**520 pages, 94 figures, 9 tables.**

Christian P. Fries

Version 1.5.12 = First Edition, First Printing.

Version 1.6.11 = First Edition, Second Printing.

Order the hardcover edition at amazon.com, amazon.co.uk, amazon.de, amazon.at, amazon.fr, amazon.jp, AbeBooks.co.uk, AbeBooks.de, Barnes & Noble.com, bol.de, buch.de, lesen.de, libri.de, or your favored book store.

Order the ebook edition at Diesel eBooks.

- Mathematical Finance Laboratory: Tutorial Applets
- Mathematical Finance Picture Book: The figures in color. Free download.
- Bibliography: A hyperlinked bibliography.
- Errata: Error corrections.
- Updates: Additional sections.
- finmath.net: Java code and Excel evaluation spreadsheets.

- Suggestions for Lecturers: Sample lecture schedule based on the book.

You may take a look at the book using the Amazon.com reader (a random page will be displayed, browsing other pages is allowed, but requires an Amazon.com account).

The full table of contents is given below.

The book arose from my lecture notes for the lectures on mathematical finance held at University of Mainz and University of Frankfurt. It tries to give a balanced representation of the theoretical foundations, state of the art models which are actually used in practice and their implementation.

In practice, none of the three aspects "theory", "modeling" and "implementation" may be considered alone. Knowledge of the theory is worthless if it isn't applied. Theory gives the tools for a consistent modeling. A model without implementation is essentially worthless. A good implementation requires a deep understanding of the model and the underlying theory.

With this in mind, the book tries to build a bridge from academia to practice and from theory to object oriented implementation.

Order the hardcover edition at AbeBooks.co.uk, AbeBooks.de, amazon.com, amazon.co.uk, amazon.de, amazon.at, amazon.fr, amazon.jp, Barnes & Noble.com, bol.de, buch.de, lesen.de, libri.de, or your favored book store.

