Engineering BGM Book

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