Fx options and structured products uwe wystup 0470011459

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FX Options and Structured Products

Uwe Wystup

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Trắc nghiệm kiến thức Forex tại : />

FX Options and Structured Products

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For other titles in the Wiley Finance Series
please see www.wiley.com/finance

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FX Options and Structured Products

Uwe Wystup

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C 2006
Copyright 

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Library of Congress Cataloging-in-Publication Data
Wystup, Uwe.
FX options and structured products / Uwe Wystup.
p. cm.
Includes bibliographical references.
ISBN-13: 978-0-470-01145-4
ISBN-10: 0-470-01145-9
1. Foreign exchange options. 2. Structured notes (Securities) 3. Derivative securities.
HG3853. W88 2006
332.4 5—dc22
2006020352

I. Title.

British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
ISBN 13 978-0-470-01145-4 (HB)
ISBN 10 0-470-01145-9 (HB)
Typeset in 10/12pt Times by TechBooks, New Delhi, India
Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire
This book is printed on acid-free paper responsibly manufactured from sustainable forestry
in which at least two trees are planted for each one used for paper production.

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To Ansua

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Contents
Preface
Scope of this Book
The Readership
About the Author
Acknowledgments

xiii
xiii
xiii
xiv
xiv

1 Foreign Exchange Options
1.1 A journey through the history of options
1.2 Technical issues for vanilla options
1.2.1 Value
1.2.2 A note on the forward
1.2.3 Greeks
1.2.4 Identities
1.2.5 Homogeneity based relationships
1.2.6 Quotation
1.2.7 Strike in terms of delta
1.2.8 Volatility in terms of delta
1.2.9 Volatility and delta for a given strike
1.2.10 Greeks in terms of deltas
1.3 Volatility
1.3.1 Historic volatility
1.3.2 Historic correlation

1.3.3 Volatility smile
1.3.4 At-the-money volatility interpolation
1.3.5 Volatility smile conventions
1.3.6 At-the-money definition
1.3.7 Interpolation of the volatility on maturity pillars
1.3.8 Interpolation of the volatility spread between maturity pillars
1.3.9 Volatility sources
1.3.10 Volatility cones
1.3.11 Stochastic volatility
1.3.12 Exercises

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Contents

1.4 Basic strategies containing vanilla options
1.4.1 Call and put spread
1.4.2 Risk reversal
1.4.3 Risk reversal flip
1.4.4 Straddle
1.4.5 Strangle
1.4.6 Butterfly
1.4.7 Seagull
1.4.8 Exercises
1.5 First generation exotics
1.5.1 Barrier options
1.5.2 Digital options, touch options and rebates
1.5.3 Compound and instalment
1.5.4 Asian options
1.5.5 Lookback options
1.5.6 Forward start, ratchet and cliquet options

1.5.7 Power options
1.5.8 Quanto options
1.5.9 Exercises
1.6 Second generation exotics
1.6.1 Corridors
1.6.2 Faders
1.6.3 Exotic barrier options
1.6.4 Pay-later options
1.6.5 Step up and step down options
1.6.6 Spread and exchange options
1.6.7 Baskets
1.6.8 Best-of and worst-of options
1.6.9 Options and forwards on the harmonic average
1.6.10 Variance and volatility swaps
1.6.11 Exercises
2 Structured Products
2.1 Forward products
2.1.1 Outright forward
2.1.2 Participating forward
2.1.3 Fade-in forward
2.1.4 Knock-out forward
2.1.5 Shark forward
2.1.6 Fader shark forward
2.1.7 Butterfly forward
2.1.8 Range forward
2.1.9 Range accrual forward
2.1.10 Accumulative forward
2.1.11 Boomerang forward
2.1.12 Amortizing forward
2.1.13 Auto-renewal forward