1 | Introduction | 1 |

1.1 | Theory, Modeling, and Implementation | 1 |

1.2 | Interest Rate Models and Interest Rate Derivatives | 1 |

1.3 | About This Book | 3 |

1.3.1 | How to Read This Book | 3 |

1.3.2 | Abridged Versions | 3 |

1.3.2.1 | Abridged version "Monte Carlo Pricing" | 3 |

1.3.2.2 | Abridged version "LIBOR Market Model" | 3 |

1.3.2.3 | Abridged version "Markov Functional Model" | 3 |

1.3.3 | Special Sections | 4 |

1.3.4 | Notation | 4 |

1.3.5 | Feedback | 5 |

1.3.6 | Resources | 5 |

I | Foundations | 7 |

2 | Foundations | 9 |

2.1 | Probability Theory | 9 |

2.2 | Stochastic Processes | 18 |

2.3 | Filtration | 20 |

2.4 | Brownian Motion | 22 |

2.5 | Wiener Measure, Canonical Setup | 24 |

2.6 | Ito Calculus | 25 |

2.6.1 | Ito Integral | 28 |

2.6.2 | Ito Process | 30 |

2.6.3 | Ito Lemma and Product Rule | 32 |

2.7 | Brownian Motion with Instantaneous Correlation | 36 |

2.8 | Martingales | 38 |

2.8.1 | Martingale Representation Theorem | 38 |

2.9 | Change of Measure | 39 |

2.10 | Stochastic Integration | 44 |

2.11 | Partial Differential Equations (PDEs) | 46 |

2.11.1 | Feynman-Kac Theorem | 46 |

2.12 | List of Symbols | 48 |

3 | Replication | 49 |

3.1 | Replication Strategies | 49 |

3.1.1 | Introduction | 49 |

3.1.2 | Replication in a Discrete Model | 53 |

3.1.2.1 | Example: Two Times, Two States, Two Assets | 53 |

3.2 | Foundations: Equivalent Martingale Measure | 58 |

3.2.1 | Challenge and Solution Outline | 58 |

3.2.2 | Steps toward the Universal Pricing Theorem | 61 |

3.2.2.1 | Self-Financing Trading Strategy | 62 |

3.2.2.2 | Relative Prices | 63 |

3.2.2.3 | Equivalent Martingale Measure | 66 |

3.2.2.4 | Payoff Replication | 67 |

3.3 | Excursus: Relative Prices and Risk-Neutral Measures | 70 |

3.3.1 | Why relative prices? | 70 |

3.3.2 | Risk-Neutral Measure | 72 |

II | First Applications | 73 |

4 | Pricing of a European Stock Option under the Black-Scholes Model | 75 |

5 | Excursus: The Density of the Underlying of a European Call Option | 81 |

6 | Excursus: Interpolation of European Option Prices | 83 |

6.1 | No-Arbitrage Conditions for Interpolated Prices | 83 |

6.2 | Arbitrage Violations through Interpolation | 85 |

6.2.1 | Example 1: Interpolation of Four Prices | 85 |

6.2.1.1 | Linear Interpolation of Prices | 85 |

6.2.1.2 | Linear Interpolation of Implied Volatilities | 86 |

6.2.1.3 | Spline Interpolation of Prices and Implied Volatilities | 87 |

6.2.2 | Example 2: Interpolation of Two Prices | 87 |

6.2.2.1 | Linear Interpolation for Decreasing Implied Volatilities | 88 |

Conclusion: | 88 | |

6.2.2.2 | Linear Interpolation for Increasing Implied Volatilities | 88 |

Conclusion: | 89 | |

6.3 | Arbitrage-Free Interpolation of European Option Prices | 89 |

7 | Hedging in Continuous and Discrete Time and the Greeks | 93 |

7.1 | Introduction | 93 |

7.2 | Deriving the Replications Strategy from Pricing Theory | 94 |

Conclusion: | 95 | |

7.2.1 | Deriving the Replication Strategy under the Assumption of a Locally Riskless Product | 96 |

7.2.2 | Black-Scholes Differential Equation | 97 |

7.2.3 | Derivative V(t) as a Function of Its Underlyings S_{i}(t) | 97 |

7.2.3.1 | Path-Dependent Options | 98 |

7.2.4 | Example: Replication Portfolio and PDE under a Black-Scholes Model | 99 |

7.2.4.1 | Interpretation of V as a Function in (t,S) | 101 |

7.3 | Greeks | 102 |

7.3.1 | Greeks of a European Call-Option under the Black-Scholes Model | 103 |

7.4 | Hedging in Discrete Time: Delta and Delta-Gamma Hedging | 103 |

7.4.1 | Delta Hedging | 105 |

7.4.2 | Error Propagation | 106 |

7.4.2.1 | Example: Time-Discrete Delta Hedge under a Black-Scholes Model | 107 |

7.4.3 | Delta-Gamma Hedging | 109 |

7.4.3.1 | Example: Time-Discrete Delta-Gamma Hedge under a Black-Scholes Model | 111 |

7.4.4 | Vega Hedging | 113 |

7.5 | Hedging in Discrete Time: Minimizing the Residual Error (Bouchaud-Sornette Method) | 113 |

7.5.1 | Minimizing the Residual Error at Maturity T | 115 |

7.5.2 | Minimizing the Residual Error in Each Time Step | 117 |

III | Interest Rate Structures, Interest Rate Products, and Analytic Pricing Formulas | 119 |

