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An Introduction to Options Trading by Frans de Weert

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Chapter 7
THE PROFIT OF OPTION TRADERS
As was shown in Section 5.1, an option can be hedged by selling δ21 underlying stocks. That is, if an investor has a long position in an option, and during the term of this option he has a constant short position of δ underlying stocks, the return on this portfolio is the interest rate. Since δ changes as the price of the underlying stock changes and with the passage of time, an investor who wants to execute this strategy has to adjust the number of shorted stocks continuously. In practice, this is not possible. The number of shorted stocks can only be adjusted at discrete points in time. This is called ‘dynamic hedging’, and whenever the number of shorted stocks is adjusted according to the value of δ, the portfolio is made delta-neutral. In this chapter the implications of dynamic hedging will be discussed, and it will be shown that dynamic hedging is exactly the reason option traders make a profit.
 
Throughout this section there might be examples where a non-integer number of stocks is bought or sold. This is practically impossible and is just for the sake of argument.

7.1 DYNAMIC HEDGING OF A LONG CALL OPTION

Consider a stock which has a price of $10. The price of a call option on this stock with a strike price of $12 is $4.
The delta of this particular call option is 0.40, the gamma is 0.02 and the theta is – 0.01 per day. The following portfolio is put together:
• long 1 call option for $4;
• short 0.40 stocks for $10 each.
The ...

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