In Chapter 3, I presented examples on how vanilla options are priced and how these pricing models are used to analyze various simple risk- management strategies. I also introduced the terminology exotic options^{[1]} (i.e., options with in-the-money payoffs that are different from those of the vanilla options). Although examples of exotic options appeared in the late 1980s, the use of exotic options became more prevalent in the early 1990s when practitioners started to better appreciate the power of exotic options to better manage risks or take market view across multiple underlyings and asset classes.

Unlike vanilla European-style options, where one had access to the accurate analytical pricing formulae (examples of which were illustrated and discussed in Chapter 3), due to the complexity of the option payoffs, it is usually difficult to obtain similar analytical expressions for pricing European-style exotic options. As a consequence of this, in the absence of cheap, high-powered computers before the early 1990s, practitioners had to resort to using simplifying assumptions, clever mathematical tricks, and specific numerical algorithms to arrive at reasonable analytical approximations to value these complex, customized options. As mentioned in Chapter 4, it is also precisely this reason why Boyle's 1977 seminal paper on the use of Monte Carlo simulations to value vanilla European-style exotic options did not get the attention it deserved.

When computer technology (and hardware) started becoming cheaper, practitioners starting gravitating towards the use of simulations to price these options. This in turn fuelled the creation and trading of even more complex options than what was already trading. As a consequence of this development, the practice of trying to uncover good mathematical approximations started to quickly fall out of favor. It is because of this reason, that I introduced simulations in Chapter 4 and showed how simulations can be used to value vanilla European-style options, among other things.

Since this book is about the use of quantitative methods to solve financial problems, instead of simply valuing exotic options using simulations, I will first attempt to derive accurate analytical pricing formulae (consistent with what was done in Chapter 3).^{[2]} In the event that I am not able to derive such formulae, I will make simplifying assumptions to help me arrive at approximate analytical solutions – consistent with what practitioners did in the past. In instances when such approximations are obtained, I will also use simulations to solve the same problem so as to give the reader a good appreciation for the effectiveness or a lack thereof of these approximations.

For ease of illustration, throughout this chapter I will restrict my discussion to valuing exotic options on indices – and, as such, assume (as in Chapter 3) that index movements are aptly described by equation (3.5)

As the reader will recall, the above characterization is equivalent to assuming that In Sr is normally distributed with a mean of

, and, variance of

[1] The origin of exotic options can be traced back to the late 1980s when academicians toyed with the idea of applying the Black-Scholes assumptions to value fancy, nontypical, or nonvanilla payoffs. The consequence of this was a flurry of publications on exotic-option valuation. Despite this, it was not until many years later when hedgers began to better appreciate the path dependency and complexity of their risk exposure and how the theoretically created exotic options could be tweaked to make them more practical – in the process making exotic options more useful.

[2] The interested reader is referred to Das (1996), Haug (2006), Hull (2012), Nelken (1995), and Ravindran (1997) for extensive discussions on the valuation of exotic options.

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