Motivation and Overview | 121 | |

8 | Interest Rate Structures | 123 |

8.1 | Introduction | 123 |

8.1.1 | Fixing Times and Tenor Times | 124 |

8.2 | Definitions | 124 |

8.3 | Interest Rate Curve Bootstrapping | 130 |

Induction Start (T_{0}): | 130 | |

Induction Step (T_{i-1} to T_{i}): | 130 | |

8.4 | Interpolation of Interest Rate Curves | 130 |

8.5 | Implementation | 131 |

9 | Simple Interest Rate Products | 133 |

9.1 | Interest Rate Products Part 1: Products without Optionality | 133 |

9.1.1 | Fix, Floating, and Swap | 133 |

9.1.2 | Money Market Account | 140 |

9.2 | Interest Rate Products Part 2: Simple Options | 142 |

9.2.1 | Cap, Floor, and Swaption | 142 |

9.2.1.1 | Example: Option on a Coupon Bond | 144 |

9.2.2 | Foreign Caplet, Quanto | 144 |

10 | The Black Model for a Caplet | 147 |

11 | Pricing of a Quanto Caplet (Modeling the FFX) | 151 |

11.1 | Choice of Numeraire | 151 |

12 | Exotic Derivatives | 155 |

12.1 | Prototypical Product Properties | 155 |

12.2 | Interest Rate Products Part 3: Exotic Interest Rate Derivatives | 157 |

12.2.1 | Structured Bond, Structured Swap, and Zero Structure | 158 |

12.2.2 | Bermudan Option | 162 |

12.2.3 | Bermudan Callable and Bermudan Cancelable | 164 |

12.2.4 | Compound Options | 166 |

12.2.5 | Trigger Products | 167 |

12.2.5.1 | Target Redemption Note | 167 |

12.2.6 | Structured Coupons | 168 |

12.2.6.1 | Capped, Floored, Inverse, Spread, CMS | 169 |

12.2.6.2 | Range Accruals | 170 |

12.2.6.3 | Path-Dependent Coupons | 170 |

Example: | 171 | |

12.2.6.4 | Flexi-Cap | 171 |

12.2.7 | Shout Options | 173 |

12.3 | Product Toolbox | 174 |

IV | Discretization and Numerical Valuation Methods | 177 |

Motivation and Overview | 179 | |

13 | Discretization of Time and State Space | 181 |

13.1 | Discretization of Time: The Euler and the Milstein Schemes | 181 |

13.1.1 | Definitions | 183 |

13.1.2 | Time Discretization of a Lognormal Process | 185 |

13.1.2.1 | Discretization via Euler Scheme | 185 |

13.1.2.2 | Discretization via Milstein scheme | 185 |

13.1.2.3 | Discretization of the Log Process | 186 |

13.1.2.4 | Exact Discretization | 186 |

13.2 | Discretization of Paths (Monte Carlo Simulation) | 186 |

13.2.1 | Monte Carlo Simulation | 187 |

13.2.2 | Weighted Monte Carlo Simulation | 187 |

13.2.3 | Implementation | 188 |

13.2.3.1 | Example: Valuation of a Stock Option under the Black-Scholes Model Using Monte Carlo Simulation | 188 |

13.2.3.2 | Separation of Product and Model | 189 |

13.2.3.3 | Model-Product Communication Protocol | 191 |

Core Object: Random Variable | 192 | |

Aggregation 1: Vector of Random Variables of Same Simulation Time | 192 | |

Aggregation 2: Time-Discrete Stochastic Process | 192 | |

Storage, Access, and Processing | 193 | |

13.2.4 | Review | 193 |

13.3 | Discretization of State Space | 195 |

13.3.1 | Definitions | 195 |

13.3.2 | Backward Algorithm | 197 |

13.3.3 | Review | 197 |

13.3.3.1 | Path Dependencies | 197 |

13.3.3.2 | Course of Dimension | 198 |

13.4 | Path Simulation through a Lattice: Two Layers | 198 |

14 | Numerical Methods for Partial Differential Equations | 199 |

15 | Pricing Bermudan Options in a Monte Carlo Simulation | 201 |

15.1 | Introduction | 201 |

15.2 | Bermudan Options: Notation | 202 |

15.2.1 | Bermudan Callable | 203 |

15.2.2 | Relative Prices | 203 |

15.3 | Bermudan Option as Optimal Exercise Problem | 204 |

15.3.1 | Bermudan Option Value as Single (Unconditioned) Expectation: The Optimal Exercise Value | 204 |

15.4 | Bermudan Option Pricing---The Backward Algorithm | 205 |

Induction start | 205 | |

Induction step | 206 | |

15.5 | Resimulation | 207 |

15.6 | Perfect Foresight | 207 |

15.7 | Conditional Expectation as Functional Dependence | 209 |

Example: | 210 | |

15.8 | Binning | 210 |

15.8.1 | Binning as a Least-Square Regression | 212 |

15.9 | Foresight Bias | 214 |

15.10 | Regression Methods---Least-Square Monte Carlo | 215 |

15.10.1 | Least-Square Approximation of the Conditional Expectation | 215 |

15.10.2 | Example: Evaluation of a Bermudan Option on a Stock (Backward Algorithm with Conditional Expectation Estimator) | 216 |