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Contents

2.2

2.3

2.4

2.5

2.6

2.1.14 Double shark forward
2.1.15 Forward start chooser forward
2.1.16 Free style forward
2.1.17 Boosted spot/forward
2.1.18 Time option
2.1.19 Exercises
Series of strategies
2.2.1 Shark forward series
2.2.2 Collar extra series
2.2.3 Exercises
Deposits and loans
2.3.1 Dual currency deposit/loan
2.3.2 Performance linked deposits
2.3.3 Tunnel deposit/loan
2.3.4 Corridor deposit/loan
2.3.5 Turbo deposit/loan
2.3.6 Tower deposit/loan
2.3.7 Exercises
Interest rate and cross currency swaps
2.4.1 Cross currency swap
2.4.2 Hanseatic swap
2.4.3 Turbo cross currency swap
2.4.4 Buffered cross currency swap
2.4.5 Flip swap
2.4.6 Corridor swap
2.4.7 Double-no-touch linked swap
2.4.8 Range reset swap
2.4.9 Exercises

Participation notes
2.5.1 Gold participation note
2.5.2 Basket-linked note
2.5.3 Issuer swap
2.5.4 Moving strike turbo spot unlimited
Hybrid FX products

3 Practical Matters
3.1 The traders’ rule of thumb
3.1.1 Cost of vanna and volga
3.1.2 Observations
3.1.3 Consistency check
3.1.4 Abbreviations for first generation exotics
3.1.5 Adjustment factor
3.1.6 Volatility for risk reversals, butterflies and theoretical value
3.1.7 Pricing barrier options
3.1.8 Pricing double barrier options
3.1.9 Pricing double-no-touch options
3.1.10 Pricing european style options

ix

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x

Contents

3.1.11 No-touch probability
3.1.12 The cost of trading and its implication on the market price
of one-touch options
3.1.13 Example
3.1.14 Further applications
3.1.15 Exercises
3.2 Bid–ask spreads
3.2.1 One touch spreads
3.2.2 Vanilla spreads
3.2.3 Spreads for first generation exotics
3.2.4 Minimal bid–ask spread
3.2.5 Bid–ask prices
3.2.6 Exercises
3.3 Settlement
3.3.1 The Black-Scholes model for the actual spot

3.3.2 Cash settlement
3.3.3 Delivery settlement
3.3.4 Options with deferred delivery
3.3.5 Exercises
3.4 On the cost of delayed fixing announcements
3.4.1 The currency fixing of the European Central Bank
3.4.2 Model and payoff
3.4.3 Analysis procedure
3.4.4 Error estimation
3.4.5 Analysis of EUR-USD
3.4.6 Conclusion
4 Hedge Accounting under IAS 39
4.1 Introduction
4.2 Financial instruments
4.2.1 Overview
4.2.2 General definition
4.2.3 Financial assets
4.2.4 Financial liabilities
4.2.5 Offsetting of financial assets and financial liabilities
4.2.6 Equity instruments
4.2.7 Compound financial instruments
4.2.8 Derivatives
4.2.9 Embedded derivatives
4.2.10 Classification of financial instruments
4.3 Evaluation of financial instruments
4.3.1 Initial recognition
4.3.2 Initial measurement
4.3.3 Subsequent measurement
4.3.4 Derecognition
4.4 Hedge accounting

4.4.1 Overview
4.4.2 Types of hedges

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Contents

4.5

4.6

4.7
4.8

4.9

4.4.3 Basic requirements
4.4.4 Stopping hedge accounting
Methods for testing hedge effectiveness
4.5.1 Fair value hedge
4.5.2 Cash flow hedge
Testing for effectiveness – a case study of the forward plus
4.6.1 Simulation of exchange rates
4.6.2 Calculation of the forward plus value
4.6.3 Calculation of the forward rates
4.6.4 Calculation of the forecast transaction’s value
4.6.5 Dollar-offset ratio – prospective test for effectiveness
4.6.6 Variance reduction measure – prospective test for effectiveness
4.6.7 Regression analysis – prospective test for effectiveness
4.6.8 Result
4.6.9 Retrospective test for effectiveness
Conclusion
Relevant original sources for accounting standards
Exercises

xi

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5 Foreign Exchange Markets
5.1 A tour through the market
5.1.1 Statement by GFI group (Fenics), 25 October 2005
5.1.2 Interview with ICY software, 14 October 2005
5.1.3 Interview with Bloomberg, 12 October 2005
5.1.4 Interview with Murex, 8 November 2005
5.1.5 Interview with SuperDerivatives, 17 October 2005
5.1.6 Interview with Lucht Probst Associates, 27 February 2006
5.2 Software and system requirements
5.2.1 Fenics
5.2.2 Position keeping
5.2.3 Pricing
5.2.4 Straight through processing
5.2.5 Disclaimers
5.3 Trading and sales
5.3.1 Proprietary trading
5.3.2 Sales-driven trading
5.3.3 Inter bank sales

