We started this chapter by introducing exotic options. In a brief theoretical summary, we explained how exotics and plain vanillas are linked together. There are many types of exotics. We showed one possible way of classification that is consistent with the fExoticOptions package. We showed how the Black-Scholes surface (a 3D chart that contains the price of a derivative dependent on time and the underlying price) can be constructed for any pricing function.

Pricing of exotic options is just the first step. Market makers keep thousands of different options in their trading books. This is possible only because each option can be decomposed into certain sensitivities, the so-called Greeks. Being partial derivatives, Greeks are additive; thus, the portfolio of derivatives has the sum of the Greeks of its elements. The next step was estimating Greeks for any derivative-pricing functions. Our numerical method can be calibrated to the real market conditions; for many parameters, we already know what the smallest possible change is. For example, the smallest change for an interbank AUDUSD fx rate is 0.0001. Even multiple partial derivatives such as gamma or vanna can be calculated with this numerical method.

In the second half of this chapter, we focused on one particular exotic option: the Double-No-Touch (DNT) binary option. The reason behind this focus is based on the popularity of DNT options and also because there are many tricks that can be shown on DNTs with conclusions relevant to many other exotic options. We showed two different ways to price DNT options. First, we implemented the Hui (1996) closed form solution, where the price is a result of an infinite sum. The speed of convergence is often very quick; however, this is not always the case. We showed a practical way of how convergence issues can be handled without wasting too much computing time. Another way to price a DNT is a static replication from one DKO call and one DKO put option. To price these DKO options, we used the fExoticOptions package. We found very little difference between the results of the two DNT pricing methods.

We showed how the DNT option behaves on real-life data by using 5 minutes frequency open-low-high-close type time series of AUDUSD fx rates from the second quarter of 2014. We estimated the survivorship probability of a DNT by simulation to show how risk premium can be included in DNTs or DOTs based on the supply-demand tensions for volatility. Finally, we showed some practical fine-tuning methods to find missing parameters for DNT with a certain price in the case of building a structured product by introducing functions to find implied parameters.