First Generation Exotics – Binary and Barrier Options
We now have the machinery to price European style options under models that provide a fuller description of the FX spot process than Black–Scholes. While European options are especially liquid, they are often seen as particularly expensive, and clients are often interested in cheaper alternatives.
One particularly common way to cheapen a European option in FX is to add a path-dependent barrier feature to it, so that the option can only be exercised if it expires in the money and has remained within a specified region of interest over its lifetime. Part of the reason these barrier options are popular in FX is that currency spot rates are often thought to be more range-bound or mean-reverting than stock prices, and clients sometimes feel that the price is unnecessarily high as a result of purchasing option protection for large but unlikely moves in the FX spot rate.
The price of a European option is just the discounted domestic risk-neutral expectation of the payout function at expiry. For a call option,
Neglecting the discount factor for now, this is just the inner product of the PDF (for ST) and the option payout. Though the option payout for a European call increases without bound beyond the strike K, the product of the PDF and the payout profile decays to zero beyond K, as the eventually dominates.
The price ...