5.3.4 Branch sales
5.3.5 Institutional sales
5.3.6 Corporate sales
5.3.7 Private banking
5.3.8 Listed FX options
5.3.9 Trading floor joke

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Bibliography

313

Index

319

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Preface
SCOPE OF THIS BOOK
Treasury management of international corporates involves dealing with cash flows in different
currencies. Therefore the natural service of an investment bank consists of a variety of money
market and foreign exchange products. This book explains the most popular products and
strategies with a focus on everything beyond vanilla options.
The book explains all the FX options, common structures and tailor-made solutions in
examples with a special focus on their application with views from traders and sales as well
as from a corporate client perspective.
The book contains actual traded deals with corresponding motivations explaining why the
structures have been traded. This way the reader gets a feel for how to build new structures to
suit clients’ needs.
Several sections deal with the quantitative aspect of FX options, such as quanto adjustment,
deferred delivery, traders’ rule of thumb and settlement issues.
One entire chapter is devoted to hedge accounting, after which the foundations of a typical
structured FX forward are examined in a case study.

The exercises are meant as practice. Several of them are actually difficult to solve
and can serve as incentives to further research and testing. Solutions to the exercises are
not part of this book. They will, however, be published on the web page for the book,
fxoptions.mathfinance.com.

THE READERSHIP
A prerequisite is some basic knowledge of FX markets, for example taken from Foreign
Exchange Primer by Shami Shamah, John Wiley & Sons, Ltd, 2003, see [1]. For the quantitative
sections some knowledge of Stochastic Calculus as in Steven E. Shreve’s volumes on Stochastic
Calculus for Finance [2] is useful. The target readers are

r
r
r

Graduate students and Faculty of Financial Engineering Programs, who can use this book
as a textbook for a course named structured products or exotic currency options.
Traders, Trainee Structurers, Product Developers, Sales and Quants with an interest in the
FX product line. For them it can serve as a source of ideas and as well as a reference guide.
Treasurers of corporates interested in managing their books. With this book at hand they can
structure their solutions themselves.

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xiv

Preface

The readers more interested in the quantitative and modeling aspects are recommended to read
Foreign Exchange Risk by J. Hakala and U. Wystup, Risk Publications, London, 2002, see [50].

This book explains several exotic FX options with a special focus on the underlying models
and mathematics, but does not contain any structures or corporate clients’ or investors’ view.

ABOUT THE AUTHOR

Uwe Wystup, Professor of Quantitative Finance at HfB Business
School of Finance and Management in Frankfurt, Germany

Uwe Wystup is also CEO of MathFinance AG, a global network of quants specializing in Quantitative Finance, Exotic Options advisory and Front Office Software Production. Previously he
was a Financial Engineer and Structurer in the FX Options Trading Team at Commerzbank.
Before that he worked for Deutsche Bank, Citibank, UBS and Sal. Oppenheim jr. & Cie. He is
founder and manager of the web site MathFinance.de and the MathFinance Newsletter. Uwe
holds a PhD in mathematical finance from Carnegie Mellon University. He also lectures on
mathematical finance for Goethe University Frankfurt, organizes the Frankfurt MathFinance
Colloquium and is founding director of the Frankfurt MathFinance Institute. He has given
several seminars on exotic options, computational finance and volatility modeling. His areas
of specialization are the quantitative aspects and the design of structured products of foreign
exchange markets. He published a book on Foreign Exchange Risk and articles in Finance
and Stochastics and the Journal of Derivatives. Uwe has given many presentations at both
universities and banks around the world. Further information on his curriculum vitae and a
detailed publication list is available at www.mathfinance.com/wystup/.

