FX Barrier Options

A Comprehensive Guide for Industry Quants

Zareer Dadachanji

• Verlag: Palgrave Macmillan
• Jahr: 2016
• ISBN:
  9781137462756
• Thema: Commerce
• Thema: Services financiers

Barrier options are a class of highly path-dependent exotic options which present particular challenges to practitioners in all areas of the financial industry. They are traded heavily as stand-alone contracts in the Foreign Exchange (FX) options market, their trading volume being second only to that of vanilla options. The FX options industry has correspondingly shown great innovation in this class of products and in the models that are used to value and risk-manage them. FX structured products commonly include barrier features, and in order to analyse the effects that these features have on the overall structured product, it is essential first to understand how individual barrier options work and behave. FX Barrier Options takes a quantitative approach to barrier options in FX environments. Its primary perspectives are those of quantitative analysts, both in the front office and in control functions. It presents and explains concepts in a highly intuitive manner throughout, to allow quantitatively minded traders, structurers, marketers, salespeople and software engineers to acquire a more rigorous analytical understanding of these products. The book derives, demonstrates and analyses a wide range of models, modelling techniques and numerical algorithms that can be used for constructing valuation models and risk-management methods. Discussions focus on the practical realities of the market and demonstrate the behaviour of models based on real and recent market data across a range of currency pairs. It furthermore offers a clear description of the history and evolution of the different types of barrier options, and elucidates a great deal of industry nomenclature and jargon.

1

Meet the Products

If barrier options did not exist, it would be necessary to invent them.

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.

There is an inextricable link between a financial product and the market or markets in which that product is traded. The markets for vanilla and barrier options are of a highly esoteric nature, and we will describe them in Chapter 4, after we have covered a number of key prerequisite concepts. In contrast, the cases of spot and forwards are fairly straightforward, and this chapter additionally describes the nature of the markets in which they trade and the way in which we value them.

1.1 Spot

Consider a European company which has received revenues in US dollars and needs to pay bills in euros in the next few days. What should it do? It could enter a contract with a wholesale bank under which it must pay an agreed amount of dollars to the bank, and the bank must pay an agreed amount of euros to the company. The amounts paid are called the principal amounts, and the date on which the two cash payments must be made is called the settlement date.

The term notional is sometimes used in place of “principal”, and one can talk of notional amounts and notional currencies. I prefer to use the term “notional” only when the amount really is notional and is not actually exchanged. For example, an FX volatility swap pays out according to a formula based on a notional amount that is not actually exchanged. Where cash flows really are exchanged (as they are for the vast majority of the products covered in this book), I will stick to the term “principal”.

This kind of trade is exceedingly common and has been completely standardized in the wholesale markets. In its standard form it is called a Foreign Exchange (FX) spot trade. The on-market exchange rate at which the trade takes place (the number of dollars per euro in this case) is called the spot rate or spot price. In this book we will denote the spot rate by S. If we need to explicitly specify that we are referring to the spot rate at a specific time t,we will write S(t). The date and time at which the trade is agreed are called the trade date and trade time.

The standardized settlement date is called the spot date, and is typically a few days after the trade date. The interval between the trade date and the spot date is referred to as the spot lag or settlement lag. The exact way in which the spot date is calculated for a given trade date depends on the currency pair, and is governed by a system of rules, known as settlement rules, which has evolved over time and which is now used by all market participants. Some details and examples are to be found in Clark [5]. Whilst the settlement rules themselves are somewhat involved, the underlying principle is plain: the time lag allows for a specified number of clear business days between the trade date and the settlement date. Typically, one clear business day is specified, with the result that the settlement date lies two business days after the trade date. This type of settlement rule is often referred to as a “T + 2” rule.

The nature of the settlement rules ensures that for any given trade date, the spot date is determined unambiguously. Most often, the opposite is also true: for any given spot date, there is only one corresponding trade date. Occasionally, though, and usually due to holidays between the trade date and the spot date, there may be more than one trade date corresponding to a given spot date.

Historically, before the electronic age, the settlement lag would have been needed for operational reasons. These days, it may be perfectly possible to settle without any time lag, but the spot lag remains by convention.

Note that the spot contract does not specify the time at which the payments must be made, so they can be made at any time during business hours in the location in which the contract has been agreed.