Induction Start | 216 | |

Induction Step | 217 | |

15.10.3 | Example: Evaluation of a Bermudan Callable | 217 |

Induction Start | 217 | |

Induction Step | 221 | |

15.10.4 | Implementation | 222 |

15.10.5 | Binning as Linear Least-Square Regression | 223 |

15.11 | Optimization Methods | 224 |

15.11.1 | Andersen Algorithm for Bermudan Swaptions | 224 |

Induction Start | 225 | |

Induction Step | 225 | |

15.11.2 | Review of the Threshold Optimization Method | 225 |

15.11.2.1 | Fitting the Exercise Strategy to the Product | 225 |

15.11.2.2 | Disturbance of the Optimizer through Discontinuities and Local Minima | 227 |

15.11.3 | Optimization of Exercise Strategy: A More General Formulation | 228 |

15.11.4 | Comparison of Optimization Method and Regression Method | 228 |

15.12 | Duality Method: Upper Bound for Bermudan Option Prices | 230 |

15.12.1 | Foundations | 230 |

15.12.2 | American Option Evaluation as Optimal Stopping Problem | 232 |

15.13 | Primal-Dual Method: Upper and Lower Bound | 235 |

16 | Pricing Path-Dependent Options in a Backward Algorithm | 237 |

16.1 | State Space Extension | 237 |

16.2 | Implementation | 238 |

16.3 | Path-Dependent Bermudan Options | 239 |

16.4 | Examples | 240 |

16.4.1 | Evaluation of a Snowball in a Backward Algorithm | 240 |

16.4.2 | Evaluation of a Autocap in a Backward Algorithm | 240 |

17 | Sensitivities (Partial Derivatives) of Monte Carlo Prices | 243 |

17.1 | Introduction | 243 |

17.2 | Problem Description | 244 |

17.2.1 | Pricing using Monte-Carlo Simulation | 244 |

17.2.2 | Sensitivities from Monte Carlo Pricing | 245 |

17.2.3 | Example: The Linear and the Discontinuous Payout | 245 |

17.2.3.1 | Linear Payout | 246 |

17.2.3.2 | Discontinuous Payout | 246 |

17.2.4 | Example: Trigger Products | 247 |

17.3 | Generic Sensitivities: Bumping the Model | 249 |

17.4 | Sensitivities by Finite Differences | 251 |

17.4.1 | Example: Finite Differences Applied to Smooth and Discontinuous Payout | 252 |

Simplified Example: | 253 | |

17.5 | Sensitivities by Pathwise Differentiation | 254 |

17.5.1 | Example: Delta of a European Option under a Black-Scholes Model | 254 |

17.5.2 | Pathwise Differentiation for Discontinuous Payouts | 255 |

17.6 | Sensitivities by Likelihood Ratio Weighting | 256 |

17.6.1 | Example: Delta of a European Option under a Black-Scholes Model Using Pathwise Derivative | 257 |

17.6.2 | Example: Variance Increase of the Sensitivity when using Likelihood Ratio Method for Smooth Payouts | 257 |

17.7 | Sensitivities by Malliavin Weighting | 258 |

17.8 | Proxy Simulation Scheme | 259 |

18 | Proxy Simulation Schemes for Monte Carlo Sensitivities and Importance Sampling | 261 |

18.1 | Full Proxy Simulation Scheme | 261 |

18.1.1 | Pricing under a Proxy Simulation Scheme | 262 |

18.1.1.1 | Basic Properties of a Proxy Simulation Scheme | 262 |

18.1.2 | Calculation of Monte Carlo Weights | 262 |

18.1.3 | Sensitivities by Finite Differences on a Proxy Simulation Scheme | 263 |

18.1.4 | Localization | 264 |

18.1.5 | Object-Oriented Design | 265 |

18.1.6 | Importance Sampling | 265 |

18.1.6.1 | Example | 265 |

18.2 | Partial Proxy Simulation Schemes | 268 |

18.2.1 | Linear Proxy Constraint | 268 |

18.2.2 | Comparison to Full Proxy Scheme Method | 269 |

18.2.3 | Nonlinear Proxy Constraint | 269 |

18.2.3.1 | Linearization of the Proxy Constraint | 270 |

18.2.3.2 | Finite Difference Approximation of the Nonlinear Proxy Constraint | 270 |