ACKNOWLEDGMENTS
I would like to thank HfB-Business School of Finance and Management in Frankfurt for
supporting this book project by allocating the necessary time.
I would like to thank my former colleagues on the trading floor, most of all Michael Braun,
Jăurgen Hakala, Tam´as Korchm´aros, Behnouch Mostachfi, Bereshad Nonas, Gustave Rieunier,
Tino Senge, Ingo Schneider, Jan Schrader, Noel Speake, Roman Stauss, Andreas Weber, and

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Preface

xv

all my colleagues and co-authors, specially Christoph Becker, Susanne Griebsch, Christoph
Kăuhn, Sebastian Krug, Marion Linck, Wolfgang Schmidt and Robert Tompkins.
I would like to thank Steve Shreve for his training in stochastic calculus and continuously
supporting my academic activities.
Chris Swain, Rachael Wilkie and many others of Wiley publications deserve respect for
dealing with my tardiness in completing this book.
Nicole van de Locht and Choon Peng Toh deserve a medal for their detailed proof reading.

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1
Foreign Exchange Options
FX Structured Products are tailor-made linear combinations of FX Options including both
vanilla and exotic options. We recommend the book by Shamah [1] as a source to learn about
FX Markets with a focus on market conventions, spot, forward and swap contracts and vanilla
options. For pricing and modeling of exotic FX options we suggest Hakala and Wystup [3] or
Lipton [4] as useful companions to this book.
The market for structured products is restricted to the market of the necessary ingredients.
Hence, typically there are mostly structured products traded in the currency pairs that can be
formed between USD, JPY, EUR, CHF, GBP, CAD and AUD. In this chapter we start with a
brief history of options, followed by a technical section on vanilla options and volatility, and
deal with commonly used linear combinations of vanilla options. Then we will illustrate the
most important ingredients for FX structured products: the first and second generation exotics.

1.1 A JOURNEY THROUGH THE HISTORY OF OPTIONS
The very first options and futures were traded in ancient Greece, when olives were sold before
they had reached ripeness. Thereafter the market evolved in the following way:
16th century Ever since the 15th century tulips, which were popular because of their exotic
appearance, were grown in Turkey. The head of the royal medical gardens in Vienna, Austria,
was the first to cultivate Turkish tulips successfully in Europe. When he fled to Holland
because of religious persecution, he took the bulbs along. As the new head of the botanical
gardens of Leiden, Netherlands, he cultivated several new strains. It was from these gardens
that avaricious traders stole the bulbs in order to commercialize them, because tulips were a
great status symbol.
17th century The first futures on tulips were traded in 1630. From 1634, people could buy
special tulip strains according to the weight of their bulbs, the same value was chosen for
the bulbs as for gold. Along with regular trading, speculators entered the market and prices
skyrocketed. A bulb of the strain “Semper Octavian” was worth two wagonloads of wheat,
four loads of rye, four fat oxen, eight fat swine, twelve fat sheep, two hogsheads of wine, four
barrels of beer, two barrels of butter, 1,000 pounds of cheese, one marriage bed with linen
and one sizable wagon. People left their families, sold all their belongings, and even borrowed
money to become tulip traders. When in 1637, this supposedly risk-free market crashed, traders
as well as private individuals went bankrupt. The government prohibited speculative trading;
this period became famous as Tulipmania.
18th century In 1728, the Royal West-Indian and Guinea Company, the monopolist in trading
with the Caribbean Islands and the African coast issued the first stock options. These were
options on the purchase of the French Island of Ste. Croix, on which sugar plantings were

