## Digital Option

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This post is based on problems 2. I was asked how to price a digital option in a job interview - and had no idea what to do! A call is only worth exercising using if the underlying price,is greater than atas the payoff from exercising is.

A digital call option with is similar - it pays off one dollar if at expiration, and pays off zero otherwise:. Suppose you have a model for pricing regular call options. How can you use to price the digital option? As a starting point, consider buying a call with and selling a call with:. This is close to the digital option, but not exactly right. We want to make the slope at steeper, so we need to buy more options. Consider buying two calls with and selling two calls at:.

As opposed to a slope of 1 between andnow we have a slope of two between and Generalizing this idea - consider a number. To get a slope ofyou buy calls at and you sell calls at. How much will the above portfolio cost? You earn from selling the calls, and pay for the calls.

The net cost is: Many complicated payoffs can be re-created as combinations of vanilla puts and calls. Digital Call Options A digital call option with is similar - it pays off one dollar if at expiration, and pays off zero otherwise: As a starting point, consider buying a call with and selling a call with: Consider buying two calls with and selling two calls at: Given that the slope isto get an infinite slope, we take the limit as goes to zero.

It might look more familiar if I re-wrote it as: Conclusion Many complicated payoffs can be re-created as combinations of vanilla puts and calls.