## Other Payoffs

This chapter has demonstrated just a few of the ways in which options can be used to produce an interesting relationship between profit and stock price. If European options expiring at time T were available with every single possible strike price, any payoff function at time T could in theory be obtained. The easiest way to see this is in terms of butterfly spreads. It will be recalled that a butterfly spread is created by buying options with strike prices Xi and X3 and selling two options with strike price X2 where Xi < X2 < X3 and X3 — X2 = X2 — X\. Figure 8.13 shows the payoff from a butterfly spread. This could be described as a "spike." As Xi and X3 become closer together, the spike becomes smaller. By judiciously combining together a large number of very small spikes, any payoff function can be approximated.

Payoff

X\ Xt Spread

Figure 8.13 Payoff from a Butterfly

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