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2

FX Options and Structured Products

planned. The project was realized in 1733 and paper stocks were issued in 1734. Along with
the stock, people purchased a relative share of the island and the possessions, as well as the
privileges and the rights of the company.
19th century In 1848, 82 businessmen founded the Chicago Board of Trade (CBOT). Today
it is the biggest and oldest futures market in the entire world. Most written documents were lost
in the great fire of 1871, however, it is commonly believed that the first standardized futures
were traded in 1860. CBOT now trades several futures and forwards, not only T-bonds and
treasury bonds, but also options and gold.
In 1870, the New York Cotton Exchange was founded. In 1880, the gold standard was
introduced.
20th century
• In 1914, the gold standard was abandoned because of the war.
• In 1919, the Chicago Produce Exchange, in charge of trading agricultural products was
renamed to Chicago Mercantile Exchange. Today it is the most important futures market for
Eurodollar, foreign exchange, and livestock.
• In 1944, the Bretton Woods System was implemented in an attempt to stabilize the currency
system.
• In 1970, the Bretton Woods System was abandoned for several reasons.
• In 1971, the Smithsonian Agreement on fixed exchange rates was introduced.
• In 1972, the International Monetary Market (IMM) traded futures on coins, currencies and
precious metal.
• In 1973, the CBOE (Chicago Board of Exchange) first traded call options; and four years
later also put options. The Smithsonian Agreement was abandoned; the currencies followed
managed floating.
• In 1975, the CBOT sold the first interest rate future, the first future with no “real” underlying
asset.
• In 1978, the Dutch stock market traded the first standardized financial derivatives.
• In 1979, the European Currency System was implemented, and the European Currency Unit
(ECU) was introduced.

• In 1991, the Maastricht Treaty on a common currency and economic policy in Europe was
signed.
• In 1999, the Euro was introduced, but the countries still used their old currencies, while the
exchange rates were kept fixed.
21st century In 2002, the Euro was introduced as new money in the form of cash.

1.2 TECHNICAL ISSUES FOR VANILLA OPTIONS
We consider the model geometric Brownian motion
d St = (rd − r f )St dt + σ St d Wt

(1.1)

for the underlying exchange rate quoted in FOR-DOM (foreign-domestic), which means that
one unit of the foreign currency costs FOR-DOM units of the domestic currency. In the case
of EUR-USD with a spot of 1.2000, this means that the price of one EUR is 1.2000 USD.
The notion of foreign and domestic does not refer to the location of the trading entity, but only

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Foreign Exchange Options

probability density

exchange rate development
1.5

1.5

1

1

0.5

0.5

0

0

0.2

0.4

0.6

3

0.8

1

0

0

4

2

Figure 1.1 Simulated paths of a geometric Brownian motion. The distribution of the spot ST at time T
is log-normal. The light gray line reflects the average spot movement.

to this quotation convention. We denote the (continuous) foreign interest rate by r f and the
(continuous) domestic interest rate by rd . In an equity scenario, r f would represent a continuous
dividend rate. The volatility is denoted by σ , and Wt is a standard Brownian motion. The sample
paths are displayed in Figure 1.1.1 We consider this standard model, not because it reflects
the statistical properties of the exchange rate (in fact, it doesn’t), but because it is widely used
in practice and front office systems and mainly serves as a tool to communicate prices in FX
options. These prices are generally quoted in terms of volatility in the sense of this model.
Applying Itˆo’s rule to ln St yields the following solution for the process St



1
St = S0 exp rd − r f − σ 2 t + σ Wt ,
(1.2)
2
which shows that St is log-normally distributed, more precisely, ln St is normal with mean
ln S0 + (rd − r f − 12 σ 2 )t and variance σ 2 t. Further model assumptions are
1. There is no arbitrage
2. Trading is frictionless, no transaction costs
3. Any position can be taken at any time, short, long, arbitrary fraction, no liquidity constraints
The payoff for a vanilla option (European put or call) is given by
F = [φ(ST − K )]+ ,

(1.3)

where the contractual parameters are the strike K , the expiration time T and the type φ, a
binary variable which takes the value +1 in the case of a call and −1 in the case of a put. The



symbol x + denotes the positive part of x, i.e., x + = max(0, x) = 0 ∨ x. We generally use the

symbol = to define a quantity. Most commonly, vanilla options on foreign exchange are of
European style, i.e. the holder can only exercise the option at time T. American style options,
1

Generated with Tino Kluge’s shape price simulator at www.mathfinance.com/TinoKluge/tools/sharesim/black-scholes.php

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4

FX Options and Structured Products

where the holder can exercise any time, or Bermudian style options, where the holder can
exercise at selected times, are not used very often except for time options, see Section 2.1.18.
1.2.1 Value
In the Black-Scholes model the value of the payoff F at time t if the spot is at x is denoted
by v(t, x) and can be computed either as the solution of the Black-Scholes partial differential
equation (see [5])
1
vt − rd v + (rd − r f )xvx + σ 2 x 2 vx x = 0,
(1.4)
2
v(T, x) = F.
(1.5)
or equivalently (Feynman-Kac-Theorem) as the discounted expected value of the payofffunction,
v(x, K , T, t, σ, rd , r f , φ) = e−rd τ IE[F].

