FOREIGN EXCHANGE DERIVATIVES: Effective Theoretical and Practical

255 Pages·2011·8.17 MB·English

FOREIGN EXCHANGE
DERIVATIVES:
Effective Theoretical and Practical
Techniques for Trading, Hedging
and Managing FX Derivatives
by
Dr. A. A. Kotze�
Financial Chaos Theory Pty. Ltd.
March 2011
http:\\www.quantonline.co.za
Philosophy is written in that great book whichever
lies before our gaze | I mean the universe |
but we cannot understand if we do not �rst learn the
language and grasp the symbols in which it is written.
The book is written in the mathematical language, and
the symbols are triangle, circles and other geometrical
�gures, without the help of which it is impossible to
conceive a single word of it, and without which one
wonders in vain through a dark labyrinth.
Galileo Galilei (1564-1642)
Abstract
Instruments traded in the financial markets are getting more and more complex.
This leads to more complex derivative structures that are harder to analyse and risk
managed. These instruments cannot be traded or managed without the relevant
systems and numerical techniques.
The global economy is becoming more and more interlinked with trading between
countries skyrocketing. Due to the world trade, foreign exchange forwards, futures,
options and exotics are becoming increasingly commonplace in today’s capital mar-
kets.
The objective of these notes is to let the reader develop a solid understanding of
the current currency derivatives used in international treasury management with an
emphasis on the African continent. This will give participants the mathematical and
practical background necessary to deal with all the products on the market.
Before I came here I was confused about the subject.
Having listened to your lecture I am still confused.
But on a higher level.
Enrico Fermi (1901-1954)
Financial Chaos Theory Pty. Ltd.
Illuminating OTC and listed Derivatives through:
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Financial Chaos Theory
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Doornfontein 2028
South Africa
But the creative principle resides in mathematics.
In a certain sense, therefore, I hold it true that
pure thought can grasp reality, as the ancients dreamed.
Albert Einstein
Contents
1 Introduction 12
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2 A History of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3 A History of Option Valuation . . . . . . . . . . . . . . . . . . . . . . 17
1.4 A Word on Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.5 Payoff Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.6 Understanding the Foreign Exchange Markets . . . . . . . . . . . . . 22
1.6.1 The Gold Standard . . . . . . . . . . . . . . . . . . . . . . . . 22
1.6.2 Uniqueness of the Market . . . . . . . . . . . . . . . . . . . . 22
1.6.3 Size of the Market . . . . . . . . . . . . . . . . . . . . . . . . 23
1.6.4 Growth in the FX Market . . . . . . . . . . . . . . . . . . . . 26
1.6.5 Emerging Markets: Growth in Derivatives Activity . . . . . . 28
1.6.6 Currency Composition of OTC Derivatives in Emerging Markets 30
1.7 Why Trade FX? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.8 Where is the Spot Traded . . . . . . . . . . . . . . . . . . . . . . . . 34
1.9 Quotation Styles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.10 Jargon: Nicknames . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.11 Settlement Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.11.1 Banking Holidays . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.11.2 Islamic Currencies . . . . . . . . . . . . . . . . . . . . . . . . 39
1.12 Pips and Big Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.13 Major, Minor and Exotic Currencies . . . . . . . . . . . . . . . . . . 41
1.13.1 Most Heavily traded Currencies . . . . . . . . . . . . . . . . . 41
1.13.2 Circulating Currencies . . . . . . . . . . . . . . . . . . . . . . 42
1.13.3 Groups of Currencies . . . . . . . . . . . . . . . . . . . . . . . 44
1.14 What is a Hard Currency? . . . . . . . . . . . . . . . . . . . . . . . . 45
1.15 Cross Currencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
1.16 Currency Indices: Rand Currency Index . . . . . . . . . . . . . . . . 49
2 Foreign Exchange Risk and Hedging 53
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.2 FX Market Participants . . . . . . . . . . . . . . . . . . . . . . . . . 55
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2.3 Risks in Forex Trading . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.4 Modern Corporate Risk Management . . . . . . . . . . . . . . . . . . 57
2.5 What is FX Risk Management? . . . . . . . . . . . . . . . . . . . . . 57
