Design a digital option as a sequence of calls, Financial Management

An individual agent thinks that there is a high probability that the Dow Jones will have a payoff (or points) between a=10000 and b=12000 at t=1.

Design a digital option (see Figure) as a sequence of calls on the Dow that converges to a pure bet on getting $1 on the interval [10000, 12000], i.e. if the Dow lies between S ? [10000, 12000] at t=1, then the portfolio of calls pays off exactly $1. The payoff is 0 otherwise.

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