34 lines
1.8 KiB
Plaintext
34 lines
1.8 KiB
Plaintext
O,apter 33: Mathematical Considerations for Index Products
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EUROPEAN EXERCISE
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647
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To account for European exercise, one basically ignores the fact that an in-the-money
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put option's minimum value is its intrinsic value. European exercise puts can trade at
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a discount to intrinsic value. Consider the situation from the viewpoint of a conver
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sion arbitrage. If one buys stock, buys puts, and sells calls, he has a conversion arbi
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trage. In the case of a European exercise option, he is forced to carry the position to
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expiration in order to remove it: He cannot exercise early, nor can he be called early.
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Therefore, his carrying costs will always be the maximum value to expiration. These
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carrying costs are the amount of the discount of the put value.
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For a deeply in-the-money put, the discount will be equal to the carrying
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charges required to carry the striking price to expiration:
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Carry = s Ji - 1 ] L (1+ r)t
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Less deeply in-the-money puts, that is, those with deltas less than - 1.00, would
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not require the full discounting factor. Rather, one could multiply the discounting
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factor by the absolute value of the put' s delta to arrive at the appropriate discounting
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factor.
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FUTURES OPTIONS
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A modified Black-Scholes model, called the Black Model, can be used to evaluate
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futures options. See Chapter 29 on futures for a futures discussion. Essentially, the
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adjustment is as follows: Use 0% as the risk-free rate in the Black-Scholes model and
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obtain a theoretical call value; then discount that result.
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Black model:
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Call value= e-rt x Black-Scholes call value [using r = 0%]
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where
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r is the risk-free interest rate
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and t is the time to expiration in years.
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The relationship between a futures call theoretical value and that of a put can
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also be discussed from the model:
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Call = Put + e-rf(J - s)
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where
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f is the futures price
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ands is the striking price. |