2.6 The Great Equiliser . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.7 How to Minimise Foreign Exchange Risks? . . . . . . . . . . . . . . . 60
2.7.1 Foreign Exchange Forward Contracts . . . . . . . . . . . . . . 60
2.7.2 Currency Options . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.7.3 Cross Currency Swaps . . . . . . . . . . . . . . . . . . . . . . 61
2.8 Hedging FX Exposures in Emerging Markets . . . . . . . . . . . . . . 61
2.9 Hedging and Profit Opportunities . . . . . . . . . . . . . . . . . . . . 64
2.10 Fundamental en Technical Analysis . . . . . . . . . . . . . . . . . . . 65
2.11 Hedging Policy Principles . . . . . . . . . . . . . . . . . . . . . . . . 66
2.12 Regulatory and Accounting Initiatives . . . . . . . . . . . . . . . . . 68
2.12.1 Basel Accords . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.12.2 Regulatory Reform of OTC Derivatives . . . . . . . . . . . . . 71
2.12.3 Accounting Reforms . . . . . . . . . . . . . . . . . . . . . . . 72
3 Capital Markets in Africa 74
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.2 Time Value of Money . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.2.1 Day Count . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.2.2 Simple Interest Rate Calculations . . . . . . . . . . . . . . . . 76
3.2.3 Zero-Coupon Bond . . . . . . . . . . . . . . . . . . . . . . . . 77
3.2.4 Discount Factors . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.2.5 Compounding Periods . . . . . . . . . . . . . . . . . . . . . . 78
3.2.6 Compounded Rates . . . . . . . . . . . . . . . . . . . . . . . . 78
3.2.7 Forward Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.2.8 Converting between Rates . . . . . . . . . . . . . . . . . . . . 81
3.2.9 Nominal and Effective Rates . . . . . . . . . . . . . . . . . . . 81
3.3 Modern Capital Markets . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.4 African Equity and Bond Markets . . . . . . . . . . . . . . . . . . . . 84
3.4.1 South Africa . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.4.2 Kenya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.4.3 Nigeria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.4.4 Mauritius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.4.5 Egypt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.5 The Zero-Coupon Yield Curve . . . . . . . . . . . . . . . . . . . . . . 94
3.6 From Spot to Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.6.1 The Spot Price . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.6.2 Simple Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 99
3.6.3 The Forward Contract . . . . . . . . . . . . . . . . . . . . . . 100
3.6.4 The Futures Contract . . . . . . . . . . . . . . . . . . . . . . . 102
2
3.6.5 Cash Flow Differences . . . . . . . . . . . . . . . . . . . . . . 102
3.7 Arbitrage-free Pricing of Forwards and Futures . . . . . . . . . . . . . 103
3.7.1 What is Arbitrage? . . . . . . . . . . . . . . . . . . . . . . . . 103
3.7.2 Box Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.7.3 Carry Cost and Forward Points . . . . . . . . . . . . . . . . . 107
3.7.4 Trading Futures on Exchange . . . . . . . . . . . . . . . . . . 110
3.8 Non-Deliverable Forwards . . . . . . . . . . . . . . . . . . . . . . . . 110
3.9 Contango and Backwardation . . . . . . . . . . . . . . . . . . . . . . 111
3.10 Forward Prices versus Futures Prices . . . . . . . . . . . . . . . . . . 113
3.11 Trading African Currency Derivatives . . . . . . . . . . . . . . . . . . 114
4 Properties of Options Prices 116
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.2 The Two Types of Derivatives . . . . . . . . . . . . . . . . . . . . . . 117
4.3 Option Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.4 Exchange Traded Options . . . . . . . . . . . . . . . . . . . . . . . . 118
4.5 Over-the-Counter Options . . . . . . . . . . . . . . . . . . . . . . . . 118
4.6 Why trade FX Options versus the Spot FX? . . . . . . . . . . . . . . 119
4.7 The Benefits of the Black & Scholes Model . . . . . . . . . . . . . . . 121
4.8 Black, Scholes and Merton did not invent any Formula! . . . . . . . . 122
4.9 The Black & Scholes Environment . . . . . . . . . . . . . . . . . . . . 125
4.10 The Black & Scholes Analysis . . . . . . . . . . . . . . . . . . . . . . 128
4.10.1 The Option Payoff . . . . . . . . . . . . . . . . . . . . . . . . 128
4.10.2 The Black & Scholes Hedging Strategy: a Replicating Portfolio 128
4.11 The Seminal Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.12 The Intuitive Black & Scholes . . . . . . . . . . . . . . . . . . . . . . 132
4.12.1 As a Replicating Portfolio - Delta Hedging . . . . . . . . . . . 132