A note on the language of foreign exchange: it is common to describe an FX spot trade as “converting dollars into euros”, but this is very loose terminology. “Conversion” implies that dollars are destroyed and euros are created, but that is not the case – an FX spot trade certainly conserves both currencies. It is far better to speak of “exchanging cash flows”. In the case above, our example company – let us suppose it is a German company called Davonda GmbH – is exchanging dollars for euros. If we want to be explicit about which way around the exchange is made, we say that Davonda is buying euros and selling dollars.There are of course times when we are merely changing the quotation of an asset value from one currency to another, as described in Section 1.1.4. In such cases, there is no exchange of cash flows, and the term “conversion” is appropriate.

1.1.1 Dollars per euro or euros per dollar?

For a given pair of currencies, C1 and C2, we may quote the exchange rate either as the number of C1 per C2 or the number of C2 per C1. The standardization of the FX spot market extends to the specification of the quote order convention –which way around the rate is quoted. For example, in the case of US dollars and euros, the convention is to quote the number of dollars per euro. For example, the spot rate in this book’s benchmark market data (reflecting the real market as at the end of September 2014) is 1.2629. Typically, the quote order convention is chosen so that the spot rate is greater than 1, but that is not always the case: for example, the exchange rate between pounds sterling and euros is quoted as the number of pounds per euro, which has never been greater than 1 – though it has got tantalizingly close! Saying “dollars per euro” is a bit too much of a mouthful when you need to say it a hundred times every day, and in the industry parlance we say “euro-dollar”1to refer to the currency pair and its exchange rate. Similarly, “pounds per euro” is spoken as “euro-sterling”. When referring to the currency pair in written form, we use the currencies’ three-letter ISO codes in place of their names, for brevity, so for our two examples above we would write “EURUSD’ and “EURGBP” respectively. Going forward in this book, I will mainly use such six-letter symbols to specify the currency pair of interest.

An alternative notation is also sometimes to be seen in which a forward slash is written between the two currencies, like so: “EUR/USD”. Confusingly, although this notation suggests EUR per USD, it actually means exactly the same as EURUSD.

For the most part in this book, we will be discussing the following three currency pairs:

1.EURUSD: US dollars per euro, spoken “euro-dollar”.

2.USDTRY: Turkish lira per US dollar, spoken “dollar-Turkey”.

3.AUDJPY: Japanese yen per Australian dollar, spoken “Aussie-yen”.

With a given quotation order convention in place, industry parlance often regards the currency pair rather like an asset, and we talk of buying it, selling it or going long/short the currency pair. So “buying EURUSD” and “going long EURUSD” both mean the exchange of cash flows that involves buying EUR and selling USD. To describe the exchange of cash flows that involves selling EUR and buying USD, we would say “selling EURUSD” or “going short EURUSD”.

We very often wish to discuss Foreign Exchange issues without having to specify a particular currency pair, and for that we require terminology that allows us to describe a general currency pair. A commonly used terminology, and the one that I shall use in this book, is to say that the spot rate describes the number of units of Domestic currency per unit Foreign currency. So, in our EURUSD example above, EUR is referred to as the Foreign currency and USD is referred to as the Domestic currency. This is a global financial convention and has nothing to do with whether a currency is regarded as domestic or foreign by an individual participant. Indeed, our example German company Davonda would regard EUR as its domestic currency and USD as a foreign currency, but that is unrelated to the market convention.

As we shall see in greater depth later on, when discussing risk-neutral valuation, the Domestic currency is the natural currency in which we measure FX derivative values. Unless otherwise stated, “value” will refer to the value in Domestic currency.

If we abbreviate the word Foreign to the three-letter code “FOR”, and the word Domestic to “DOM”, the currency pair can be written as “FORDOM”. The following equation then serves as a reminder of our conventions:

1.1.2 Big figures and small figures

The precision with which exchange rates are quoted in the spot market is subject to market conventions, which, like so m...

Inhaltsverzeichnis

• HalfTitle
• Title
• Copyright
• Dedication
• Contents
• List of Figures
• List of Tables
• Preface
• Acknowledgements
• Foreword
• Glossary of Mathematical Notation
• Contract Types
• 1  Meet the Products
• 2  Living in a Black–Scholes World
• 3  Black–Scholes Risk Management
• 4  Smile Pricing
• 5  Smile Risk Management
• 6  Numerical Methods
• 7  Further Topics
• Appendix A: Derivation of the Black–Scholes Pricing Equations Vanilla Options
• Appendix B: Normal and Lognormal Probability Distributions
• Appendix C: Derivation of the Local Volatility Function
• Appendix D: Calibration of Mixed Local/Stochastic Volatility (LSV) Models
• Appendix E: Derivation of Fokker–Planck Equation for the Local Volatility Model
• Bibliography
• Index