(1.6)

This is the reason why basic financial engineering is mostly concerned with solving partial
differential equations or computing expectations (numerical integration). The result is the
Black-Scholes formula
v(x, K , T, t, σ, rd , r f , φ) = φe−rd τ [ f N (φd+ ) − K N (φd− )].

(1.7)

We abbreviate
•
•
•
•

x: current price of the underlying

τ = T − t: time to maturity

f = IE[ST |St = x] = xe(rd −r f )τ : forward price of the underlying
 r −r
θ± = d σ f ± σ2

•
•

2

+σ θ± τ

ln f ± σ τ
√
= Kσ √τ2
σ τ
1 2

n(t) = √12π e− 2 t = n(−t)
 x
N (x) = −∞ n(t) dt = 1 −
 ln

• d± =

x
K

N (−x)

The Black-Scholes formula can be derived using the integral representation of Equation (1.6)
v = e−rd τ IE[F]
= e−rd τ IE[[φ(ST − K )]+ ]
 +∞ 

+
√
1 2
−rd τ
=e
φ xe(rd −r f − 2 σ )τ +σ τ y − K
n(y) dy.

−∞

(1.8)

Next one has to deal with the positive part and then complete the square to get the BlackScholes formula. A derivation based on the partial differential equation can be done using
results about the well-studied heat-equation.
1.2.2 A note on the forward
The forward price f is the strike which makes the time zero value of the forward contract
F = ST − f

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(1.9)


Foreign Exchange Options

5

equal to zero. It follows that f = IE[ST ] = xe(rd −r f )T , i.e. the forward price is the expected
price of the underlying at time T in a risk-neutral setup (drift of the geometric Brownian motion
is equal to cost of carry rd − r f ). The situation rd > r f is called contango, and the situation
rd < r f is called backwardation. Note that in the Black-Scholes model the class of forward
price curves is quite restricted. For example, no seasonal effects can be included. Note that
the value of the forward contract after time zero is usually different from zero, and since one
of the counterparties is always short, there may be risk of default of the short party. A futures
contract prevents this dangerous affair: it is basically a forward contract, but the counterparties
have to maintain a margin account to ensure the amount of cash or commodity owed does not
exceed a specified limit.
1.2.3 Greeks
Greeks are derivatives of the value function with respect to model and contract parameters.

They are an important information for traders and have become standard information provided
by front-office systems. More details on Greeks and the relations among Greeks are presented
in Hakala and Wystup [3] or Reiss and Wystup [6]. For vanilla options we list some of them
now.
(Spot) delta
∂v
= φe−r f τ N (φd+ )
∂x

(1.10)

∂v
= φe−rd τ N (φd+ )
∂f

(1.11)

Forward delta

Driftless delta
φN (φd+ )

(1.12)

∂ 2v
n(d+ )
= e−r f τ √
2
∂x
xσ τ

(1.13)

Gamma

Speed
∂ 3v
n(d+ )
= −e−r f τ 2 √
∂x3
x σ τ



d+
√ +1
σ τ


(1.14)

Theta
∂v
n(d+ )xσ
= −e−r f τ
√
∂t
2 τ
+ φ[r f xe−r f τ N (φd+ ) − rd K e−rd τ N (φd− )]
Charm

√
2(rd − r f )τ − d− σ τ
∂ 2v
−r f τ
−r f τ
N (φd+ ) + φe
n(d+ )
= −φr f e
√
∂ x∂τ
2τ σ τ

(1.15)

(1.16)

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FX Options and Structured Products

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