4.12.2 As a Present Value of Risk-neutral Expectation . . . . . . . . 133
4.13 A Currency Option Model . . . . . . . . . . . . . . . . . . . . . . . . 133
4.14 Options on Forwards and Futures . . . . . . . . . . . . . . . . . . . . 134
4.15 Settlement Adjustments . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.16 Option Pricing in Excel . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.17 Option Sensitivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5 The Mechanics of Option Prices 139
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.2 Payoff Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.3 Building Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.4 Put-Call-Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.5 Option Dynamics and Risk Managements . . . . . . . . . . . . . . . . 143
5.5.1 Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.5.2 The Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
3
5.5.3 The Gamma and Theta Relationship . . . . . . . . . . . . . . 149
5.5.4 Intrinsic and Time Value . . . . . . . . . . . . . . . . . . . . . 149
5.6 Some more Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . 152
5.7 Useful Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.7.1 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.7.2 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.7.3 Put-Call-Delta Parity . . . . . . . . . . . . . . . . . . . . . . . 154
5.7.4 Symmetries for Currency Options . . . . . . . . . . . . . . . . 154
6 American Options 156
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.2 Pricing American Options . . . . . . . . . . . . . . . . . . . . . . . . 156
6.3 Optimal Early Exercies . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.4 The Early Exercise Boundary . . . . . . . . . . . . . . . . . . . . . . 158
6.5 American versus European Options . . . . . . . . . . . . . . . . . . . 160
6.6 Binomial Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.6.1 The Stock Price Tree . . . . . . . . . . . . . . . . . . . . . . . 161
6.6.2 Valuing Options on Currencies . . . . . . . . . . . . . . . . . . 164
6.6.3 Advantages of the Binomial Tree . . . . . . . . . . . . . . . . 167
6.6.4 Risk Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.7 Convergence of the Binomial Model . . . . . . . . . . . . . . . . . . . 168
6.8 The Trinomial Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.9 Options on Currency Futures on Safex/YieldX . . . . . . . . . . . . 171
7 Volatility 174
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7.2 Volatility Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.3 Volatility Defined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
7.4 The Variance Rate of Return . . . . . . . . . . . . . . . . . . . . . . 178
7.5 Empirical Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
7.6 Importance of Volatility . . . . . . . . . . . . . . . . . . . . . . . . . 180
7.7 Different Types of Volatility . . . . . . . . . . . . . . . . . . . . . . . 181
7.8 Trading or Nontrading Days . . . . . . . . . . . . . . . . . . . . . . . 181
7.9 Estimation of Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . 182
7.9.1 Historical Volatility . . . . . . . . . . . . . . . . . . . . . . . . 182
7.9.2 Noncentered Volatility . . . . . . . . . . . . . . . . . . . . . . 185
7.9.3 Realized/Actual Volatility . . . . . . . . . . . . . . . . . . . . 185
7.9.4 Implied Volatility . . . . . . . . . . . . . . . . . . . . . . . . . 186
7.10 Extreme-Value Estimators . . . . . . . . . . . . . . . . . . . . . . . . 187
7.10.1 Parkinson Estimator . . . . . . . . . . . . . . . . . . . . . . . 187
7.10.2 Garman-Klass Estimator . . . . . . . . . . . . . . . . . . . . . 188
7.11 Higher Moments of the Normal Distribution . . . . . . . . . . . . . . 189
4
7.12 Skewness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
7.13 Kurtosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
7.14 The Volatility Measure? . . . . . . . . . . . . . . . . . . . . . . . . . 190
7.15 Sample Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
7.15.1 Volatility Plots . . . . . . . . . . . . . . . . . . . . . . . . . . 192
7.15.2 Introducing Filtering . . . . . . . . . . . . . . . . . . . . . . . 193
7.16 The Volatility Skew . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
7.16.1 Universality of the Skew . . . . . . . . . . . . . . . . . . . . . 195
7.16.2 Why do we observe a Skew? . . . . . . . . . . . . . . . . . . . 196
7.16.3 Shapes of the skew . . . . . . . . . . . . . . . . . . . . . . . . 199
7.16.4 Delta Hedging and the Skew . . . . . . . . . . . . . . . . . . . 202