Example: | 271 | |

18.2.4 | Transition Probability from a Nonlinear Proxy Constraint | 271 |

18.2.4.1 | The Proxy Constraint Revisited | 271 |

18.2.4.2 | Transition Probabilities for General Proxy Constraints | 272 |

18.2.4.3 | Example | 273 |

18.2.4.4 | Approximating an F_{ti+1}-measurable Proxy Constraint by an F_{ti}-measurable Proxy Constraint | 274 |

18.2.5 | Sensitivity with Respect to the Diffusion Coefficients---Vega | 274 |

18.2.6 | Example: LIBOR Target Redemption Note | 274 |

18.2.7 | Example: CMS Target Redemption Note | 276 |

18.2.7.1 | Delta and Gamma of a CMS TARN | 277 |

18.2.7.2 | Vega of a CMS TARN | 278 |

18.3 | Localized Proxy Simulation Schemes | 279 |

18.3.1 | Problem Description | 279 |

18.3.2 | Solution | 282 |

18.3.3 | Partial Proxy Simulation Scheme (revisited) | 282 |

18.3.3.1 | Reference Scheme and Target Scheme | 282 |

18.3.3.2 | Transition Probabilities | 283 |

18.3.3.3 | Proxy Constraint and Proxy Scheme | 283 |

18.3.3.4 | Calculating Expectations using a Proxy Simulation Scheme | 283 |

18.3.3.5 | Example: Euler Schemes | 284 |

18.3.4 | Localized Proxy Simulation Scheme | 285 |

18.3.5 | Example: Euler Schemes | 286 |

18.3.6 | Implementation | 286 |

18.3.7 | Examples and Numerical Results | 287 |

18.3.7.1 | Localizers | 287 |

18.3.7.2 | Model | 287 |

18.3.7.3 | Example: Digital Caplet | 287 |

18.3.7.4 | Example: Target Redemption Note (TARN) | 288 |

18.3.7.5 | Proxy Constraint and Localizer for the Target Redemption Note | 291 |

V | Pricing Models for Interest Rate Derivatives | 293 |

Motivation and Overview | 295 | |

19 | LIBOR Market Model | 297 |

19.1 | Derivation of the Drift Term | 299 |

19.1.1 | Derivation of the Drift Term under the Terminal Measure | 299 |

19.1.2 | Derivation of the Drift Term under the Spot LIBOR Measure | 301 |

19.1.3 | Derivation of the Drift Term under the T_{k}-Forward Measure | 303 |

19.2 | The Short Period Bond P(T_{m(t)+1};t) | 304 |

19.2.1 | Role of the Short Bond in a LIBOR Market Model | 304 |

19.2.2 | Link to Continuous Time Tenors | 304 |

19.2.3 | Drift of the Short Bond in a LIBOR Market Model | 304 |

19.3 | Discretization and (Monte Carlo) Simulation | 305 |

19.3.1 | Generation of the (Time-Discrete) Forward Rate Process | 305 |

19.3.2 | Generation of the Sample Paths | 306 |

19.3.3 | Generation of the Numeraire | 306 |

19.4 | Calibration---Choice of the Free Parameters | 307 |

19.4.1 | Choice of the Initial Conditions | 308 |

19.4.1.1 | Reproduction of Bond Market Prices | 308 |

19.4.2 | Choice of the Volatilities | 308 |

19.4.2.1 | Reproduction of Caplet Market Prices | 308 |

19.4.2.2 | Reproduction of Swaption Market Prices | 309 |

19.4.2.3 | Functional Forms | 311 |

19.4.3 | Choice of the Correlations | 311 |

19.4.3.1 | Factors | 311 |

19.4.3.2 | Functional Forms | 312 |

19.4.3.3 | Factor Reduction | 312 |

19.4.3.4 | Calibration | 313 |

19.4.4 | Covariance Structure | 313 |

19.4.5 | Analytic Evaluation of Caplets, Swaptions and Swap Rate Covariance | 314 |

