Pricing the Double-no-touch option

A Double-no-touch (DNT) option is a binary option that pays a fixed amount of cash at expiry. Unfortunately, the fExoticOptions package does not contain a formula for this option at present. We will show two different ways to price DNTs that incorporate two different pricing approaches. In this section, we will call the function dnt1, and for the second approach, we will use dnt2 as the name for the function.

Hui (1996) showed how a one-touch double barrier binary option can be priced. In his terminology, "one-touch" means that a single trade is enough to trigger the knock-out event, and "double barrier" binary means that there are two barriers and this is a binary option. We call this DNT as it is commonly used on the FX markets. This is a good example for the fact that many popular exotic options are running under more than one name. In Haug (2007a), the Hui-formula is already translated into the generalized framework. S, r, b, σ, and T have the same meaning...