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The cash-or-nothing call option has a payoff that is equal to the strike price. All three options have the same expiry date.

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Is this homework? –  Bob Jansen Dec 19 '13 at 13:44
    
No, I am preparing for an exam I have tomorrow and I do not understand this section of the chapters that will be examined. –  user35777 Dec 19 '13 at 13:56
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Did you plot the payoff diagrams? –  Bob Jansen Dec 19 '13 at 14:16
    
yes i have drawn them but i still don't see the relation between them. –  user35777 Dec 19 '13 at 16:14
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2 Answers

The payoff of an European call option is $(S_T-K)^+$. At maturity, if the spot price is greater than (or equal to) the strike price, then holding an asset-or-nothing call option has payoff $S_T$, writing a cash-or-noting call option $K$, which together give the payoff of the European call in this scenario. If the spot price is less than the strike, then all the three assets have 0 payoff. These suffice to show that the two are equivalent.

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Formally, a long call payoff can be split as follows:

$$(S_T-K)^+ = (S_T-K)\cdot 1_{\{S_T>K\}} $$ $$= S_T\cdot 1_{\{S_T>K\}} - K\cdot 1_{\{S_T>K\}},$$ that is, long an asset-or-nothing digital call payoff and short a cash-or-nothing digital call payoff.

Here, $1_A$ is $1$ if event $A$ takes place, and it is $0$ otherwise.

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