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This book is a quantitative quide to barrier options in FX environments.



1. Meet the Products

Were the financial products that we use today designed? Or did they evolve by natural selection? In today’s complex world of financial products, it’s very much a mixture of both. Structured products, such as FX accumulators, are the way they are very much by design: product developers have sat down together to discuss how to synthesize a product which matches their needs (which usually corresponds to the needs of their clients). In contrast, stand-alone barrier options are better seen as having evolved. This chapter traces that evolution process, via the simpler products of spot, forwards and vanilla options.
Zareer Dadachanji

2. Living in a Black-Scholes World

How many people in the world have heard of “the Black-Scholes model”? And how many people in the world really understand how it works? Whilst these questions are not well defined, they nonetheless illustrate a point: there is enormous variation in the depth of familiarity people have with this model. The aim of this chapter is to explore the model in sufficient depth to understand its mathematical and behavioural subtleties. Such an understanding is not only very valuable in its own right, but it also paves the way for a clearer understanding of the more advanced models which we will be covering later in the book.
Zareer Dadachanji

3. Black-Scholes Risk Management

Risk management takes many forms, but common to all of them is the following set of aims:
  • identify risk;
  • measure risk;
  • reduce risk;
  • report risk.
Zareer Dadachanji

4. Smile Pricing

Let’s cut to the chase: what value should the volatility σ in the Black-Scholes model take? To see why we should home in on this particular question, consider the pricing formula for a vanilla option, Equation 2.72. Its inputs are:
  • spot;
  • Domestic interest rate;
  • Foreign interest rate;
  • volatility;
  • strike;
  • maturity.
Zareer Dadachanji

5. Smile Risk Management

In Chapter 3, we discussed how we can manage market risk when our valuation model is Black-Scholes. In this chapter we will go through some of the models that we described in the chapter on smile pricing, Chapter 4, and discuss the new or modified risk management issues that arise in each case. Specifically, we will discuss risk management under the following models:
  • Black-Scholes with term structure (bsts): see Section 4.2
  • Local volatility (lv): see Section 4.9
  • Mixed local/stochastic volatility (lsv): see Section 4.12
In each case, we will take care to clarify the nature of the risk factors.
Zareer Dadachanji

6. Numerical Methods

My intention in this chapter is to highlight a variety of specific numerical techniques that are of great value in the types of calculations and analysis required for FX barrier options. Of high importance are the two broad classes of numerical methods that are used for calculation of option values: finite-difference methods and Monte Carlo simulation. There is a lot of very good literature available on these two extensive subjects, so rather than re-introducing the subjects here, I will give references to published material and then point out some specific aspects of these subjects that are of particular importance.
Zareer Dadachanji

7. Further Topics

All of the currencies in the benchmark currency pairs that we have used for illustrating results — EUR, USD, TRY, AUD and JPY — are (at the time of writing) examples of free-floating currencies: their exchange rates with respect to other currencies are determined by market forces, rather than being set by monetary authorities, such as the central bank of the country which issues the currency. A currency which is not free-floating is described as managed. Subtle distinctions between different types of managed currencies can additionally be made [58].
Zareer Dadachanji


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