19.4.5.1 | Analytic Evaluation of a Caplet in the LIBOR Market Model | 314 |

19.4.5.2 | Analytic Evaluation of a Swaption in the LIBOR Market Model | 314 |

19.4.5.3 | Analytic Calculation of Swap Rate Covariance in the LIBOR Market Model | 318 |

19.5 | Interpolation of Forward Rates in the LIBOR Market Model | 319 |

19.5.1 | Interpolation of the Tenor Structure {T_{i}&rculb; | 319 |

19.5.1.1 | Assumption 1: No Stochastic Shortly Before Maturity. | 320 |

19.5.1.2 | Assumption 2: Linearity Shortly Before Maturity. | 321 |

19.6 | Object-Oriented Design | 323 |

19.6.1 | Reuse of Implementation | 324 |

19.6.2 | Separation of Product and Model | 324 |

19.6.3 | Abstraction of Model Parameters | 324 |

19.6.4 | Abstraction of Calibration | 325 |

20 | Swap Rate Market Models | 329 |

20.1 | The Swap Measure | 330 |

20.2 | Derivation of the Drift Term | 331 |

20.3 | Calibration---Choice of the Free Parameters | 332 |

20.3.1 | Choice of the Initial Conditions | 332 |

20.3.1.1 | Reproduction of Bond Market Prices or Swap Market Prices | 332 |

20.3.2 | Choice of the Volatilities | 332 |

20.3.2.1 | Reproduction of Swaption Market Prices | 332 |

21 | Excursus: Instantaneous Correlation and Terminal Correlation | 335 |

21.1 | Definitions | 335 |

21.2 | Terminal Correlation Examined in a LIBOR Market Model Example | 336 |

21.2.1 | Decorrelation in a One-Factor Model | 337 |

21.2.2 | Impact of the Time Structure of the Instantaneous Volatility on Caplet and Swaption Prices | 339 |

21.2.3 | Swaption Value as a Function of Forward Rates | 340 |

21.3 | Terminal Correlation Is Dependent on the Equivalent Martingale Measure | 342 |

21.3.1 | Dependence of the Terminal Density on the Martingale Measure | 342 |

22 | Heath-Jarrow-Morton Framework: Foundations | 345 |

22.1 | Short-Rate Process in the HJM Framework | 346 |

22.2 | The HJM Drift Condition | 347 |

23 | Short-Rate Models | 351 |

23.1 | Introduction | 351 |

23.2 | The Market Price of Risk | 352 |

23.3 | Overview: Some Common Models | 354 |

23.4 | Implementations | 355 |

23.4.1 | Monte Carlo Implementation of Short-Rate Models | 355 |

23.4.2 | Lattice Implementation of Short-Rate Models | 355 |

24 | Heath-Jarrow-Morton Framework: Immersion of Short-Rate Models and LIBOR Market Model | 357 |

24.1 | Short-Rate Models in the HJM Framework | 357 |

24.1.1 | Example: The Ho-Lee Model in the HJM Framework | 358 |

24.1.2 | Example: The Hull-White Model in the HJM Framework | 359 |

24.2 | LIBOR Market Model in the HJM Framework | 360 |

24.2.1 | HJM Volatility Structure of the LIBOR Market Model | 360 |

24.2.2 | LIBOR Market Model Drift under the Q^{B} Measure | 362 |

24.2.3 | LIBOR Market Model as a Short Rate Model | 364 |

25 | Excursus: Shape of the Interest Rate Curve under Mean Reversion and a Multifactor Model | 365 |

25.1 | Model | 365 |

25.2 | Interpretation of the Figures | 366 |

25.3 | Mean Reversion | 367 |

25.4 | Factors | 368 |

25.5 | Exponential Volatility Function | 369 |

25.6 | Instantaneous Correlation | 371 |

26 | Ritchken-Sakarasubramanian Framework: HJM with Low Markov Dimension | 373 |

26.1 | Introduction | 373 |

26.2 | Cheyette Model | 374 |

26.3 | Implementation: PDE | 375 |

27 | Markov Functional Models | 377 |

27.1 | Introduction | 377 |

27.1.1 | The Markov Functional Assumption (Independent of the Model Considered) | 378 |

