We start this chapter by showing some examples for exotic options, giving one possible classification. We will show examples from the fExoticOptions package and how the so-called Black-Scholes surface can be created for any derivative-pricing function. Afterwards, we will focus on the numerical estimation of the Greeks of any exotic derivative. Next, we will show the pricing of an exotic option that is not yet included in the fExoticOptions package.

We have chosen the Double-no-touch (DNT) binary option mainly because of its popularity on the foreign exchange (FX) markets and the many conclusions that are relevant even for other exotics. We will use AUDUSD as underlying because at the time of writing this chapter, there is a significant interest differential between the AUD and the USD interest rates, and we can show how to put these rates into the pricing functions. We will show a second way of calculating the price of a DNT by using static option replication arguments. We will show a real-life example of a DNT, and in a simulation, we will show a way to estimate the survivorship probability of a DNT. Using this, we can discuss the relationship of real-world and risk-neutral probabilities and the role of risk premium. Finally, we will show some practical fine-tuning tricks to embed exotic options into structured products.

Besides seeing examples to implement complex exotic option-pricing functions and simulations in R, as a side effect, understanding the Greeks as links between exotics and vanillas will be the learning outcome of this chapter. We will use the same terminology that was introduced in Chapter 5, FX Derivatives, which also includes much more about currencies and plain vanilla options.