7.16.5 The Term Structure of Volatility . . . . . . . . . . . . . . . . 203
7.16.6 What is a Volatility Surface? . . . . . . . . . . . . . . . . . . . 203
7.16.7 Skews in South Africa . . . . . . . . . . . . . . . . . . . . . . 204
7.17 Linear Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
7.18 Sticky Volatilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
7.18.1 Sticky Delta . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
7.18.2 Sticky Strike . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
7.18.3 Which is better: sticky strike or sticky delta? . . . . . . . . . 210
8 Option Strategies 212
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
8.2 Basic Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
8.2.1 Buying Calls . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
8.2.2 Buying Puts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
8.2.3 Bull and Bear Spreads . . . . . . . . . . . . . . . . . . . . . . 215
8.2.4 Risk Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
8.3 Neutral Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
8.3.1 Straddles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
8.3.2 Strips and Straps . . . . . . . . . . . . . . . . . . . . . . . . . 222
8.3.3 Strangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
8.3.4 Butterfly Spreads . . . . . . . . . . . . . . . . . . . . . . . . . 223
8.3.5 The Iron Condor . . . . . . . . . . . . . . . . . . . . . . . . . 225
8.3.6 Fence or Seagull . . . . . . . . . . . . . . . . . . . . . . . . . . 225
8.4 Trading Strategies and Market Conditions . . . . . . . . . . . . . . . 227
8.4.1 Key Points to Remember . . . . . . . . . . . . . . . . . . . . . 228
8.4.2 BULLISH Market Strategies . . . . . . . . . . . . . . . . . . . 228
8.4.3 BEARISH Market Strategies . . . . . . . . . . . . . . . . . . . 229
8.4.4 NEUTRAL Market Strategies . . . . . . . . . . . . . . . . . . 230
8.4.5 Special Market Situations . . . . . . . . . . . . . . . . . . . . 231
A Create an Effective Risk Management Policy 232
5
B Solution to the BS Differential Equation 236
C Probabilistic Solution 238
D Calculating the Implied Volatility 240
E Black-Scholes or Black Holes? 241
F Connecting Africa 246
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List of Figures
1.1 Payoff profiles for futures and forwards. . . . . . . . . . . . . . . . . . 20
1.2 Payoff profiles for Calls and Puts. . . . . . . . . . . . . . . . . . . . . 21
1.3 Payoff profile for a capped collar. . . . . . . . . . . . . . . . . . . . . 21
1.4 Foreign exchange instruments by turnover. . . . . . . . . . . . . . . . 24
1.5 Regional breakdown of FX turnover by instrument. . . . . . . . . . . 24
1.6 Turnover by currency pair. . . . . . . . . . . . . . . . . . . . . . . . . 25
1.7 Increase in global FX market turnover by counterparty. . . . . . . . . 26
1.8 Growth in algorithmic trading. . . . . . . . . . . . . . . . . . . . . . . 27
1.9 Graphs showing the increase in global FX market turnover. . . . . . . 28
1.10 Derivatives turnover in advanced and emerging markets. . . . . . . . 29
1.11 Derivatives turnover in EMEs during 2010 showing the ratio between
OTC and exchange traded derivatives. . . . . . . . . . . . . . . . . . 33
1.12 Top 10 currency traders . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.13 Execution method in FX trades . . . . . . . . . . . . . . . . . . . . . 36
1.14 A: OTC foreign exchange traded in South Africa. B: South Africa’s
Rand traded against other currencies. . . . . . . . . . . . . . . . . . . 37
1.15 Settlement date and day count convention for some major currencies
[Ca 10]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.16 Most traded currencies. Note that the total adds up to 200% due to
each currency being a pair. . . . . . . . . . . . . . . . . . . . . . . . . 41
1.17 FX market turnover by currency pair. . . . . . . . . . . . . . . . . . . 42
1.18 African currency rates shown on Reuters 3000. . . . . . . . . . . . . . 43
1.19 Historical FX rates for African currencies: ZAR, KES, NGN and EGP. 46
1.20 RAIN since inception. . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.1 Corporate risks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.2 Objectives of Risk Management. . . . . . . . . . . . . . . . . . . . . . 58
2.3 Hedging should stabilise cash flows. . . . . . . . . . . . . . . . . . . . 59
2.4 Principles of a Policy and Process. . . . . . . . . . . . . . . . . . . . . 67
2.5 Steps to hedge accounting. . . . . . . . . . . . . . . . . . . . . . . . . 73
3.1 Cash flow diagram for a zero-coupon bond. . . . . . . . . . . . . . . . 77
3.2 Determining the forward rate. . . . . . . . . . . . . . . . . . . . . . . 81
7