27.1.2 | Outline of This Chapter | 379 |

27.2 | Equity Markov Functional Model | 379 |

27.2.1 | Markov Functional Assumption | 379 |

27.2.2 | Example: The Black-Scholes Model | 380 |

27.2.3 | Numerical Calibration to a Full Two-Dimensional European Option Smile Surface | 381 |

27.2.3.1 | Market Price | 382 |

27.2.3.2 | Model Price | 382 |

27.2.3.3 | Solving for the Functional | 383 |

27.2.4 | Interest Rates | 383 |

27.2.4.1 | A Note on Interest Rates and the No-Arbitrage Requirement | 383 |

27.2.4.2 | Where Are the Interest Rates? | 383 |

27.2.5 | Model Dynamics | 384 |

27.2.5.1 | Introduction | 384 |

27.2.5.2 | Interest Rate Dynamics | 386 |

27.2.5.3 | Forward Volatility | 388 |

27.2.6 | Implementation | 390 |

27.3 | LIBOR Markov Functional Model | 390 |

27.3.1 | LIBOR Markov Functional Model in Terminal Measure | 390 |

27.3.1.1 | Evaluation within the LIBOR Markov Functional Model | 392 |

27.3.1.2 | Calibration of the LIBOR Functional | 393 |

Induction start | 393 | |

Induction step | 393 | |

Induction start | 396 | |

Induction step | 396 | |

27.3.2 | LIBOR Markov Functional Model in Spot Measure | 396 |

27.3.2.1 | Calibration of the Markov Functional Model under Spot Measure | 397 |

27.3.2.2 | Forward Induction Step | 397 |

27.3.2.3 | Dealing With the Path Dependency of the Numeraire | 398 |

27.3.2.4 | Efficient Calculation of the LIBOR Functional From Given Market Prices | 398 |

27.3.3 | Remark on Implementation | 400 |

27.3.3.1 | Fast Calculation of Price Functionals | 400 |

27.3.3.2 | Discussion on the Implementation of the Markov Functional Model under Terminal and Spot Measure | 401 |

