Engineering BGM

Also known as the Libor market model, the Brace-Gatarek-Musiela (BGM) model is becoming an industry standard for pricing interest rate derivatives. Written by one of its developers, Engineering BGM builds progressively from simple to more sophisticated versions of the BGM model, offering a range of methods that can be programmed into production code to suit readers' requirements.After introducing the standard lognormal flat BGM model, the book focuses on the shifted/displaced diffusion version. Using this version, the author develops basic ideas about construction, change of measure, correlation, calibration, simulation, timeslicing, pricing, delta hedging, barriers, callable exotics (Bermudans), and vega hedging. Subsequent chapters address cross-economy BGM, the adaptation of the BGM model to inflation, a simple tractable stochastic volatility version of BGM, and Brazilian options suitable for BGM analysis. An appendix provides notation and an extensive array of formulae.The straightforward presentation of various BGM models in this handy book will help promote a robust, safe, and stable environment for calibrating, simulating, pricing, and hedging interest rate instruments.

Preface	p. xiii
Introduction	p. 1
Background HJM	p. 2
The first 'correct' Black caplet	p. 6
Forward BGM construction	p. 8
Bond and Swap Basics	p. 11
Zero coupon bonds - drifts and volatilities	p. 11
Swaps and swap notation	p. 14
Forward over several periods	p. 18
Current time	p. 19
Shifted BGM	p. 21
Definition of shifted model	p. 21
Several points worth noting	p. 22
Backward construction	p. 24
Swaprate Dynamics	p. 27
Splitting the swaprate	p. 28
The shift part	p. 29
The stochastic part	p. 31
Swaption values	p. 34
Multi-period caplets	p. 35
Swaprate models	p. 36
Properties of Measures	p. 39
Changes among forward and swaprate measures	p. 40
Terminal measure	p. 41
Spot Libor measure	p. 42
Jumping measure	p. 44
Historical Correlation and Volatility	p. 45
Flat and shifted BGM off forwards	p. 48
Gaussian HJM off yield-to-maturity	p. 49
Flat and shifted BGM off swaprates	p. 50
Calibration Techniques	p. 55
Fitting the skew	p. 57
Maturity only fit	p. 58
Homogeneous spines	p. 59
Piecewise linear	p. 59
Rebonato's function	p. 60
Bi-exponential function	p. 60
Sum of exponentials	p. 60
Separable one-factor fit	p. 61
Separable multi-factor fit	p. 63
Alternatively	p. 65
Pedersen's method	p. 66
Cascade fit	p. 69
Extension	p. 71
Exact fit with semidefinite programming	p. 71
Interpolating Between Nodes	p. 75
Interpolating forwards	p. 75
Dead forwards	p. 76
Interpolation of discount factors	p. 77
Consistent volatility	p. 78
Simulation	p. 79
Glasserman type simulation	p. 79
Under the terminal measure P[subscript n]	p. 80
Under the spot measure P[subscript 0]	p. 80
Big-step simulation	p. 81
Volatility approximation	p. 81
Drift approximation	p. 82
Big-stepping under the terminal measure P[subscript n]	p. 84
Big-stepping under a tailored spot measure P[subscript 0]	p. 84
Timeslicers	p. 87
Terminal measure timeslicer	p. 88
Intermediate measure timeslicer	p. 89
A spot measure timeslicer is problematical	p. 90
Some technical points	p. 91
Node placement	p. 91
Cubics against Gaussian density	p. 92
Splining the integrand	p. 92
Alternative spline	p. 93
Two-dimensional timeslicer	p. 93
Pathwise Deltas	p. 95
Partial derivatives of forwards	p. 96
Partial derivatives of zeros and swaps	p. 97
Differentiating option payoffs	p. 98
Vanilla caplets and swaptions	p. 99
Barrier caps and floors	p. 100
Bermudans	p. 103
Backward recursion	p. 104
Alternative backward recursion	p. 106
The Longstaff-Schwartz lower bound technique	p. 106
When to exercise	p. 107
Regression technique	p. 108
Comments on the Longstaff-Schwartz technique	p. 109
Upper bounds	p. 110
Bermudan deltas	p. 111
Vega and Shift Hedging	p. 113
When calibrated to coterminal swaptions	p. 114
The shift part	p. 115
The volatility part	p. 116
When calibrated to liquid swaptions	p. 118
Cross-Economy BGM	p. 121
Cross-economy HJM	p. 121
Forward FX contracts	p. 123
In the HJM framework	p. 124
In the BGM framework	p. 125
Cross-economy models	p. 127
Model with the spot volatility deterministic	p. 128
Cross-economy correlation	p. 131
Pedersen type cross-economy calibration	p. 135
Inflation	p. 141
TIPS and the CPI	p. 141
Dynamics of the forward inflation curve	p. 143
Futures contracts	p. 145
The CME futures contract	p. 146
Stochastic Volatility BGM	p. 149
Construction	p. 149
Swaprate dynamics	p. 153
Shifted Heston options	p. 155
Characteristic function	p. 155
Option price as a Fourier integral	p. 158
Simulation	p. 160
Simulating V (t)	p. 160
Interpolation, Greeks and calibration	p. 162
Interpolation	p. 162
Greeks	p. 162
Caplet calibration	p. 163
Swaption calibration	p. 164
Options in Brazil	p. 165
Overnight DI	p. 165
Pre-DI swaps and swaptions	p. 166
In the HJM framework	p. 168
In the BGM framework	p. 168
DI index options	p. 169
In the HJM framework	p. 169
DI futures contracts	p. 170
Hedging with futures contracts	p. 172
DI futures options	p. 172
Notation and Formulae	p. 175
Swap notation	p. 175
Gaussian distributions	p. 176
Conditional expectations	p. 176
Density shift	p. 176
Black formula	p. 178
Gaussian density derivatives	p. 179
Gamma and vega connection	p. 181
Bivariate distribution	p. 182
Ratio of cumulative and density distributions	p. 182
Expected values of normals	p. 183
Stochastic calculus	p. 185
Multi-dimensional Ito	p. 185
Brownian bridge	p. 185
Product and quotient processes	p. 185
Conditional change of measure	p. 186
Girsanov theorem	p. 186
One-dimensional Ornstein Uhlenbeck process	p. 188
Generalized multi-dimensional OU process	p. 188
SDE of a discounted variable	p. 188
Ito-Venttsel formula	p. 189
Linear Algebra	p. 189
Cholesky decomposition	p. 189
Singular value decomposition	p. 190
Semidefinite programming (SDP)	p. 192
Some Fourier transform technicalities	p. 195
The chi-squared distribution	p. 198
Miscellaneous	p. 201
Futures contracts	p. 201
Random variables from an arbitrary distribution	p. 201
Copula methodology	p. 201
References	p. 203
Table of Contents provided by Ingram. All Rights Reserved.

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