expected to change for a small
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change in the stock price. The delta was described in Chapter 3 on call buying, and
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is more formally known as the hedge ratio.
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Delta= N(d1)
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The formula is so simple to use that it can fit quite easily on most programmable cal
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culators. In fact, some of these calculators can be observed on the exchange floors as
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the more theoretical floor traders attempt to monitor the present value of option pre
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miums. Of course, a computer can handle the calculations easily and with great
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speed. A large number of Black-Scholes computations can be performed in a very
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short period of time.
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The cumulative normal distribution function can be found in tabular form in
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most statistical books. However, for computation purposes, it would be wasteful to
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repeatedly look up values in a table. Since the normal curve is a smooth curve (it is
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the "bell-shaped" curve used most commonly to describe population distributions),
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the cumulative distribution can be approximated by a formula:
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x = l-z(l.330274y 5 - l.821256y 4 + l.781478y 3 - .356538y 2 + .3193815y)
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where y 1 and z = .3989423e-<r212
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= 1 + .2316419lcrl
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Then N(cr) = x if cr > 0 or N(cr) = 1- x if cr < 0 |