27.3.4 | Change of Numeraire in a Markov Functional Model | 401 |

27.4 | Implementation: Lattice | 403 |

27.4.1 | Convolution with the Normal Probability Density | 404 |

27.4.1.1 | Piecewise Constant Approximation | 404 |

27.4.1.2 | Piecewise Polynomial Approximation | 405 |

27.4.2 | State Space Discretization | 407 |

27.4.2.1 | Equidistant Discretization | 407 |

VI | Extended Models | 409 |

28 | Credit Spreads | 411 |

28.1 | Introduction---Different Types of Spreads | 411 |

28.1.1 | Spread on a Coupon | 411 |

28.1.2 | Credit Spread | 411 |

28.2 | Defaultable Bonds | 412 |

28.3 | Integrating Deterministic Credit Spread into a Pricing Model | 414 |

28.3.1 | Deterministic Credit Spread | 415 |

28.3.2 | Implementation | 416 |

28.4 | Receiver's and Payer's Credit Spreads | 418 |

28.4.1 | Example: Defaultable Forward Starting Coupon Bond | 419 |

28.4.2 | Example: Option on a Defaultable Coupon Bond | 420 |

29 | Hybrid Models | 421 |

29.1 | Cross-Currency LIBOR Market Model | 421 |

29.1.1 | Derivation of the Drift Term under Spot Measure | 422 |

29.1.1.1 | Dynamic of the Domestic LIBOR under Spot Measure | 422 |

29.1.1.2 | Dynamic of the Foreign LIBOR under Spot Measure | 423 |

29.1.1.3 | Dynamic of the FX Rate under Spot Measure | 425 |

29.1.2 | Implementation | 426 |

29.2 | Equity Hybrid LIBOR Market Model | 426 |

29.2.1 | Derivation of the Drift Term under Spot-Measure | 426 |

29.2.1.1 | Dynamic of the Stock Process under Spot Measure | 427 |

29.2.2 | Implementation | 428 |

29.3 | Equity Hybrid Cross-Currency LIBOR Market Model | 428 |

29.3.1 | Dynamic of the Foreign Stock under Spot Measure | 429 |

29.3.2 | Summary | 430 |

29.3.3 | Implementation | 431 |

VII | Implementation | 433 |

30 | Object-Oriented Implementation in Java™ | 435 |

30.1 | Elements of Object-Oriented Programming: Class and Objects | 435 |

30.1.1 | Example: Class of a Binomial Distributed Random Variable | 436 |

30.1.2 | Constructor | 438 |

30.1.3 | Methods: Getter, Setter, and Static Methods | 438 |

30.1.3.1 | Calling Convention, Signatures | 438 |

30.1.3.2 | Getter, Setter | 439 |

30.1.3.3 | Static Methods | 439 |

30.2 | Principles of Object Oriented Programming | 440 |

30.2.1 | Encapsulation and Interfaces | 440 |

30.2.1.1 | Encapsulation | 441 |

Example of Encapsulation: Offering Alternative Methods: | 442 | |

Example of Encapsulation: Performance Improvement by Adding a Cache to the Internal Data Modell: | 442 | |

30.2.1.2 | Interfaces | 443 |

30.2.2 | Abstraction and Inheritance | 444 |

30.2.3 | Polymorphism | 447 |

30.3 | Example: A Class Structure for One-Dimensional Root Finders | 449 |

30.3.1 | Root Finder for General Functions | 449 |

30.3.1.1 | Interface | 449 |

30.3.1.2 | Bisection Search | 451 |

30.3.2 | Root Finder for Functions with Analytic Derivative: Newton's Method | 451 |

30.3.2.1 | Interface | 451 |

30.3.2.2 | Newton Method | 451 |

30.3.3 | Root Finder for Functions with Derivative Estimation: Secant Method | 452 |

30.3.3.1 | Inheritance | 452 |

30.3.3.2 | Polymorphism | 454 |

30.4 | Anatomy of a Java™ Class | 458 |

30.5 | Libraries | 460 |

30.5.1 | Java™ 2 Platform, Standard Edition (j2se) | 460 |

30.5.2 | Java™ 2 Platform, Enterprise Edition (j2ee) | 460 |

30.5.3 | Colt | 460 |

30.5.4 | Commons-Math: The Jakarta Mathematics Library | 461 |

30.6 | Some Final Remarks | 461 |

30.6.1 | Object-Oriented Design (OOD)/Unified Modeling Language (UML) | 461 |

VIII | Appendices | 463 |

A | A Small Collection of Common Misconceptions | 465 |

B | Tools (Selection) | 467 |

B.1 | Generation of Random Numbers | 467 |

B.1.1 | Uniform Distributed Random Variables | 467 |

B.1.1.1 | Mersenne Twister | 467 |

B.1.2 | Transformation of the Random Number Distribution via the Inverse Distribution Function | 468 |

B.1.3 | Normal Distributed Random Variables | 468 |

B.1.3.1 | Inverse Distribution Function | 468 |

B.1.3.2 | Box-Muller Transformation | 468 |

B.1.4 | Poisson Distributed Random Variables | 468 |

B.1.4.1 | Inverse Distribution Function | 468 |

B.1.5 | Generation of Paths of an n-Dimensional Brownian Motion | 469 |

B.2 | Factor Decomposition---Generation of Correlated Brownian Motion | 471 |

B.3 | Factor Reduction | 472 |

B.4 | Optimization (One-Dimensional): Golden Section Search | 475 |

B.5 | Linear Regression | 476 |

B.6 | Convolution with Normal Density | 477 |

C | Exercises | 479 |

D | Java™ Source Code (Selection) | 487 |

D.1 | Java™ Classes for Chapter 30 | 487 |

List of Symbols | 493 | |

List of Figures | 495 | |

List of Tables | 499 | |

List of Listings | 500 | |

Bibliography | 503 | |

Index | 511 |

- Created: March 2003 - today
- LaTeX Source: >1 MB
- Tools: Operating System: Mac OS X. LaTeX IDE: TeXShop. Drawings: OmniGraffle. Graphs: Java, Eclipse. Ordering Process: Tomact, JSP
- Downloads: >5000 since Januar 2005